Rémi Bonnefoi
email: remi.bonnefoi@centralesupelec.fr

Christophe Moy
email: christophe.moy@centralesupelec.fr

Haïfa Farès
email: haifa.fares@centralesupelec.fr

Jacques Palicot
email: jacques.palicot@centralesupelec.fr

Power Allocation for Minimizing Energy Consumption of OFDMA Downlink with Cell DTx

With Cell DTx, a base station can dynamically be switched to sleep mode during very short periods (inferior to frame duration) in order to reduce its energy consumption. In this paper, we study the problem of energy consumption minimization with Cell DTx for the downlink of an Orthogonal Frequency Division Multiple Access (OFDMA) base station. We show that this problem can be rewritten as a convex problem. Then, we derive interpretable expressions for the transmit power and sleep mode duration which minimize the energy consumption. Besides, we show that resource allocation algorithms already proposed for transmit power minimization have near optimal performance. Simulation results show that up to 38% of base station energy consumption can be saved with Cell DTx.

I. INTRODUCTION

Nowadays, base station use around 40-50% of the total energy consumed by wireless mobile networks [START_REF] Fehske | The global footprint of mobile communications: The ecological and economic perspective[END_REF]. Therefore, an improvement of base stations energy efficiency will greatly reduce the carbon footprint of cellular networks.

Cell DTx has recently been proposed for reducing the base energy consumption [START_REF] Frenger | Reducing Energy Consumption in LTE with Cell DTX[END_REF]. With this solution, a base station can be switched to sleep mode during very short periods (a few milliseconds) to save energy. During these short sleeps, some of the elements of the RF chain such as the power amplifier are switched off. However, for a given throughput (or capacity) with Cell DTx, an increase in sleep mode duration increases the instantaneous transmit power and vice versa. Consequently, there is a trade-off between the dynamic energy consumption, which depends on the transmit power, and the static power consumption, which is reduced by increasing sleep mode duration. Finding the power allocation which minimizes the base station energy consumption aims at finding the trade-off between dynamic and static energy consumption.

The problem of optimal power allocation with Cell DTx was first set in Time Division Multiple Access (TDMA) in [START_REF] Holtkamp | On Minimizing Base Station Power Consumption[END_REF]. The closed form expression of the optimal power allocation in TDMA has been proposed in [START_REF] Wang | Sleep mode design for green base stations[END_REF]. This problem was further investigated in [START_REF] Bonnefoi | Dynamic Sleep Mode for Minimizing a Femtocell Power Consumption[END_REF] for femtocells and in [START_REF] Bonnefoi | New Macrocell Downlink Energy Consumption Minimization with Cell DTx and Power Control[END_REF] for macrocells. In [START_REF] Holtkamp | Minimizing Base Station Power Consumption[END_REF], authors use the optimal power allocation in TDMA to propose an efficient power allocation in Orthogonal Frequency Division Multiple Access (OFDMA). However, the optimal power allocation has not been studied in this paper. In [START_REF] Ren | Energy-efficient resource allocation for OFDMA networks with sleep mode[END_REF], authors prove that the problem of optimal power allocation with Cell DTx is quasi-convex and show that a water-filling algorithm can be used to find the optimal power allocation. However, the authors did not provide interpretable expressions of the transmit power. In this paper, we first focus on the problem of power allocation and we show that this problem can be rewritten as a convex optimization problem with a variable change. Then we solve this problem, and we show that the optimal power allocation can be computed using only a water-filling algorithm and single variable root-finding algorithms. We finally investigate the problem of resource allocation and we show that resource allocation strategies already proposed for power control [START_REF] Kivanc | Computationally efficient bandwidth allocation and power control for OFDMA[END_REF] (i.e. for minimizing the RF transmit power) give a near optimal performance with Cell DTx.

The rest of this paper is organized as follows. In section II, we introduce the system model and the problem of energy consumption minimization. The problem of power allocation is rewritten in a convex form in section III and solved in section IV. The problem of resource allocation is studied in section V. Some numerical results are provided in section VI and section VII concludes this paper.

II. SYSTEM MODEL AND PROBLEM FORMULATION

We suppose a base station which serves N u users in N c subchannels of bandwidth B c using OFDMA. The base station allocates each subchannel to a user and adjusts the transmit power in order to reduce its energy consumption. We denote S k the set of subchannels used for user k, and P k,n T X the transmit power allocated to user k in channel n. As illustrated in figure 1, the time is divided into frames of duration T. In each frame, the base station is active and serves the users during t a and then switches to sleep mode until the end of the frame. We denote µ a = t a T the proportion of time used to serve the users. Moreover, the base station provides each user k ∈ 1, N u a given capacity C k :

C k = B c µ a N k n=1 log 2 1 + G k,n P k,n T X B c , (1)
In this equation,

N k = |S k | denotes the cardinal of S k , and G k,n = |h k , n | 2 N 0
is the channel gain of user k in subchannel n (attenuation, antenna gain, thermal noise, noise figure). Moreover, we use a linear model for the instantaneous base station power consumption [START_REF] Auer | How much energy is needed to run a wireless network?[END_REF]:

P supply = P 0 + mP T X
when active P supply = P s during sleep mode [START_REF] Frenger | Reducing Energy Consumption in LTE with Cell DTX[END_REF] Where P 0 is the static power consumption when the base station is active (processing, cooling, etc.). P s is the base station power consumption during sleep mode, m is the coefficient of the load dependence. P T X ∈]0; P max ] is the total instantaneous RF transmit power which is the sum of the transmit powers allocated in subchannels. P max denotes the maximum RF transmit power of the base station. In [START_REF] Auer | How much energy is needed to run a wireless network?[END_REF] authors list some realistic values for P 0 , P s , P max , and m. In table I, we mention the characteristics used in this paper for numerical evaluations. We now set the problem of energy consumption minimization. In a first step, we only consider the problem of power allocation. We suppose that the sets S k , ∀k ∈ 1; N u have been allocated. To minimize the base station energy consumption, we minimize the average power consumption during a frame whose expression can be deduced from [START_REF] Frenger | Reducing Energy Consumption in LTE with Cell DTX[END_REF].

min µ a , P T X (1 -µ a )P s + P 0 + m N u k=1 N k n=1 P k,n T X µ a (3a) S.t. µ a ∈ [0; 1] (3b) C k -B c µ a N k n=1 log 2 1 + G k,n P k,n T X B c = 0, ∀k (3c) N u k=1 N k n=1 P k,n T X ≤ P max (3d) P k,n T X ≥ 0, ∀k, n (3e)
III. PROBLEM REFORMULATION As proven in appendix A, the problem introduced in equation (3) is not convex in that form. In order to rewrite the problem of energy consumption minimization as a convex problem, we first proceed a variable change and we introduce the capacities per subchannel C k,n :

C k,n = B c µ a log 2 1 + G k,n P k,n T X B c . (4)
With these new variables, the objective function of equation (3a) becomes:

(1 -µ a )P s + P 0 + m N u k=1 N k n=1 B c G k,n 2 C k , n Bc µa -1 µ a (5)
Proposition 1. The objective function introduced in equation ( 5) is convex.

Proof. µ a → µ a (P 0 -P s ) is linear. Moreover, the Hessian matrix of the functions g k,n :

µ a , C k,n → mB c µ a G k , n 2 C k , n
Bc µa -1 is equal to:

H k,n = m ln(2) 2 G k,n B c µ a 2 C k , n Bc µa C 2 k , n µ 2 a - C k , n µ a - C k , n µ a 1 (6)
λ 1 = 0 and λ 2 = m ln(2) 2 G k , n B c µ a 1 + C 2 k , n µ 2 a 2 C k , n Bc µa
are the two eigenvalues of H k,n . These eigenvalues are positive, and consequently, g k,n are convex functions. Thus, equation ( 5) is convex as the sum of convex functions.

Moreover, constraint (3d) becomes:

N u k=1 N k n=1 B c G k,n 2 C k , n Bc µa -1 ≤ P max . (7)
An analysis of the Hessian matrix of equation [START_REF] Holtkamp | Minimizing Base Station Power Consumption[END_REF] shows that this constraint is not convex. We can change it into a convex constraint without changing the minimization problem. Indeed, the optimal power allocation can't be obtained for µ a = 0, as a consequence, the constraint of ( 7) can be rewritten:

µ a N u k=1 N k n=1 B c G k,n 2 C k , n Bc µa -1 ≤ µ a P max . (8)
Moreover, this new constraint is convex (the proof is identical to the one done for the objective function). Finally, we obtain a convex problem for the optimal power allocation with Cell DTx:

min µ a ,C k , n (1 -µ a )P s + P 0 + m N u k=1 N k n=1 B c G k,n 2 C k , n Bc µa -1 µ a (9a) S.t. µ a ∈ [0; 1] (9b) C k - N k n=1 C k,n = 0, ∀k (9c)
µ a N u k=1 N k n=1 B c G k,n 2 C k , n Bc µa -1 ≤ µ a P max (9d) C k,n ≥ 0, ∀k, n (9e)
The Karush-Kuhn and Tucker (KKT) conditions can finally be used to solve this problem.

IV. OPTIMAL POWER ALLOCATION

We denote µ opt the optimal value of µ a . We calculate the optimal capacity per subchannel using the derivative of the Lagrangian of the studied problem:

C k,n = C k N k + B c µ opt log 2 G k,n N k n=1 G k,n 1 N k † , (10)
where N k denotes the number of subchannels used for user k in which the capacity is non-zero and a † = max(a, 0).

Once the expression of the capacity per subchannel is computed, we use equation ( 4) to derive the optimal transmit power per subchannel:

P k,n T X = B c 2 C k N k Bc µ o pt N k n=1 G k,n 1 N k - 1 G k,n † (11)
We now compute the expression of µ opt the optimal service time. With the KKT conditions, each inequality constraint can be either considered as an equality (the constraint is binding) or as a strict inequality (the constraint is non-binding). In the studied problem, the two constraints of (9b) and (9d) can either be binding or non-binding. If the constraint of (9d) is binding, the base station energy consumption is minimized by maximizing the instantaneous transmit power and so minimizing the service time and maximizing sleep mode duration. On the contrary, if the constraint of (9b) is binding the energy consumption is minimized without switching to sleep mode during the frame. Thus, these two constraints can't simultaneously be binding.

As a consequence, if one of the two constraints of (9b) or (9d) is binding, the other will be non-binding. This leads to three different cases:

1) None of these two constraints are binding.

2) Constraint of (9b) is binding (µ a = 1).

3) Constraint of (9d) is binding.

We have a different value of µ opt for each case. 1) Case 1: In this case, none of the two constraints (9b) and (9d) is binding. We denote:

f (µ a ) = N u k=1 2 C k k Bc µa N k n=1 G k,n 1 N k C k ln(2) B c µ a -N k - P 0 -P s mB c + N u k=1 N k n=1 1 G k,n , (12)
and µ opt is the solution of:

f (µ a ) = 0. ( 13
)
The proof is given in appendix B.

It may be noted that in the single-user case, the optimal service time has the following expression:

µ opt = C ln(2) N B c × 1 W e -1 N n=1 G n 1 N P 0 -P s mN B c -1 N N n=1 1 G n + 1 , (14)
where W denotes the W-Lambert function [START_REF] Corless | On the LambertW function[END_REF] which is the reciprocal bijection of x → xe x . This formula is very similar to the optimal TDMA power allocation [START_REF] Bonnefoi | New Macrocell Downlink Energy Consumption Minimization with Cell DTx and Power Control[END_REF].

2) Case 2: If (9b) is binding, µ opt = 1. We obtain the power allocation which minimizes the transmit power without sleep mode [START_REF] Seong | Optimal Resource Allocation for OFDMA Downlink Systems[END_REF]:

P k,n T X = B c 2 C k N k Bc N k n=1 G k,n 1 N k - 1 G k,n + (15)
3) Case 3: If the constraint of (9d) is binding, µ opt is the solution of:

B c N u k=1 N k 2 C k N k Bc µ o p t N k n=1 G k,n 1 N k - N k n=1 1 G k,n = P max ( 16
)
The proof is similar to the one done in case 1.

In this case, the instantaneous transmit power is maximized when the base station is active, µ a is minimized and the sleeping time is maximized. In other words, at each iteration of the water-filling algorithm described by equation ( 11), we know that the RF transmit power can't be higher than a maximum transmit power P max . Consequently, the service time can't be lower than a minimum service time, which is the solution of equation ( 16). In the following we denote µ min this minimum service time.

Finally, to compute the power allocation which minimizes the base station energy consumption, we use the water-filling algorithm described by equation [START_REF] Corless | On the LambertW function[END_REF]. At each iteration, we first compute µ min solving equation ( 16), then we compute µ opt solving equation ( 13). This can be done using single variable root finding algorithms (bisection method, Newton's method, etc.). Then, if µ opt < µ min (resp. µ opt > 1), the service time is set to µ min (resp. to 1). Each iteration ends with a computation of the transmit power in subchannels with equation [START_REF] Corless | On the LambertW function[END_REF].

V. RESOURCE ALLOCATION

Once the problem of power allocation is solved, we analyse the problem of resource allocation. Indeed, with N u users and N c subchannels, the number of resource allocations for which each user has at least one subchannel, is equal to [START_REF] Loehr | Bijective Combinatorics[END_REF]:

N res = N u i=0 (-1) i N u i (N u -i) N c = O(N N c u ) (17)
In many cases, N res is too large to propose an exhaustive search for the optimal resource allocation. That is why in the following, we propose sub-optimal resource allocations.

Equation ( 11) and (15) give the power allocations which minimize respectively the energy consumption with and without Cell DTx. These two formulas are very similar. That is why, in this section, we use exhaustive searches for a small number of users to compare two strategies. We first analyse the average power consumption with the optimal resource allocation. To do that, we use the algorithm described in the previous section to compute the transmit power with Cell DTx for all the possible resource allocations and we select the one for which the energy consumption is minimized. With the second strategy, we use equation (15) to compute the power allocation which minimizes the transmit power for all resource allocations and we choose the resource allocation for which the transmit power is minimized. Then, we use the algorithm described in the previous section to compute the users' transmit power and the service time of the base station with Cell DTx for this resource allocation. In that case, we use a suboptimal resource allocation, the power consumption with this second strategy will be higher than the one obtained with the first strategy. In the following, we numerically evaluate the maximum loss caused by this suboptimal resource allocation.

As in LTE standard, we suppose a band divided in 180kHz subchannels composed by 12 subcarriers. For each user, we set an average maximum SNR denoted ρ i max which is the average SNR that would have user i if he was served with all the available transmit power. We use the ETU Extended Typical Urban channel model to generate fading. We make an iterative search over all the possible resource allocations and we analyse the error committed if we use DTx with the optimal resource allocation without DTx.

We first consider scenarios in which the base station serves two users in 6 (1.4 MHz LTE bandwidth) or 15 (3 MHz LTE bandwidth) subchannels. Table II shows the maximum loss obtained for different values of ρ max and for different base station types. This loss is the ratio between the extra energy consumed by the suboptimal resource allocation and the gain in energy provided by DTx compared to the optimal resource allocation without DTx. We can see in table II that, in the cases studied, the loss never exceeds 10.5%. Furthermore, we made some simulations with 4 users and 6 subchannels with macrocells and femtocells and the average loss was always below 1%. We show in figure 2 the average power consumption with the optimal and suboptimal strategies in one of the worst case scenarios (7 th scenario). We can see that the optimal and suboptimal resource allocations provide similar results.

These numerical evaluations show that for all base station types, there is little interest in developing new algorithms for resource allocations. Indeed, resource allocation algorithms developed for transmit power minimization (power control) will have near optimal performance. Moreover, many solutions have been proposed for transmit power minimization such as Power consumption (W) Optimal power allocation without DTx Optimal power and resource allocation DTx with suboptimal resource allocation Fig. 2. Average femtocell power consumption in the 7 t h scenario. We can see that the gap between the optimal resource allocation and the resource allocation which minimizes the transmit power (suboptimal) increases and then decreases as the capacity constraint increases.

in [START_REF] Seong | Optimal Resource Allocation for OFDMA Downlink Systems[END_REF], [START_REF] Wong | Multiuser OFDM with adaptive subcarrier, bit, and power allocation[END_REF]. In this paper, without loss of generality, we use the Bandwidth Assignment Based on SNR (BABS) algorithm to compute the number of subchannels per user and the Amplitude-Craving Greedy (ACG) algorithm for subchannel assignment. Both algorithms have been proposed in [START_REF] Kivanc | Computationally efficient bandwidth allocation and power control for OFDMA[END_REF]. The main advantage of the BABS+ACG algorithm is its low complexity which is O(N u N c ).

VI. NUMERICAL RESULTS

In a first step, we consider some of the scenarios introduced in the previous section. We suppose that a base station serves two users with a 1.4 MHz band divided in 6 subchannels of 12 subcarriers. We compare the base station average power consumption with the optimal resource allocation and with the BABS+ACG algorithm. All the results are displayed in figure 3. For low capacity constraint, the BABS+ACG algorithm is near optimal and DTx saves up to 37.5% of a femtocell energy consumption and 38.5% of a macrocell energy consumption. Furthermore, we can see in figure 3a and 3c that when the two users have the same SNR, the BABS+ACG resource allocation is optimal with and without DTx. Furthermore, when one user has a low SNR and the other a high SNR, the energy consumption with the BABS+ACG resource allocation moves away from the optimal average power consumption as the capacity constraint increases. We can see in figure 3b that for a macrocell, the gap between the optimal power consumption and the power consumption with the BABS+ACG algorithm is the same with and without DTx. In this case, this gap is only caused by the resource allocation and can be reduced with more efficient resource allocations (with a higher complexity). On the contrary, we observe in figure 3d that when the BABS+ACG is used for femtocells with Cell DTx, There is a gap caused by the suboptimality of the resource allocation which minimizes the transmit power. The loss observed in figure 2 is amplified by the suboptimality of the BABS+ACG algorithm. However, the dynamic part of the power consumption is less important for femtocells. Consequently, we have approximately the same loss caused by the suboptimal resource allocation for femtocells (5% of (d) Femtocell, ρ 1 ma x = 5dB, ρ 2 ma x = 25dB Fig. 3. Comparison of the proposed approach with the optimal solution when 2 users are served in 6 subchannels (1.4MHz band). the total power consumption) and for macrocells (6% of the total power consumption).

We now study the performance of Cell DTx in a realistic scenario. We suppose that a macrocell serves some users distributed according to a Poisson Point Process (PPP) in a 20MHz band divided in 100 subchannels. The central frequency is 2GHz. Moreover, the average number of users in the cell coverage is equal to 10 and the distance between the users and the base station varies between 300 and 1500m. The pathloss is computed with the Winner II suburban macrocell pathloss model. The antenna gain is set to 10dBi. The ETU Extended Pedestrian A model is used to generate fading and a 2dB noise figure is added. We compare the optimal power allocations with the BABS+ACG algorithm with and without Cell DTx.

We can see the evolution of the average power consumption with and without DTx in figure 4. With the BABS+ACG algorithm, DTx saves up to 38% of the base station energy consumption. The energy saved with DTx decreases as the capacity constraint increases. Indeed, when the capacity constraint increases, the dynamic part of the power consumption (which is proportional to the transmit power) increases and the optimal strategy becomes closer to the optimum without DTx. For a capacity constraint of 8 Mbits/s DTx saves 2% of the base station energy consumption.

VII. CONCLUSION

In this paper, we study the problem of power and resource allocation in OFDMA with Cell DTx. We prove that the problem of power allocation can be rewritten as a convex problem and can be solved using a water-filling algorithm and single variable root-finding algorithms. Moreover, we show that the algorithms developed for power control without DTx have good performance when DTx is used. We use the BABS+ACG algorithm and we show that DTx can save up to 38% of base station energy consumption. In our future work, we will study the problem of power allocation with Cell DTx in multicell scenarios.

ACKNOWLEDGEMENT

Part of this work is supported by the project SOGREEN, which is funded by the French national research agency, under the grant agreement coded: N ANR-14-CE28-0025-02 and by Région Bretagne, France.

APPENDIX A NON-CONVEXITY OF THE FIRST PROBLEM

To prove the non-convexity of the problem introduced in equation ( 3), we can either prove that the objective function of equation (3a) or the constraints of equation (3c) are not convex as function of µ a and the transmit powers P k,n T X for all k, n. The proof is made for the constraints of equation (3c). We denote:

α n = 1 ln(2) G k,n 1 + G k , n P k , n T X B c ≥ 0 (18)
β n = G 2 k,n µ a B c ln(2) 1 1 + G k , n P k , n T X B c 2 ≥ 0 (19)
Capacity per user (Mbits/s) For k ∈ 1; N u , the calculation of the Hessian matrix of (3c) gives:

H k = 0 -α 1 • • • -α N k -α 1 β 1 • • • 0 . . . . . . . . . . . . -α N k 0 • • • β N k (20)
Tr(H k ) > 0, thus, at least one eigenvalue of H k is strictly positive. Expanding along the first row, we can compute the determinant of this matrix:

det(H k ) = - N k n=1 α 2 n N k j=1, j n β j < 0 (21)
Thus the product of the eigenvalues of H k is negative and H k has at least one strictly negative eigenvalue. H k has at least a strictly negative and a strictly positive eigenvalue. As a consequence, the constraint (3c) is not convex.

APPENDIX B OPTIMAL SERVICE TIME AND POWER ALLOCATION

The proof is done for the first case. The Lagrangian of the problem studied is:

L = P s + µ a (P 0 -P s ) + mµ a B c N u k=1 N k n=1 1 G k,n 2 C k , n Bc µa -1 - N u k=1 λ k N k n=1 C k,n -C k (22)
The derivative of the Lagrangian with respect to C k,n is

∂L ∂C k,n = m ln(2) G k,n 2 C k , n Bc µa -λ k (23)
At the minimum, this derivative is zero, thus:

C k,n = B c µ a log 2 λ k G k,n m ln(2) (24)
Since N k n=1 C k,n = C k , we have:

B c µ a log 2 λ k m ln(2) = C k N k - B c µ a N k log 2 N k n=1 G k,n (25)
Thus,

C k,n = C k N k + B c µ a log 2 G k,n N k n=1 G k,n 1 N k (26)
Furthermore, the differential of the Lagrangian with respect to µ a is:

∂L ∂ µ a = N u k=1 N k n=1 mB c G k,
N u k=1 2 C k N k Bc µa N k n=1 G k,n 1 N k C k ln(2) B c µ a -N k - P 0 -P s mB c + N u k=1 N k n=1 1 G k,n = 0 (28)
Then, by successively setting to zero all the capacities below zero, according to water-filling principle, we obtain the optimal capacity per subchannel. We can finally compute the optimal power allocation per subchannel.

Fig. 1 .

 1 Fig. 1. Frame composed of 6 subchannels each of bandwidth B c , two users are served by the base station. For each user and in each subchannel, the power is allocated in order to minimize the base station energy consumption.

Fig. 4 .

 4 Fig. 4. Average base station power consumption with and without DTx with the BABS+ACG algorithm. DTx allows to save up to 38% of the base station energy consumption.


Romain Joly
email: romain.joly@univ-grenoble-alpes.fr

Geneviève Raugel
email: genevieve.raugel@math.u-psud.fr

A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations

Keywords: finite-and infinite-dimensional dynamical systems, vector fields, scalar parabolic equation, Kupka-Smale property, genericity AMS Subject Classification: 35-02, 37-02, 35B05, 35B41, 35K57, 37C10, 37C20

The results

Introduction

In this paper, we want to point out the similarities between the dynamics of vector fields in R d and those of reaction-diffusion equations on bounded domains. More precisely, we consider the following classes of equations.

Class of vector fields

Let d ≥ 1 and r ≥ 1 and let g ∈ C r (R d , R d ) be a given vector field. We consider the ordinary differential equation ẏ(t) = g(y(t)) t > 0 y(0) = y 0 ∈ R d (1.1) where ẏ(t) denotes the time-derivative of y(t).

The equation (1.1) defines a local dynamical system T g (t) on R d by setting T g (t)y 0 = y(t).

We assume that there exists M > 0 large enough such that ∀y ∈ R d , y ≥ M ⇒ y|g(y) < 0 .

This condition ensures that T g (t) is a global dynamical system. Moreover, the ball B(0, M) attracts the bounded sets of R d . Therefore, T g (t) admits a compact global attractor 1 A g . The attractor A g contains the most interesting trajectories such as periodic, homoclinic and heteroclinic orbits 1 and any α-or ω-limit set 1 . Therefore, if one neglects the transient dynamics, the dynamics on A g is a good representation of the whole dynamics of T g (t).

Class of scalar parabolic equations

Let d ′ ≥ 1 and let Ω be either a regular bounded domain of R d ′ , or the torus T d ′ . We choose p > d ′ and α ∈ ((p + d ′ )/2p, 1). We denote X α ≡ D((-∆ N ) α ) the fractional power space 1 associated with the Laplacian operator ∆ N on L p (Ω) with homogeneous Neumann boundary conditions. It is well-known 1 that X α is continuously embedded in the Sobolev space W 2α,p (Ω) and thus it is compactly embedded in C 1 (Ω). Let r ≥ 1 and

f ∈ C r (Ω × R × R d ′ , R).
We consider the parabolic equation    u(x, t) = ∆u(x, t) + f (x, u(x, t), ∇u(x, t)) (x, t) ∈ Ω × (0, +∞)

∂u ∂ν (x, t) = 0 (x, t) ∈ ∂Ω × (0, +∞) u(x, 0) = u 0 (x) ∈ X α (1.2)
where u(t) is the time-derivative of u(t). Eq. (1.2) defines a local dynamical system S f (t) on X α (see [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF]) by setting S f (t)u 0 = u(t). We assume moreover that there exist c ∈ C 0 (R + , R + ), ε > 0 and κ > 0 such that f satisfies

∀R > 0, ∀ξ ∈ R d ′ , sup (x,z)∈Ω×[-R,R] |f (x, z, ξ)| ≤ c(R)(1 + |ξ| 2-ε )
and ∀z ∈ R, ∀x ∈ Ω, |z| ≥ κ ⇒ zf (x, z, 0) < 0 .

Then, Eq. (1.2) defines a global dynamical system in X α which admits a compact global attractor A (see [START_REF] Poláčik | Parabolic equations: asymptotic behavior and dynamics on invariant manifolds[END_REF]).

The reader, which is not familiar with partial differential equations, may neglect all the technicalities about X α , the Sobolev spaces and the parabolic equations in a first reading. The most important point is that S f (t) is a dynamical system defined on an infinitedimensional function space. Compared with the finite-dimensional case, new difficulties arise. For example, the existence of a compact global attractor requires compactness properties, coming here from the smoothing effect of (1.2). We also mention that, even if the backward uniqueness property holds, backward trajectories do not exist in general for (1.2). The reader interested in the dynamics of (1.2) may consult [START_REF] Fiedler | Spatio-temporal dynamics of reaction-diffusion patterns[END_REF], [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF], [START_REF] Hale | Dynamics of a scalar parabolic equation[END_REF], [START_REF] Poláčik | Parabolic equations: asymptotic behavior and dynamics on invariant manifolds[END_REF] or [START_REF] Raugel | Global attractors in partial differential equations[END_REF].

The purpose of this paper is to emphasize the different relationships between the dynamics of (1.1) and (1.2). The correspondence is surprisingly perfect. It can be summarized by Table 1. This correspondence has already been noticed for some of the properties of the table. We complete here the correspondence for all the known properties of the dynamics of the parabolic equation. Table 1 will be discussed in more details in Section 2 and, for cooperative systems, in Section 4. Some of the properties presented in the table concerning finite-dimensional dynamical systems are trivial, other ones are now well-known.

Ω f (x, u, ∇u) ≡ f (x, u)
Table 1: the correspondence between the dynamics of vector fields and the ones of parabolic equations.

However, the corresponding results for the parabolic equation are more involved and some of them are very recent. These properties are mainly based on Sturm-Liouville arguments and unique continuation properties for the parabolic equations as explained in Section 3. The study of the dynamics generated by vector fields in dimension d ≥ 3 is still a subject of research. Taking into account the correspondence presented in Table 1 should give a guideline for research on the dynamics of the parabolic equations. Some examples of open questions are given in Section 5.

We underline that we only consider the dynamics on the compact global attractors. Hence, we deal with dynamical systems on compact sets. It is important to be aware of the fact that, even if the dimension of the compact global attractor A of the parabolic equation (1.2) is finite, it can be made as large as wanted by choosing a suitable function f . This is true even if Ω is one-dimensional. Therefore, all the possible properties of the dynamics of (1.2) do not come from the low dimension of A but from properties, which are very particular to the flow of the parabolic equations.

Finally, we remark that most of the results described here also hold in more general frames than (1.1) and (1.2). For example, R d could be replaced by a compact orientable manifold without boundary. We could also choose for (1.2) more general boundary conditions than Neumann ones, or less restrictive growing conditions for f . The domain Ω may be replaced by a bounded smooth manifold. Finally, notice that the case Ω = T d ′ can be seen as Ω = (0, 1) d ′ with periodic boundary conditions.

Details and comments about the correspondence table

We expect the reader to be familiar with the basic notions of the theory of dynamical systems and flows. Some definitions are briefly recalled in the glossary at the end of this paper. For more precisions, we refer for example to [START_REF] Katok | Introduction to the modern theory of dynamical systems[END_REF], [START_REF] Newhouse | Lectures on dynamical systems[END_REF], [START_REF] Palis | Geometric theory of dynamical systems. An introduction[END_REF], [START_REF] Robinson | Dynamical systems. Stability, symbolic dynamics, and chaos[END_REF] or [START_REF] Ruelle | Elements of differentiable dynamics and bifurcation theory[END_REF] for finitedimensional dynamics and to [START_REF] Hale | An introduction to infinite dimensional dynamical systems[END_REF], [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF], [START_REF] Robinson | Infinite-dimensional dynamical systems. An introduction to dissipative parabolic PDEs and the theory of global attractors[END_REF] or [36] for the infinite-dimensional ones. We first would like to give short comments and motivations concerning the properties appearing in Table 1. Notice that we do not deal in this section with the cooperative systems of ODEs. The properties of these systems are discussed in Section 4.

A generic property of the dynamics is a property satisfied by a countable intersection of open dense subsets of the considered class of dynamical systems. Generic dynamics represent the typical behaviour of a class of dynamical systems. For finite-dimensional flows, we mainly consider classes of the form (T g (t)) g∈C 1 (R d ,R d ) . The parameter is the vector field g, which belongs to the space C 1 (R d , R d ) endowed with either the classical C 1 or the C 1 Whitney topology. Notice that the question wether or not a property is generic for g ∈ C r (R d , R d ) for some r ≥ 2 may be much more difficult than C 1 genericity. We will not discuss this problem here. In some cases, we restrict the class of vector fields to subspaces of C 1 (R d , R d ) such as radially symmetric, gradient vector fields or cooperative systems. In a similar way, for infinite-dimensional dynamics, we consider families of the type (S f (t)

) f ∈C 1 (Ω×R×R d ′ ,R) , where C 1 (Ω × R × R d ′ , R
) is endowed with either the classical C 1 or the C 1 Whitney topology. For some results, we restrict the class of nonlinearities f to homogeneous ones or to ones, which are independent of the last variable ξ.

Poincaré-Bendixson property and the convergence to an equilibrium or a periodic orbit are properties related to the following question: how simple are the α-and ω-limit sets of the trajectories ? For vector fields, the restriction of the complexity of the limit sets may come from the restriction of freedom due to the low dimension of the flow. As said above, there is no restriction on the dimension of the global attractor for the parabolic equations. The possible restrictions of the complexity of the limit sets come from particular properties of the parabolic equations, see Section 3.

Hyperbolicity of equilibria and periodic orbits, tranversality of stable and unstable manifolds, Kupka-Smale and Morse-Smale properties are properties related to the question of stability of the local and global dynamics respectively. Morse-Smale property is the strongest one. It implies the structural stability of the global dynamics: if the dynamical system T g (t) satisfies the Morse-Smale property, then for g close enough to g, the dynamics of T g(t), restricted to its attractor A g, are qualitatively the same as the ones of T g (t) on A g , see [START_REF] Palis | On Morse-Smale dynamical systems[END_REF], [START_REF] Palis | Structural stability theorems[END_REF] and [START_REF] Palis | Geometric theory of dynamical systems. An introduction[END_REF]. The same structural stability result holds for parabolic equations satisfying the Morse-Smale property, see [START_REF] Hale | An introduction to infinite dimensional dynamical systems[END_REF], [36] and [START_REF] Oliva | Morse-Smale semiflows. Openess and A-stability[END_REF]. It is natural to wonder if almost all the dynamics satisfy these properties, that is if these properties are generic.

The fact that the knowledge of the equilibria and the periodic orbits implies the knowledge of the whole dynamics may be studied at different levels. Two equilibria or periodic orbits being given, can we know if they are connected or not by a heteroclinic orbit ? Are two dynamics with the same equilibria and periodic orbits equivalent ? Is there a simple algorithm to determine the global dynamics from the position of the equilibria and the periodic orbits ? These questions are among the rare dynamical questions coming from the study of partial differential equations and not from the study of vector fields. Indeed, for finite-dimensional dynamical systems, the answers, either positive or negative, are too simple. In contrast, such kinds of results are probably among the most amazing ones for the dynamics of the parabolic equations.

The persistent chaotic dynamics and the fact that the dimension of the attractor is equal to the largest dimension of the unstable manifolds, are related to the following question: how complicated may be the dynamics ? In general, the dimension of the attractor of a dynamical system may be larger than the largest dimension of the unstable manifolds. The classes of systems, where these dimensions automatically coincide, are strongly constrained, which in some sense implies a simple behaviour. On the contrary, chaotic dynamics have very complicated behaviour. Chaotic dynamics may occur through several phenomena, and the notion of chaotic behaviour depends on the authors. In this paper, "persistent chaotic dynamics" refers to the presence of a tranversal homoclinic orbit generating a Smale horseshoe (see [START_REF] Smale | Diffeomorphisms with many periodic points[END_REF]). The persistent chaotic dynamics provide complicated dynamics, which cannot be removed by small perturbations of the system. Such an open set of chaotic dynamics is a counter-example to the genericity of the Morse-Smale systems.

The question of the realization of vector fields in the parabolic equations is as follows: a vector field g ∈ C r (R d , R d ) being given, can we find a function f and an invariant manifold M ⊂ L p (Ω) such that the dynamics of the parabolic equation (1.2) restricted to M is equivalent to the dynamics generated by the vector field g ? A positive answer to this question implies that the dynamics of the considered class of parabolic equations is at least as complicated as the dynamics of the considered class of vector fields. Such a realization result is very interesting since, on the opposite, the other properties stated in Table 1 roughly say that the dynamics of the parabolic equation (1.2) cannot be much more complicated than the ones of the corresponding class of finite-dimensional flows. One has to keep in mind that the manifold M, on which the finite-dimensional dynamics are realized, is not necessarily stable with respect to the dynamics of the parabolic equation. Typically, M cannot be stable if the finite-dimensional system contains a stable periodic orbit, since all periodic orbits of (1.2) are unstable (see for example [START_REF] Hirsch | Stability and convergence on strongly monotone dynamical systems[END_REF]). Now, we give short comments and references for the correspondences stated in Table 1.

• d = 1 and Ω = (0, 1)
The dynamics generated by a one-dimensional vector field is very simple. Its attractor consists in equilibrium points and heteroclinic orbits connecting two of them. The existence of these heteroclinic orbits is easily deduced from the positions of the equilibrium points. Moreover, these heteroclinic connections are trivially transversal. Finally, (1.1) is clearly a gradient system with associated Lyapounov functional F (y) = -y 0 g(s)ds. As a consequence, the Morse-Smale property is equivalent to the hyperbolicity of all the equilibrium points, which holds for a generic one-dimensional vector field.

The dynamics S f (t) generated by (1.2) for Ω = (0, 1) is richer since its attractor may have a very large dimension. However, these dynamics satisfy similar properties. These similarities are mainly due to the constraints coming from the non-increase of the number of zeros of solutions of the linear parabolic equation (see Theorem 3.1). Zelenyak has proved in [START_REF] Zelenyak | Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable[END_REF] that S f (t) admits an explicit Lyapounov function and thus that it is gradient. He also showed that the ω-limit sets of the trajectories consist in single equilibrium points. In Proposition 3.2, we give a short proof of this result, due to Matano. The fact that the stable and unstable manifolds of equilibrium points always intersect transversally comes from Theorem 3.1 and the standard Sturm-Liouville theory. This property has been first proved by Henry in [START_REF] Henry | Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations[END_REF] and later by Angenent [START_REF] Angenent | The Morse-Smale property for a semilinear parabolic equation[END_REF] in the weaker case of hyperbolic equilibria. As a consequence of the previous results, the Morse-Smale property is equivalent to the hyperbolicity of the equilibrium points and is satisfied by the parabolic equation on (0, 1) generically with respect to f . The most surprising result concerning (1.2) on Ω = (0, 1) is the following one. Assuming that every equilibrium point is hyperbolic and that the equilibrium points e 1 ,...,e p are known, one can say if two given equilibria e i and e j are connected or not by a heteroclinic orbit. This property has been proved by Brunovský and Fiedler in [START_REF] Brunovský | Connecting orbits in scalar reaction diffusion equations. II. The complete solution[END_REF] for f = f (u) and by Fiedler and Rocha in [START_REF] Fiedler | Heteroclinic orbits of semilinear parabolic equations[END_REF] in the general case. The description of the heteroclinic connections is obtained from the Sturm permutation which is a permutation generated by the respective positions of the values e i (0) and e i (1) of the equilibrium points at the endpoints of Ω = (0, 1). The importance of Sturm permutation has been first underlined by Fusco and Rocha in [START_REF] Fusco | A permutation related to the dynamics of a scalar parabolic PDE[END_REF]. We also refer to the work of Wolfrum [START_REF] Wolfrum | A sequence of order relations: encoding heteroclinic connections inscalar parabolic PDE[END_REF] , which presents a very nice formalism for this property. Fiedler and Rocha showed in [START_REF] Fiedler | Orbit equivalence of global attractors of semilinear parabolic differential equations[END_REF] that the Sturm permutation characterizes the global dynamics of (1.2) on (0, 1). They proved in [START_REF] Fiedler | Realization of meander permutations by boundary value problems[END_REF] that it is possible to give the exact list of all the permutations which are Sturm permutations for some nonlinearity f and thus to give the list of all the possible dynamics of the parabolic equation on (0, 1). The fact that the dimension of the attractor is equal to the largest dimension of the unstable manifolds has been shown by Rocha in [START_REF] Rocha | Properties of the attractor of a scalar parabolic PDE[END_REF]. The previous works of Jolly [START_REF] Jolly | Explicit construction of an inertial manifold for a reaction diffusion equation[END_REF] and Brunovský [START_REF] Brunovský | The attractor of the scalar reaction diffusion equation is a smooth graph[END_REF] deal with the particular case f ≡ f (u), but show a stronger result: the attractor can be embedded in a C 1 invariant graph of dimension equal to the largest dimension of the unstable manifolds. Finally, let us mention that it is easy to realize any one-dimensional flow in an invariant manifold of the semi-flow generated by the one-dimensional parabolic equation. For example, in the simplest case of Neumann boundary conditions as in (1.2), one can realize the flow of any vector field g as the restriction of the dynamics of the equation u = ∆u + g(u) to the subspace of spatially constant functions.

• d = 2 and Ω = T 1 , general case Even if they are richer than in the one-dimensional case, the flows generated by vector fields on R 2 are constrained by the Poincaré-Bendixson property (see the original works of Poincaré [START_REF] Poincaré | Sur les courbes définies par une équation différentielle[END_REF] and Bendixson [START_REF] Bendixson | Sur les courbes définies par des équations différentielles[END_REF] or any textbook on ordinary differential equations). In particular, this constraint precludes the existence of non-trivial non-wandering points in Kupka-Smale dynamics. Due to the low dimension of the dynamics, the stable and unstable manifolds of hyperbolic equilibria or periodic orbits always intersect transversally if either one of the manifold corresponds to a periodic orbit or if the invariant manifolds correspond to two equilibrium points with different Morse indices. Moreover, there is no homoclinic trajectory for periodic orbits. Using these particular properties, Peixoto proved in [START_REF] Peixoto | Structural stability on two-dimensional manifolds[END_REF] that the Morse-Smale property holds for a generic two-dimensional vector field.

The first correspondence between two-dimensional flows and the dynamics of the parabolic equation (1.2) on the circle Ω = T 1 has been obtained by Fiedler and Mallet-Paret in [START_REF] Fiedler | A Poincaré-Bendixson theorem for scalar reactiondiffusion equations[END_REF]. They proved that the Poincaré-Bendixson property holds for (1.2) on T 1 , by using the properties of the zero number (see Theorem 3.1). The realization of any two-dimensional flow in a two-dimensional invariant manifold of the parabolic equation on the circle has been proved by Sandstede and Fiedler in [START_REF] Sandstede | Dynamics of periodically forced parabolic equations on the circle[END_REF]. Very recently, Czaja and Rocha have shown in [START_REF] Czaja | Transversality in scalar reaction-diffusion equations on a circle[END_REF] that the stable and unstable manifolds of two hyperbolic periodic orbits always intersect transversally and that there is no homoclinic connection for a periodic orbit. The other automatic transversality results and the proof of the genericity of the Morse-Smale property have been completed by the authors in [START_REF] Joly | Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle[END_REF] and [START_REF] Joly | Generic Morse-Smale property for the parabolic equation on the circle[END_REF].

• d = 2 and Ω = T 1 , radial symmetry and T 1 -equivariance When the vector field g satisfies a radial symmetry, the dynamics of the two-dimensional flow generated by (1.1) becomes roughly one-dimensional. The closed orbits consist in 0, circles of equilibrium points and periodic orbits being circles described with a constant rotating speed. The dynamics are so constrained that the closed orbits being given, it is possible to describe all the heteroclinic connections. Notice that no homoclinic connection is possible. We also underline that the Morse-Smale property is generic in the class of radially symmetric vector fields.

If the two-dimensional radial vector fields are too simple to attract much attention, the corresponding case for the parabolic equation (1.2) on Ω = T 1 with homogeneous nonlinearity f (x, u, ∂ x u) ≡ f (u, ∂ x u) has been extensively studied. Since Theorem 3.1 holds for (1.2) with any one-dimensional domain Ω, it is natural to expect results for (1.2) on Ω = T 1 similar to the ones obtained for (1.2) on Ω = (0, 1). In particular, one may wonder if it is possible to describe the global dynamics of (1.2) knowing the equilibria and the periodic orbits only. However, this property is still open for general non-linearities f (x, u, ∂ x u) in the case Ω = T 1 . Moreover, if one believes in the correspondence stated in this paper, one can claim that it is in fact false for a general nonlinearity f (x, u, ∂ x u). Therefore, it was natural to first study the simpler case of homogeneous nonlinearities f ≡ f (u, ∂ x u). Indeed, the dynamics in this case are much simpler, in particular the closed orbits are either homogeneous equilibrium points e(x) ≡ e ∈ R, or circles of non-homogeneous equilibrium points, or periodic orbits consisting in rotating waves u(x, t) = u(xct) (notice the correspondence with the closed orbits of a radially symmetric two-dimensional flow). This property is a consequence of the zero number property of Theorem 3.1 and has been proved in [START_REF] Angenent | The dynamics of rotating waves in scalar reaction diffusion equations[END_REF] by Angenent and Fiedler. The works of Matano and Nakamura [START_REF] Matano | The global attractor of semilinear parabolic equations on T 1[END_REF] and of Fiedler, Rocha and Wolfrum [START_REF] Fiedler | Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle[END_REF] show that the unstable and stable manifolds of the equilibria and the periodic orbits always intersect transversally and that no homoclinic orbit can occur. Moreover, in [START_REF] Fiedler | Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle[END_REF], the authors give an algorithm for determining the global dynamics of the parabolic equation (1.2) on Ω = T 1 with homogeneous nonlinearity f ≡ f (u, ∂ x u). This algorithm uses the knowledge of the equilibria and the periodic orbits only. In [START_REF] Rocha | Realization of period maps of planar Hamiltonian systems[END_REF], Rocha also characterized all the dynamics, which may occur. Due to the automatic transverse intersection of the stable and unstable manifolds and due to the possibility of transforming any circle of equilibrium points into a rotating periodic orbit (see [START_REF] Fiedler | Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle[END_REF]), one can show that the Morse-Smale property holds for the parabolic equation on T 1 for a generic homogeneous nonlinearity f (u, ∂ x u) (see [START_REF] Joly | Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle[END_REF]). Finally, the realization of any radially symmetric twodimensional flow in the dynamics of (1.2) on T 1 for some f ≡ f (u, ∂ x u) and the fact that the dimension of the attractor is equal to the largest dimension of the unstable manifolds are shown in [36].

• d ≥ 3 and dim(Ω)≥ 2
The genericity of the Kupka-Smale property for vector fields in R d , d ≥ 3, has been proved independently by Kupka in [START_REF] Kupka | Contribution à la théorie des champs génériques[END_REF] and by Smale in [START_REF] Smale | Stable manifolds for differential equations and diffeomorphisms[END_REF]. Their proofs have been simplified by Peixoto in [START_REF] Peixoto | On an approximation theorem of Kupka and Smale[END_REF] (see [START_REF] Abraham | Transversal mappings and flows[END_REF] and [START_REF] Palis | Geometric theory of dynamical systems. An introduction[END_REF]). The strong difference with the lower dimensional vector fields is that, when d ≥ 3, (1.1) may admit transversal homoclinic orbits consisting in the transversal intersection of the stable and unstable manifolds of a hyperbolic periodic orbit. The existence of such an intersection is stable under small perturbations and yields a Smale horseshoe containing an infinite number of periodic orbits and chaotic dynamics equivalent to the dynamics of the shift operator, see [START_REF] Smale | Diffeomorphisms with many periodic points[END_REF]. Therefore, the Morse-Smale property cannot be dense in vector fields on R d with d ≥ 3. Even worse, the set of vector fields, whose dynamics are structurally stable under small perturbations, is not dense (notice that this set contains the vector fields satisfying the Morse-Smale property). Indeed, as shown in [START_REF] Guckenheimer | Structural stability of Lorenz attractors[END_REF], there exists an open set U of vector fields of R 3 and a foliation of U by 2-codimensional leaves (U λ ) λ∈R 2 such that each g ∈ U admits a Lorenz attractor A g and such that the dynamics of two attractors A g and A g are qualitatively equivalent if and only if g and g belong to the same leave U λ . The possible presence of other chaotic dynamics such as Anosov systems or wild dynamics is also noteworthy, see [START_REF] Anosov | Geodesic flows on closed Riemannian manifolds of negative curvature[END_REF], [START_REF] Newhouse | Diffeomorphisms with infinitely many sinks[END_REF] and [START_REF] Bonatti | Connexions hétéroclines et généricité d'une infinité de puits et de sources[END_REF]. For the interested reader, we refer to [START_REF] Robinson | Dynamical systems. Stability, symbolic dynamics, and chaos[END_REF] or [START_REF] Smale | Differentiable dynamical systems[END_REF].

In [START_REF] Brunovský | Generic transversality of heteroclinic and homoclinic orbits for scalar parabolic equations[END_REF], Brunovský and both authors proved that the stable and unstable manifolds of hyperbolic equilibria or periodic orbits of the parabolic equation (1.2) are generically transversal. To obtain the genericity of Kupka-Smale property, it remains to obtain the generic hyperbolicity of periodic orbits. Even if this problem is still open, we may strongly believe that the Kupka-Smale property is generic for the parabolic equation (1.2). There exist several results concerning the embedding of the finite-dimensional flows into the parabolic equations. Poláčik has shown in [START_REF] Poláčik | Imbedding of any vector field in a scalar semilinear parabolic equation[END_REF] that any ordinary differential equation on R d can be embedded into the flow of (1.2) for some f and for some domain Ω ⊂ R d . The constraint that the dimension of Ω is equal to the dimension of the imbedded flow is removed in [START_REF] Poláčik | High-dimensional ω-limit sets and chaos in scalar parabolic equations[END_REF], however the result concerns a dense set of flows only. A similar result has been obtained by Dancer and Poláčik in [START_REF] Dancer | Realization of vector fields and dynamics of spatially homogeneous parabolic equations[END_REF] for homogeneous nonlinearities f (u, ∇u) (see also [START_REF] Prizzi | Complicated dynamics of parabolic equations with simple gradient dependence[END_REF]). These realization results imply the possible existence of persistent chaotic dynamics in the flow of the parabolic equation (1.2) as soon as Ω has a dimension larger than one: transversal homoclinic orbits, Anosov flows on invariant manifolds of any dimension, Lorenz attractors etc...

• Gradient case

When g is a gradient vector field ∇G with G ∈ C 2 (R d , R), then -G is a strict Lyapounov function and (1.1) is a gradient system. In this case, the Kupka-Smale property is equivalent to the Morse-Smale property. The genericity of the Morse-Smale property for gradient vector fields has been obtained by Smale in [START_REF] Smale | On gradient dynamical systems[END_REF].

In the case where the nonlinearity f ∈ C r (Ω × R, R) (that is, f ≡ f (x, u) does not depend on ∇u), the parabolic equation (1.2) admits a strict Lyapounov function given by E

(u) = Ω 1 2 |∇u(x)| 2 -F (x, u(x)) dx, where F (x, u) = u 0 f (x, s)
ds is a primitive of f , and hence generates a gradient system. Brunovský and Poláčik have shown in [START_REF] Brunovský | The Morse-Smale structure of a generic reaction-diffusion equation in higher space dimension[END_REF] that the Morse-Smale property holds for the parabolic equation, generically with respect to f (x, u). It is noteworthy that the Morse-Smale property is no longer generic if one restricts the nonlinearities to the class of homogeneous functions f ≡ f (u) (see [START_REF] Poláčik | Persistent saddle connections in a class of reaction-diffusion equations[END_REF]). Poláčik has shown in [START_REF] Poláčik | Reaction-diffusion equations and realization of gradient vector fields[END_REF] that any generic gradient vector field of R d can be realized in the flow of the parabolic equation (1.2) on a bounded domain of R 2 with an appropriate nonlinearity f (x, u). The paper [START_REF] Poláčik | Reaction-diffusion equations and realization of gradient vector fields[END_REF] also contains the realization of particular dynamics such as non-transversal intersections of stable and unstable manifolds.

• Caveat: general ODEs or cooperative systems? In Table 1, we have given the striking correspondence between the flow generated by Eq. (1.1) and the semiflow generated by the parabolic equation (1.2). In addition, we have pointed out that some classes of cooperative systems are also involved in this correspondence. In fact, the reader should be aware that the dynamics of (1.2) is much closer to the ones of a cooperative system than to the ones of the general vector field (1.1). Indeed, the semiflow S f (t) generated by the parabolic equation (1.2) belongs to the class of strongly monotone semiflows, which means that this semiflow has more constraints than the flow T g (t) generated by a general vector field g (see Section 4). That is why, it could be more relevant to write Table 1 in terms of cooperative systems only (for example, by replacing the case of the general ODE with d ≥ 3 by the case of a cooperative system of ODEs in dimension d ≥ 4). However, we have chosen to mainly write Table 1 in terms of general ODEs for several reasons: -as far as the properties stated in Table 1 are concerned, there is no difference between the dynamics of a general ODE and the ones of a parabolic PDE, -the dynamics of general ODEs are common knowledge, whereas speaking in terms of cooperative systems may not give a good insight of the dynamics of (1.2), -not all the properties stated in Table 1 are known for the class of cooperative systems (for example the genericity of Kupka-Smale property is not yet known for d ≥ 4).

Zero number and unique continuation properties for the scalar parabolic equation

The results presented in Table 1 and in Section 2 strongly rely on properties specific to the parabolic equations. The purpose of this section is to give a first insight of these particular properties and of their use, to the reader. Dynamical systems generated by vectors fields are flows on R d , whereas the phase-space of the parabolic equation is an infinite-dimensional space X α . It is important to be aware of the fact that the parabolic equations generate only a small part of all possible dynamical systems on the Banach space X α . On one hand, this implies less freedom in perturbing the dynamics and hence in obtaining density results. In particular, whereas one can easily construct perturbations of a vector field g which are localized in the phase space R d , the perturbations of the nonlinearity f act in a non local way on X α (many different functions u can have the same values of u and ∇u at a given point x). Therefore, it is important to obtain unique continuation results in order to find values (x, u, ∇u), which are reached only once by a given periodic, heteroclinic or homoclinic orbit. On the other hand, the small class of dynamics generated by the parabolic equations admits special properties. These properties may in particular yield the constraints, which make the dynamics similar to the ones of low-dimensional vector fields.

The scalar parabolic equation in space dimension one (Ω = (0, 1) or T 1 ) satisfies a very strong property: the number of zeros of the solutions of the linearized equation is nonincreasing in time. This property is often called Sturm property since its idea goes back to Sturm [START_REF] Sturm | Sur une classe d'équations à différences partielles[END_REF] in 1836. There are different versions of this result, which have been proved by Nickel [START_REF] Nickel | Gestaltaussagen über Lösungen parabolischer Differentialgleichungen[END_REF], Matano [START_REF] Matano | Convergence of solutions of one-dimensional semilinear parabolic equations[END_REF][START_REF] Matano | Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation[END_REF], Angenent and Fiedler [START_REF] Angenent | The zero set of a solution of a parabolic equation[END_REF][START_REF] Angenent | The dynamics of rotating waves in scalar reaction diffusion equations[END_REF] and Chen [START_REF] Chen | A strong unique continuation theorem for parabolic equations[END_REF] (see also [START_REF] Galaktianov | Sturm's theorems on zero sets in nonlinear parabolic equations[END_REF] for a survey). By similar technics, a geometrical result on braids formed by solutions of the one-dimensional parabolic equation is obtained in [START_REF] Ghrist | Scalar parabolic PDEs and braids[END_REF].

Theorem 3.1. Let Ω = (0, 1) with Neumann boundary conditions or Ω = T 1 . Let T > 0, a ∈ W 1,∞ (Ω × [0, T ], R) and b ∈ L ∞ (Ω × [0, T ], R). Let v : Ω × (0, T ) → R be a bounded non-trivial classical solution of ∂ t v = ∂ 2 xx v + a(x, t)∂ x v + b(x, t)v , (x, t) ∈ Ω × (0, T ) .
Then, for any t ∈ (0, T ), the number of zeros of the function x ∈ Ω → v(x, t) is finite and non-increasing in time. Moreover, it strictly decreases at t = t 0 if and only if x → v(x, t 0 ) has a multiple zero.

Theorem 3.1 is the fundamental ingredient of almost all the results given in Table 1 in the cases Ω = (0, 1) and Ω = T 1 . It can be used either as a strong comparison principle or as a strong unique continuation property, as shown in the following examples of applications. General surveys can be found in [START_REF] Fiedler | Spatio-temporal dynamics of reaction-diffusion patterns[END_REF], [START_REF] Hale | Dynamics of a scalar parabolic equation[END_REF] and [36].

In the first application presented here, Theorem 3.1 is used as a strong maximum principle. In some sense, it yields an order on the phase space which is preserved by the flow. This illustrates how Theorem 3.1 may imply constraints similar to the ones of lowdimensional vector fields. The following result was first proved in [START_REF] Zelenyak | Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable[END_REF] and the proof given here comes from [START_REF] Matano | Convergence of solutions of one-dimensional semilinear parabolic equations[END_REF] (see also [START_REF] Fiedler | Spatio-temporal dynamics of reaction-diffusion patterns[END_REF]). Proposition 3.2. Let Ω = (0, 1), let u 0 ∈ X α and let u(x, t) be the corresponding solution of the parabolic equation (1.2) with homogeneous Neumann boundary conditions. The ω-limit set of u 0 consists of a single equilibrium point.

Proof: We first notice that v(x, t) = ∂ t u(x, t) satisfies the equation

∂ t v(x, t) = ∂ 2 xx v(x, t) + f ′ u (x, u(x, t), ∂ x u(x, t))v(x, t) + f ′ ∂xu (x, u(x, t), ∂ x u(x, t))∂ x v(x, t) .
Due to the Neumann boundary conditions, we have ∂ x u(0, t) = ∂ 2 xt u(0, t) = ∂ x v(0, t) = 0 for all t > 0. In particular, as soon as v(0, t) = 0, v(t) has a double zero at x = 0. Due to Theorem 3.1, either v is a trivial solution, that is v ≡ 0 for all t, and u is an equilibrium point, or v(0, t) vanishes at most a finite number of times since v(t) can have a multiple zero only a finite number of times. Assume that u is not an equilibrium, then u(0, t) must be monotone for large times and thus converges to a ∈ R. Any trajectory w in the ω-limit set of u 0 must hence satisfy w(0, t) = a for all t. Therefore, ∂ t w(0, t) = 0 for all t and ∂ t w(0, t) has a multiple zero at x = 0 for all times. Using Theorem 3.1, we deduce as above that w is an equilibrium point of (1.2). But there exists at most one equilibrium w satisfying w(0) = a and the Neumann boundary condition ∂ x w(0) = 0. Therefore, the ω-limit set of u 0 is a single equilibrium point w.

The second application comes from [START_REF] Joly | Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle[END_REF]. It shows how Theorem 3.1 can be used as a unique continuation property. This kind of property roughly says that, if two solutions coincide too much near a point (x 0 , t 0 ), then they must be equal everywhere. The motivation beyond this example of application is the following. We consider a time-periodic solution of (1.2) on Ω = T 1 . The problem is to find a perturbation of the nonlinearity f , which makes this periodic orbit hyperbolic. As enhanced above, such a perturbation is nonlocal in the phase space of (1.2). To be able to perform perturbation arguments, it is important to show that one can find a perturbation of f which acts only locally on the periodic orbit. To this end, one proves the following result. Proposition 3.3. Let p(x, t) be a periodic orbit of (1.2) on Ω = T 1 . Let T > 0 be its minimal period. Then, the map

(x, t) ∈ T 1 × [0, T ) → (x, p(x, t), ∂ x p(x, t))
is one to one.

Proof: Assume that this map is not injective. Then there exist x 0 , t 0 ∈ [0, T ) and

t 1 ∈ [0, T ), t 0 = t 1 such that p(x 0 , t 0 ) = p(x 0 , t 1 ) and ∂ x p(x 0 , t 0 ) = ∂ x p(x 0 , t 1 ) . The function v(x, t) = p(x, t + t 1 -t 0 ) -p(x, t) is a solution of the equation ∂ t v(x, t) = ∂ 2 xx v(x, t) + a(x, t)v(x, t) + b(x, t)∂ x v(x, t) ,
where a(x, t) =

1 0 f ′ u (x, p(x, t) + s(p(x, t + t 1 -t 0 ) -p(x, t)), ∂ x p(x, t + t 1 -t 0 ))ds and b(x, t) = 1 0 f ′ ux (x, p(x, t), ∂ x (p(x, t) + s(p(x, t + t 1 -t 0 ) -p(x, t)))ds.
Moreover, the function v(x, t) satisfies v(x 0 , t 0 ) = 0 and ∂ x v(x 0 , t 0 ) = 0 and does not vanish everywhere since |t 1t 0 | < T . Due to Theorem 3.1, the number of zeros of v(t) drops strictly at t = t 0 and never increases. However, v(t) is a periodic function of period T , and thus, its number of zeros is periodic. This leads to a contradiction and proves the proposition.

In a domain Ω of dimension d ′ ≥ 2, there is no known counterpart for Theorem 3.1 as shown in [START_REF] Fusco | Order structures and the heat equation[END_REF]. In particular, Proposition 3.3 does no longer hold. However, to be able to construct relevant perturbations of periodic orbits, one needs a result similar to Proposition 3.3, even if weaker. The following result can be found in [START_REF] Brunovský | Generic transversality of heteroclinic and homoclinic orbits for scalar parabolic equations[END_REF]. Its proof is based on a generalization of the arguments of [START_REF] Hardt | Nodal sets for solutions of elliptic equations[END_REF] and on unique continuations properties of the parabolic equations.

Theorem 3.4. Let p(x, t) be a periodic orbit of (1.2) with minimal period T > 0. There exists a generic set of points (x 0 , t 0 ) ∈ Ω × [0, T ) such that if t ∈ [0, T ) satisfies p(x 0 , t) = p(x 0 , t 0 ) and ∇p(x 0 , t) = ∇p(x 0 , t 0 ), then t = t 0 .

Cooperative systems of ODEs

We consider a system of differential equations

ẏ(t) = g(y(t)) , y(0) = y 0 ∈ R N , (4.1)
where g = (g i ) i=1...N is a C 1 vector field. Due to the analogy with biological models, the following definitions are natural. We say that (4.1) is a cooperative (resp. competitive) system if for any y ∈ R N and i = j, ∂g i ∂y j (y) is non-negative (resp. non-positive) and the matrix ( ∂g i ∂y j )(y) is irreducible i.e. it is not a block diagonal matrix (the simpler assumption that all the coefficients ∂g i ∂y j (y) are positive is sometimes made instead of the irreducibility). We say that (4.1) is a tridiagonal system if ∂g i ∂y j = 0 for |i -j| ≥ 2 and a cyclic tridiagonal system if the indices i and j are considered modulo N, i.e. if, in addition, we allow ∂g 1 ∂y N and ∂g N ∂y 1 to be non-zero. For the reader interested in cooperative systems, we refer to [START_REF] Smith | Monotone dynamical systems. An introduction to the theory of competitive and cooperative systems[END_REF].

In this section, we only consider cooperative systems. However, notice that, by changing t into -t or y i into -y i , we obtain similar results for competitive systems and for systems with different sign conditions.

The dynamics of cooperative systems may be as complicated as the dynamics of general vector fields. Indeed, Smale has shown in [START_REF] Smale | On the differential equations of species in competition[END_REF] that any vector field in R N -1 can be realized in a invariant manifold of a cooperative system in R N . Notice that this realization result implies that any one-dimensional vector field can be imbedded in a tridiagonal cooperative system and any two-dimensional vector field can be imbedded in a cyclic tridiagonal cooperative system. This explains why we present the tridiagonal cooperative systems in Table 1 as generalization of one-and two-dimensional vector fields.

However, the dynamics of a cooperative system (4.1) is really different from the ones of the general ODE (1.1) since a cooperative system generates a strongly monotone flow, that is, a flow which preserves a partial order. It is noteworthy that the semiflow S f (t) generated by the parabolic equation (1.2) also belongs to the class of strongly monotone semiflows (it preserves the order of X α induced by the classical order of C 0 (Ω)). Therefore, the semiflow of (1.2) is much closer to the one of the cooperative system (4.1), both admitting more constraints than the flow T g (t) generated by a general vector field g. In [START_REF] Hirsch | Differential equations and convergence almost everywhere in strongly monotone semiflows[END_REF] and [START_REF] Hirsch | Stability and convergence on strongly monotone dynamical systems[END_REF] for example, Hirsch has shown that almost all bounded trajectories of a strongly monotone semiflow are quasiconvergent, that is, their ω-limit sets consist only of equilibria. More precisely, all initial data, which have bounded nonquasiconvergent trajectories, form a meager subset (that is, the complement of a generic subset) of the phase space. Later, in [START_REF] Poláčik | Convergence in smooth strongly monotone flows defined by semilinear parabolic equations[END_REF], Poláčik has proved that the set of all initial data u 0 ∈ X α , which have bounded nonconvergent trajectories in the semiflow of the parabolic equation (1.2), is meager in

X α .
Moreover, since the works of Hirsch and Smillie, it is known that the dynamics of cooperative systems, which are in addition tridiagonal, are very constrained in any dimension N. Indeed, in [START_REF] Hirsch | Systems of differential equations which are competitive or cooperative. I. Limit sets[END_REF], [START_REF] Hirsch | Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere[END_REF] and [START_REF] Smillie | Competitive and cooperative tridiagonal systems of differential equations[END_REF], strong properties of the limit sets of cooperative systems are proved. In particular, any three-dimensional cooperative system satisfies the Poincaré-Bendixson property and the trajectory of any tridiagonal cooperative system converges to a single equilibrium point. Inspired by the articles of Henry and Angenent about the parabolic equation on (0, 1), Fusco and Oliva (see [START_REF] Fusco | Jacobi matrices and transversality[END_REF]) showed a theorem similar to Theorem 3.1 (see [START_REF] Smith | A discrete Lyapunov function for a class of linear differential equations[END_REF] for a more general statement). Theorem 4.1. Let N be the set of vector y ∈ R N such that, for all i = 1 . . . N, either y i = 0 or y i = 0 and y i-1 y i+1 < 0 (where y 0 = y N +1 = 0). For every y ∈ N , we set N(y) to be the number of sign changes for y i , when i goes from 1 to N. Let y(t) = 0 be a solution of

ẏ(t) = A(t)y(t) , (4.2)
where A ∈ C 0 (R, M N (R)) satisfies A ij (t) > 0 for all t ∈ R and all i = j. Then, the times t where y(t) ∈ N are isolated and, if y(t 0 ) ∈ N , then, for every ε > 0 small enough, N(y(t + ε)) < N(y(tε)).

In other words, the number of sign changes of the solutions of the linear equation (4.2) is non-increasing in time and strictly drops at t 0 if and only if y(t 0 ) has in some sense a multiple zero. The parallel with Theorem 3.1 is of course striking. Using Theorem 4.1, Fusco and Oliva have shown that the stable and unstable manifolds of equilibrium points of a tridiagonal cooperative system always intersect transversally. As a consequence, the Morse-Smale property is generic in the class of tridiagonal cooperative systems.

Theorem 4.1 also holds for cyclic tridiagonal cooperative systems, see [START_REF] Fusco | Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems[END_REF] and [START_REF] Smith | A discrete Lyapunov function for a class of linear differential equations[END_REF]. Using this fundamental property, Fusco and Oliva have shown in [START_REF] Fusco | Transversality between invariant manifolds of periodic orbits for a class of monotone dynamical systems[END_REF] that the stable and unstable manifolds of periodic orbits of cyclic tridiagonal cooperative systems always intersect transversally. In addition, Mallet-Paret and Smith have shown in [START_REF] Mallet-Paret | The Poincaré-Bendixson theorem for monotone cyclic feedback systems[END_REF] that cyclic tridiagonal cooperative systems satisfy the Poincaré-Bendixson property. Notice that, following [START_REF] Joly | Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle[END_REF] and [START_REF] Joly | Generic Morse-Smale property for the parabolic equation on the circle[END_REF], one should be able to prove the genericity of the Morse-Smale property for cyclic tridiagonal cooperative systems. This has been proved very recently by Percie du Sert (see [66]) Considering all these results, it is not surprising that there exists a parallel between tridiagonal cooperative systems and the parabolic equation on (0, 1). Indeed, consider a solution v of the linear one-dimensional parabolic equation

v(x, t) = ∂ 2 xx v(x, t) + a(x, t)∂ x v(x, t) + b(x, t)v(x, t) (x, t) ∈ R × R + (4.3)
We discretize the segment (0, 1) by a sequence of points x k = (k -1)/(N -1) with k = 1 . . . N. The natural approximation of v is given by

y k ≈ v(x k ) solution of ẏk (t) = y k+1 (t) -2y k (t) + y k-1 (t) h 2 + a k (t) y k+1 -y k h + b k (t)y k (t) (4.4)
where a k (t) = a(x k , t), b k (t) = b(x k , t) and h = 1/(N -1). If h is small enough, (4.4) is a tridiagonal cooperative system. The relation between Theorems 3.1 and 4.1 is obvious in this framework.

Kupka-Smale property and other open problems

One of the main goals of the study of dynamical systems is to understand the behaviour of a generic dynamical system. The most recent results concerning the parabolic equations are related to the genericity of Kupka-Smale property. In [START_REF] Brunovský | Generic transversality of heteroclinic and homoclinic orbits for scalar parabolic equations[END_REF], the generic transversality of homoclinic and heteroclinic orbits of hyperbolic equilibria or periodic orbits of (1.2) is proved. However, the generic hyperbolicity of periodic orbits is still an open problem (see the discussion in [START_REF] Brunovský | Generic transversality of heteroclinic and homoclinic orbits for scalar parabolic equations[END_REF]). Its resolution would complete the whole correspondance of Table 1 and is one of the most important open problem concerning the qualitative dynamics of the parabolic equation (1.2). However, even if the generic Kupka-Smale property is obtained, it cannot give a good insight of the complex and chaotic dynamics that may be generated by homoclinic connections. For this reason, the study of finite-dimensional flows has been pursued further the Kupka-Smale property and is still in progress. The corresponding results should serve as a guideline for the study of the flow generated by the parabolic equation (1.2). For vector fields, one of the main steps beyond Kupka-Smale property is Pugh's closing lemma: if p is a non-wandering point of the dynamical system T g (t) generated by (1.1), then there exists a C 1 -perturbation g of g such that p is a periodic point of T g(t) (the case of a C r -perturbation with r ≥ 2 is still open). The proof of Pugh in [START_REF] Pugh | The closing lemma[END_REF] concerns discrete dynamical systems. It has been adapted to the case of flows by Pugh and Robinson in [START_REF] Pugh | The C 1 closing lemma, including Hamiltonians[END_REF] (see also [START_REF] Robinson | Introduction to the closing lemma, in The structure of attractors in dynamical systems[END_REF] for an introduction to the proof). A direct consequence of Pugh closing lemma is the general density theorem: for a generic finite-dimensional flow, the non-wandering points are is the closure of the periodic points (see [START_REF] Pugh | An improved closing lemma and a general density theorem[END_REF] and [START_REF] Robinson | Dynamical systems. Stability, symbolic dynamics, and chaos[END_REF]). Other connecting lemmas have been proved by Hayashi [START_REF] Hayashi | Connecting invariant manifolds and the solution of the C 1 stability and Ω-stability conjectures for flows[END_REF] and Bonatti and Crovisier [START_REF] Bonatti | Récurrence et généricité[END_REF]. They enable a better understanding of generic dynamics. For example, the class of finite-dimensional dynamical systems which either satisfy the Morse-Smale property or admit a transversal homoclinic connection is generic (see [START_REF] Pujals | Homoclinic tangencies and hyperbolicity for surface diffeomorphisms[END_REF], [START_REF] Bonatti | On the existence of non-trivial homoclinic classes[END_REF] and [START_REF] Crovisier | Birth of homoclinic intersections: a model for the central dynamics of partially hyperbolic systems[END_REF] for discrete dynamical systems in dimensions d = 2, d = 3 and d ≥ 4 respectively, and see [START_REF] Arroyo | Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows[END_REF] for three-dimensional flows). Obtaining similar results for the flow of the parabolic equation should be a very interesting and difficult challenge.

Other interesting open problems concern the realization of finite-dimensional dynamics in the semiflow of parabolic equations. Indeed, we only know that one can realize the dynamics of a dense set of general ODEs in the flow of a parabolic equation (1.2) on a twodimensional domain. One may wonder if it is possible to realize the dynamics of all ODEs. Since the parallel between parabolic equations and cooperative systems is stronger, the following strong realization conjecture may be more plausible: any flow of a cooperative system of ODEs can be realized in an invariant manifold of the flow of a parabolic equation (1.2) on a two-dimensional domain.

Finally, the genericity of the Morse-and Kupka-Smale properties is also an interesting problem for other classes of partial differential equations. The genericity of the Morse-Smale property is known for the wave equations ü + γ u = ∆u + f (x, u) with constant damping γ > 0 (see [START_REF] Brunovský | Genericity of the Morse-Smale property for damped wave equations[END_REF]) and with variable damping γ(x) ≥ 0 in space dimension one (see [START_REF] Joly | Generic transversality property for a class of wave equations with variable damping[END_REF]). We recall that, in both cases, the associated dynamical system is gradient. Nothing is known for other classes of PDEs, in particular for the equations of fluids dynamics and for systems of parabolic equations U = ∆U + f (x, U), with U(x, t) ∈ R N . In all these cases, the main problem consists in understanding how the perturbations act on the phase plane of the PDE. Either one proves unique continuation results similar to Theorem 3.4 in order to be able to use local perturbations of the flow (as in [START_REF] Joly | Generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle[END_REF], [START_REF] Joly | Generic Morse-Smale property for the parabolic equation on the circle[END_REF] and [START_REF] Brunovský | Generic transversality of heteroclinic and homoclinic orbits for scalar parabolic equations[END_REF]), or one uses particular non-local perturbations in a very careful way (as in [START_REF] Brunovský | The Morse-Smale structure of a generic reaction-diffusion equation in higher space dimension[END_REF], [START_REF] Brunovský | Genericity of the Morse-Smale property for damped wave equations[END_REF] and [START_REF] Joly | Generic transversality property for a class of wave equations with variable damping[END_REF]).

Glossary

In this section, S(t) denotes a general continuous dynamical system on a Banach space X. An orbit of S(t) is denoted by x(t) = S(t)x 0 with t ∈ I, where I = [0, +∞), I = (-∞, 0] or I = (-∞, +∞) in the case of a positive, negative or global trajectory respectively. Compact global attractor: if it exists, the compact global attractor A of S(t) is a compact invariant set which attracts all the bounded sets of X. Notice that A is then the set of all the bounded global trajectories. See [START_REF] Hale | Asymptotic behavior of dissipative systems[END_REF]. αand ω-limit sets: let x 0 ∈ X. The α-limit set α(x 0 ) and the ω-limit set ω(x 0 ) of x 0 are the sets of accumulation points of the negative and positive orbits coming from x 0 respectively. More precisely,

α(x 0 ) = {x ∈ X / ∃(t n ) n∈N , t n ---→ n→∞ -∞ and a negative trajectory x(t) such that x(0) = x 0 and x(t n ) ---→ n→∞ x} ω(x 0 ) = {x ∈ X / ∃(t n ) n∈N , t n ---→ n→∞ +∞ such that S(t n )x 0 ---→ n→∞ x}
The limit sets α(x 0 ) and ω(x 0 ) are non-empty connected compact sets. Homoclinic or heteroclinic orbit: let x(t) = S(t)x 0 be a global trajectory of S(t). Assume that the α-and ω-limit sets of x 0 exactly consists in one orbit, denoted x -(t) and x + (t) respectively, this orbit being either an equilibrium point or a periodic orbit. The trajectory x(t) is said to be a homoclinic orbit if x -(t) = x + (t) and a heteroclinic orbit if x -(t) = x + (t). Backward uniqueness property: S(t) satisfies the backward uniqueness property if for any time t 0 > 0 and any trajectories x 1 (t) and x 2 (t), x 1 (t 0 ) = x 2 (t 0 ) implies x 1 (t) = x 2 (t) for all t ∈ [0, t 0 ]. Notice that this does not mean that S(t) admits negative trajectories. Hyperbolic equilibrium points or periodic orbits: an equilibrium point e of S(t) is hyperbolic if the linearized operator x → D e S(1)x has no spectrum on the unit circle. Let p(t) be a periodic solution of S(t) with minimal period T . For each x ∈ X, we denote t → Π(t, 0)x the corresponding trajectory of the linearization of S(t) along the periodic solution p(t). Then, p(t) is said hyperbolic if the linear map x → Π(T, 0)x has no spectrum on the unit circle except the eigenvalue 1 which is simple. Remark that then, for any integer k = 0, the linear map x → Π(kT, 0)x has no spectrum on the unit circle except the eigenvalue 1 (which is simple). Stable and unstable manifolds: let e be a hyperbolic equilibrium point of S(t). There exists a neighbourhood N of e such that the set W u loc (e) = {x 0 ∈ X , ∃ a negative trajectory x(t) with x(0) = x 0 and, ∀t ≤ 0, x(t) ∈ N } is a submanifold of X, in which all negative trajectories converge to e when t goes to -∞.

The manifold W u loc (e) is called the local unstable manifold of e. Pushing W u loc (e) by the flow S(t), one can define the (global) unstable set W u (e) = ∪ t≥0 S(t)W u loc (e), which consists in all the negative trajectories converging to e when t goes to -∞. This unstable set W u (e) is an immersed submanifold under suitable properties. For instance, backward uniqueness properties are needed to extend the manifold structure. In the same way, one defines the local stable manifold W s loc (e) = {x 0 ∈ X , ∀t ≥ 0, S(t)x 0 ∈ N } = {x 0 ∈ X , ∀t ≥ 0, S(t)x 0 ∈ N and S(t)x 0 ----→ t→+∞ e} .

General partial differential equations (and parabolic equations in particular) do not admit negative trajectories. Therefore, it is less easy to extend the local stable manifold to a global stable manifold. However, one can define the stable set W s (e) of e as follows W s (e) = {x 0 ∈ X , S(t)x 0 ----→ t→+∞ e} .

Under suitable additional properties (which are satisfied by the parabolic equation (1.2)), one can show that W s (e) is also an immersed submanifold. For instance, backward uniqueness properties of the adjoint dynamical system S * (t) on X * and finite-codimensionality of W s loc (e) are needed (see [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF] for more details). If p(t) is a hyperbolic periodic orbit, one defines its unstable and local stable manifolds in the same way. See for example [START_REF] Palis | Geometric theory of dynamical systems. An introduction[END_REF] for more details. Non-wandering set: a point x 0 ∈ X is non-wandering if for any neighbourhood N ∋ x 0 and any time t 0 > 0, there exists t ≥ t 0 such that S(t)N ∩ N = ∅. The Kupka-Smale and Morse-Smale properties: S(t) satisfies the Kukpa-Smale property if all its equilibrium points and periodic orbits are hyperbolic and if their stable and unstable manifolds intersect transversally. It satisfies the Morse-Smale property if in addition its non-wandering set consists only in a finite number of equilibrium points and periodic orbits. We refer to [START_REF] Palis | Geometric theory of dynamical systems. An introduction[END_REF] for more precisions on these notions. Gradient dynamical systems: S(t) is gradient if it admits a Lyapounov functional, that is a function Φ ∈ C 0 (X, R) such that, for all x 0 ∈ X, t → Φ(S(t)x 0 ) is non-increasing and is constant if and only if x 0 is an equilibrium point. We recall that a gradient dynamical system does not admit periodic or homoclinic orbits. Cooperative system of ODEs: see Section 4. Generic set and Baire space: a generic subset of a topological space Y is a set which contains a countable intersection of dense open subsets of Y . A property is generic in Y if it is satisfied for a generic set of Y . The space Y is called a Baire space if any generic set is dense in Y . In particular a complete metric space is a Baire space. Whitney topology: let k ≥ 0 and let M be a Banach manifold. The Whitney topology on C k (M, R) is the topology generated by the neighbourhoods {g ∈ C k (M, R) , |D i f (x) -D i g(x)| ≤ δ(x), ∀i ∈ {0, 1, . . . , k}, ∀x ∈ M} , where f is any function in C k (M, R) and δ is any positive function in C k (M, (0, +∞)). Notice that C k (M, R) endowed with the Whitney topology is a Baire space even if it is not a metric space when M is not compact. We refer for instance to [START_REF] Golubitsky | Stable mapping and their singularities[END_REF]. The fractional power space X α : let A be a positive self-adjoint operator with compact inverse on L 2 (Ω). Let (λ n ) be the sequence of its eigenvalues, which are positive, and let (ϕ n ) be the corresponding sequence of eigenfunctions, which is an orthonormal basis of L 2 (Ω). For each α ∈ R, we define the fractional power of A by A( n c n ϕ n ) = n c n λ α n ϕ n . In particular, A 0 = Id and A 1 = A. The space X α is the domain of A α that is

X α = {ϕ ∈ L 2 (Ω) , ϕ = n c n ϕ n such that (c n λ α n ) ∈ ℓ 2 (N)} .
It is possible to define fractional powers of more general operators, called sectorial operators, see [START_REF] Henry | Geometric theory of semilinear parabolic equations[END_REF].

The Sobolev space W s,p (Ω): if s is a positive integer, W s,p (Ω) is the space of (classes of) functions f ∈ L p (Ω), which are s times differentiable in the sense of distributions and whose derivatives up to order s belong to L p (Ω). It is possible to extend this notion to positive numbers s which are not integers by using interpolation theory.

Unique continuation properties: let us consider a partial differential equation on Ω and let u(x, t) be any solution of it. A unique continuation property for this PDE is a result stating that if u(x, t) vanishes on a subset of Ω × R + which is too large in some sense, then u(x, t) must vanish for all (x, t) in Ω × R + .

•• 3 •

 3 Gradient dynamics • Convergence to an equilibrium point • Automatic transversality of stable and unstable manifolds • Genericity of Morse-Smale property • Knowledge of the equilibrium points implies knowledge of the whole dynamics • Dimension of the attractor equal to the largest dimension of the unstable manifolds • Realisation of the ODE in the PDE Ω = (0, 1) Poincaré-Bendixson property • Automatic transversality of stable and unstable manifolds of two orbits if one of them is a hyperbolic periodic orbit or if both are equilibrium points with different Morse indices. • Non-existence of homoclinic orbits for periodic orbits • Genericity of Morse-Smale property • Realisation of the ODE in the PDE Ω = T 1 General case d = 2 g radially symmetric • Automatic transversality of stable and unstable manifolds of equilibrium points and periodic orbits • No homoclinic orbit • Knowledge of the equilibrium points and of the periodic orbits implies knowledge of the whole dynamics • Genericity of the Morse-Smale property. • Dimension of the attractor equal to the largest dimension of the unstable manifolds • Realisation of the ODE in the PDE Ω = T 1 f (x, u, ∇u) ≡ f (u, ∇u) d ≥ Existence of persistent chaotic dynamics • Genericity of Kupka-Smale property (ODE) • Generic transversality of homoclinic and heteroclinic orbits (PDE) • Realisation of the ODE in the PDE dim(Ω)≥ 2 Any d g ≡ ∇G • Gradient dynamics • Genericity of the Morse-Smale property • Realisation of a generic ODE in the PDE Any

To make the reading of this article easier for the reader, who is not familiar with dynamical systems theory or with the study of PDEs, we add a short glossary at the end of the paper.

Acknowledgments

The authors are very grateful to Sylvain Crovisier and Lucien Guillou for fruitful discussions.