Adrien Nguyen Huu
email: adrien.nguyenhuu@gmail.com

Bernardo Costa-Lima

Orbits in a stochastic Goodwin-Lotka-Volterra model

Keywords: Lotka-Volterra model, Goodwin model, Brownian motion, Random perturbation, Business cycles, Stochastic Lyapunov techniques

This paper examines the cycling behavior of a deterministic and a stochastic version of the economic interpretation of the Lotka-Volterra model, the Goodwin model. We provide a characterization of orbits in the deterministic highly non-linear model. We then study the cycling behavior for a stochastic version, where a Brownian noise is introduced via an heterogeneous productivity factor. Sufficient conditions for existence of the system are provided. We prove that the system produces cycles around an equilibrium point in finite time for general volatility levels, using stochastic Lyapunov techniques for recurent domains. Numerical insights are provided.

Introduction

The Lotka-Volterra equation is at the heart of population dynamics, but also possesses a famous economic interpretation. Introduced by Richard Goodwin [START_REF] Goodwin | A growth cycle, Socialism, capitalism and economic growth[END_REF] in 1967, the model in its modern form [START_REF] Desai | A clarification of the Goodwin model of the growth cycle[END_REF] reduces to the planar oscillator on a subset D of R + :

dx t = x t (Φ(y t ) -α) dt dy t = y t (κ(x t ) -γ) dt , (1)
where x t denotes the wage share of the working population and y t the employment rate, α and γ are constant, and the following assumption is made on κ and Φ.

Assumption 1. Consider system (1).

(i) Φ ∈ C 2 ([0, 1)), Φ (y) > 0, Φ (y) ≥ 0 for all y ∈ [0, 1), Φ(0) < α and lim y→1 -Φ(y) = +∞.

(ii) κ ∈ C 2 (R + ), -∞ < κ (x) < 0 for all x ∈ R + , κ(0) > γ and lim x→+∞ κ(x) = -∞.

Lemma 1 below asserts that Assumption 1 is sufficient to have (x t , y t ) ∈ D := R * + × (0, 1) for any t ≥ 0 if (x 0 , y 0 ) ∈ D. This property preserves the above interpretation for x and y: the employment rate cannot exceed one for obvious reasons, but the wage share can, depending on the chosen economic assumptions, see [START_REF] Grasselli | An analysis of the Keen model for credit expansion, asset price bubbles and financial fragility[END_REF]. This distinctive feature of the economic version (1) on its biological counterpart follows from a construction based on assumptions describing a closed capitalist economy. It can be done in three steps:

(I) Assume a Leontief production function P t = min(K t /ν; a t y t N t ) with full utilization of capital, i.e., K t /ν = a t y t N t . Here, P t is the yearly output, K t the invested capital, ν > 0 a capital-to-output ratio, a t := a 0 exp(αt) is the average productivity of workers and N t := N 0 exp(βt) is the size of the labor class.

(II) The capital depreciates and receives investment, i.e., dK t /dt = (κ(x t )-δ)K t , where δ > 0 is the depreciation rate and κ the investment function. Goodwin [START_REF] Goodwin | A growth cycle, Socialism, capitalism and economic growth[END_REF] originally invokes Say's law, i.e., κ : x ∈ R + → (1 -x)/ν.

(III) Assume a reserve army effect for wage negotiation of the form dw t = Φ(y t )w t dt where w t := a t x t represents the real wage of the total working population, and Φ is the Phillips curve.

Defining γ := α+β +δ allows to retrieve (1) for (x t , y t ) := (w t /a t , K t /(νa t N t )). The class-struggle model [START_REF] Arató | A famous nonlinear stochastic equation (Lotka-Volterra model with diffusion)[END_REF] has been extensively studied because it allows to generate endogenous real business cycles affecting the production level P t , e.g. [START_REF] Flaschel | Some stability properties of Goodwin's growth cycle a critical elaboration[END_REF][START_REF] Glombowski | Generalizations of Goodwin's growth cycle model[END_REF][START_REF] Grasselli | An analysis of the Keen model for credit expansion, asset price bubbles and financial fragility[END_REF]15,[START_REF] Velupillai | Some stability properties of Goodwin's growth cycle[END_REF][START_REF] Veneziani | Structural stability and Goodwin's growth cycle[END_REF]. On this matter, Goodwin himself conceded that the model is "starkly schematized and hence quite unrealistic" [START_REF] Goodwin | A growth cycle, Socialism, capitalism and economic growth[END_REF]. It hardly connects with irregular observed trajectories, see [START_REF] Harvie | Testing Goodwin: growth cycles in ten OECD countries[END_REF][START_REF] Mohun | Goodwin cycles and the US economy[END_REF]. The objective of this paper is thus to study the following perturbed version of (1) by a standard Brownian motion (W t ) t≥0 on a stochastic basis (Ω, F, P):

dx t = x t (Φ(y t ) -α + σ 2 (y t ))dt + σ(y t )dW t dy t = y t (κ(x t ) -γ + σ 2 (y t ))dt + σ(y t )dW t , (2)
where σ is a positive function of y bounded by σ 0 > 0, and the filtration F t is generated by paths of W . The form of σ is discussed in Remark 2 after. A stronger condition, Assumption 3, is assumed later on the behavior of σ to ensure that solutions of (2) remain in D. The example of Section 5 will also illustrate how such condition can hold. We modify the economic development (I), (II) and (III) by introducing the perturbation on one assumption, namely we assume that for t ≥ 0, da t := a t dα t = a t (αdtσ(y t )dW t ) , a 0 ≥ 0 ,

instead of da t = a t αdt. Using Itô formula with (3) in the previous reasoning retrieves [START_REF] Bahar | Stochastic delay LotkaVolterra model[END_REF]. Productivity is one of the few exogenous parameters of the model, and one of those that were significantly invoked as influencial over business cycles, e.g. [START_REF] Evans | Productivity shocks and real business cycles[END_REF][START_REF] Hansen | Indivisible labor and the business cycle[END_REF]. Without arguing for the pertinence of that particular assumption, we simply suggest here that a standard continuous perturbation in this crucial parameter seems a good starting point.

To our knowledge, this is the first attempt to consider random noise in Goodwin interpretation of the famous prey-predator model. To stay in the spirit of the economic application, the present paper studies the cyclical behavior of the deterministic system (1) and the stochastic version [START_REF] Bahar | Stochastic delay LotkaVolterra model[END_REF]. Namely, our contribution are as follows, developed in the present order:

• In Section 2, we fully characterize solutions of (1) and the period of their orbits. This generalizes standard results on Lotka-Volterra systems to bounded domains of existence.

• In Section 3, we provide existence conditions for regular solutions of (2). We use the entropy of (1) to estimate the deviation induced by [START_REF] Bartlett | On theoretical models for competitive and predatory biological systems[END_REF]. We provide a definition of stochastic orbits for [START_REF] Bahar | Stochastic delay LotkaVolterra model[END_REF]. The proof that solutions of (2) draw stochastic orbits in finite time around a unique point is given in Section 4.

Our contribution has to be put in contrast with numerous studies of random perturbations of the Lotka-Volterra system. Apart from the obviously different origin of perturbations in the model, attention was mainly given to systems like (2) for its asymptotic behavior (e.g., [START_REF] Khasminskii | Long term behavior of solutions of the Lotka-Volterra system under small random perturbations[END_REF][START_REF] Mao | Asymptotic behaviour of the stochastic Lotka-Volterra model[END_REF][START_REF] Nguyen Huu | Dynamics of a stochastic Lotka-Volterra model perturbed by white noise[END_REF]), regularity, persistence and extinction of species (e.g., [START_REF] Cai | Stochastic analysis of the Lotka-Volterra model for ecosystems[END_REF][START_REF] Mao | Environmental Brownian noise suppresses explosions in population dynamics[END_REF][START_REF] Nguyen Huu | Dynamics of a stochastic Lotka-Volterra model perturbed by white noise[END_REF][START_REF] Reichenbach | Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model[END_REF]), and the addition of regimes, jumps or delay (e.g., [START_REF] Bahar | Stochastic delay LotkaVolterra model[END_REF][START_REF] Liu | Stochastic LotkaVolterra systems with Lévy noise[END_REF][START_REF] Zhu | On hybrid competitive Lotka-Volterra ecosystems[END_REF]). Here, we attempt to provide a relevant description of trajectories (x t , y t ) and indirectly P t , namely a cyclical behavior. This is done using stochastic Lyapunov techniques for recurrent domains as described in [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF][START_REF] Thygesen | A survey of Lyapunov techniques for stochastic differential equations[END_REF]. By conveniently dividing the domain D, we obtain that almost every trajectory "cycles" around a point in finite time. The L 1 -boundedness is out of reach with our method, but numerical simulations are presented in Section 5, not only to provide expectation of cycles, but also allow to conjecture a limit cycle phenomenon for the expectation of (x t , y t ). It is somewhat unclear how Assumption 1 and late Assumption 3 on Φ, κ and σ, are relevant in these results. We show below thatthey are sufficient to obtain existence of regular solutions to [START_REF] Bahar | Stochastic delay LotkaVolterra model[END_REF]. This actually emphasizes the role played by the entropy of the deterministic system in the well-posedness of the stochastic system and as a natural measure for perturbation.

Deterministic orbits

According to Assumption 1, there exists only one non-hyperbolic equilibrium point to (1) in D given by (x, ŷ) := (κ -1 (γ), Φ -1 (α)). On the boundary of D, there exists also an additional equilibrium (0, 0) which is eluded along the paper.

Definition 1. Let V 1 , V 2 and V be three functions defined by V : (x, y) ∈ R * + × (0, 1) → V 1 (x) + V 2 (y) and V 1 : x ∈ R * + → x x κ(x) -κ(s) s ds , V 2 : y ∈ (0, 1) → y ŷ Φ(s) -Φ(ŷ) s ds .
Lemma 1. Let (x 0 , y 0 ) ∈ D. Let Assumption 1 hold. Then a solution (x t , y t ) to (1) starting at (x 0 , y 0 ) at t = 0 describes closed orbits given by the set of points {(x, y) ∈ D : V (x, y) = V (x 0 , y 0 )}, and (x t , y t ) ∈ D for all t ≥ 0.

Proof. It is well-known [START_REF] Grasselli | An analysis of the Keen model for credit expansion, asset price bubbles and financial fragility[END_REF] that V is a Lyapunov function and a constant of motion for system (1): V 1 and V 2 take non-negative values with V 1 (x) = V 2 (ŷ) = 0, and dV /dt(x, y) = 0. Additionally, under Assumption 1.(i)-(ii),

lim x↑+∞ V 1 (x) = lim x↓0 + V 1 (x) = lim y↑1 - V 2 (y) = lim y↓0 + V 2 (y) = +∞ , (4)
so that for any (x 0 , y 0 ) ∈ D, V (x 0 , y 0 ) < +∞ and the solution stays in D.

The value of V characterizing an orbit, it is in bijection with its period. The following generalizes [START_REF] Hsu | A remark on the period of the periodic solution in the Lotka-Volterra system[END_REF].

Theorem 1. Let (x t , y t ) t≥0 be a solution to (1) satisfying Assumption 1, with (x 0 , x 0 ) ∈ D\{(x, ŷ)}. Let V 0 := V (x 0 , y 0 ), and x < x the two solutions to equation

V 1 (x) = V 0 . Define three functions F 1 , F 2 , G by F 1 : u ∈ R → V 2 (Φ -1 (u + +α)) , F 2 : u ∈ R → V 2 (Φ -1 (-u -+α)) , G : z ∈ R → V 0 -V 1 (e z ) .
Then (x T , y T ) = (x 0 , y 0 ) for T defined by

T (V 0 ) := log(x) log(x) 1 F -1 1 (G(z)) - 1 F -1 2 (G(z)) dz . (5)
Proof. Let (x 0 , y 0 ) ∈ D\{(x, ŷ)}, V 0 = V (x 0 , y 0 ) > 0 and (x t , y t ) a solution to (1) starting at (x 0 , y 0 ). According to Lemma 1,

V 1 (x t ) = V 0 implies V 2 (y t ) = 0. Then {x ∈ R + : x t =
x for some t ≥ 0} = [x, x] =: I. Homogeneity of (1) allows to set (x 0 , y 0 ) = (x, ŷ) without loss of generality. Let T 1 := inf{t ≥ 0 :

x t = x}. For t ∈ [0, T 1 ], (x t , y t ) ∈ [x, x] × [ŷ, ȳ]
, with ȳ such that V 2 (ȳ) = V 0 . Let z t := log(x t ) for t ≥ 0. Then (1) rewrites dz = (Φ(y) -α)dt, y = Φ -1 (dz/dt + α) and we get

d 2 z dt 2 = Φ (y) dy dt = Φ Φ -1 dz dt + α Φ -1 dz dt + α [κ(e z ) -γ] .
Let u := Φ(y) -α and define Ψ :

= Φ • Φ -1 × Φ -1 , to rewrite again dz/dt = u du/dt = Ψ (u + α) [κ(e z ) -γ] . (6)
Since

z t ∈ [z, z] := [log(x), log(x)] and u t ∈ [0, Φ(ȳ) -α] for t ∈ [0, T 1 ]
, separation of variables in [START_REF] Desai | A clarification of the Goodwin model of the growth cycle[END_REF] provides two quantities F and G:

F (u) := u 0 s Ψ (s + α) ds = z z [κ(e s ) -γ]ds =: G(z) . (7)
The function F verifies F (0) = G(z) = 0, is increasing on [0, Φ(ȳ) -α] and decreasing on [Φ(y) -α, 0] with y < ŷ so that V 2 (y) = 0. Coming back to y = Φ -1 (u + α) we get

F (u) = u 0 s Φ (Φ -1 (s + α))Φ -1 (s + α) ds = Φ -1 (u+α) ŷ Φ(s) -Φ(ŷ) s ds = V 2 (Φ -1 (u + α)) , implying that F (u) ∈ [0, V (x, ŷ)] for u ∈ [Φ(y) + α, Φ(ȳ) + α]. We can write F = F 1 + F 2
= V 0 -V 1 (e z ) Since sign(κ(x) -γ) = sign(x -x) we have max z∈[z,z] G(z) = G(log(x)) = V 1 (x) = V (x, ŷ
), while minimums are given by G(z) = G(z) = 0. This sums up with G([z, z]) ⊂ [0, V 0 ], so we can write on this interval F -1 1 (G(z)) = u = dz/dt which finally gives

T 1 = z z dz F -1 1 (G(z))
.

We apply the same method for the other half orbit, taking (x 0 , y 0 ) = (x, ŷ) and T 2 := inf{t ≥ 0 : x t = x}, to reach the other half of expression (5), i.e., T

(V 0 ) = T 1 + T 2 .
Remark 1. A first order approximation of (1) at (x, ŷ) provides a linear homogeneous system, which solution is trivially given by a linear combination of sines and cosines of (-xΦ (ŷ)ŷκ (x)t). It follows that lim

V 0 →0 T (V 0 ) = 2π -xΦ (ŷ)ŷκ (x) > 0 .
3

Stochastic Goodwin model

We study a specific case of (2) where the deterministic part cancels at a unique point in D defined by (x, ỹ) := (κ -1 (γσ 2 (ỹ)), ỹ) where ỹ comes from the following.

Assumption 2. There is a unique ỹ ∈ (0, 1) such that Φ(ỹ) -α + σ 2 (ỹ) = 0

For a stochastic differential equation to have a unique global solution for any given initial value, functions Φ and κ are generally required to satisfy linear growth and local Lipschitz conditions, see [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF]. We can however consider the following Theorem of Khasminskii [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF], which is a reformulation of Theorem 3.4, Theorem 3.5 and Corollary 3.1 of [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF] to our context. Theorem 2. Consider the following stochastic differential equation for z taking values in R 2 + :

dz t = µ(z t )dt + σ(z t )dW t . (8)
Let (D n ) n≥1 be an increasing sequence of open sets, and

(K n ) n≥1 a sequence of constants, verifying (a) Dn ⊂ D for all n ≥ 1, (b) n D n = D.
(c) For any n ≥ 1, functions µ and σ are Lipschitz on D n and verify

|µ(z)|+|σ(z)| ≤ K n (1+|z|) for any z ∈ D n . Let ϕ ∈ C 1,2,2 (R + × D) and (K, k) ∈ R 2 + be such that, denoting L z the generator associated with (8), (d) L z ϕ(t, z t ) ≤ Kϕ(t, z t ) + k on the set R + × D, (e) lim n inf D\Dn ϕ(t, z) = +∞ for any t ≥ 0.
Then for any z ∈ D, there exists a regular adapted solution to [START_REF] Flaschel | Some stability properties of Goodwin's growth cycle a critical elaboration[END_REF], unique up to null sets, with the Markov property and verifying z t ∈ D for all t ≥ 0 almost surely.

To satisfy conditions (a) to (e), we study (2) under the additional sufficient growth conditions. Assumption 3. There exist two positive constants K, k such that (i) σ 2 (y)Φ (y) ≤ KV 2 (y) + k for all y ∈ (0, 1),

(ii) -xκ (x) -κ(x) ≤ KV 1 (x) + k for all x ∈ R * + .
Remark 2. Assumption 3.(i) involves both Φ and σ to ensure that y t ∈ (0, 1) for all t ≥ 0 almost surely. Assumption 3.(ii) holds for polynomial growth of κ, suiting the classical conditions of existence on R + for x t . The dependence of σ could be generalized to x in full generality, implying a stronger condition than (ii). We refrain from doing this easy extension, emphasizing the unavoidable dependence in y.

For ϕ ∈ C 1,2,2 (R + × D), we recall the diffusion operator associated with (2) by

Lϕ(t, x, y) := ∂ϕ ∂t + ∂ϕ ∂x x(Φ(y) -α + σ 2 (y)) + ∂ϕ ∂y y(κ(x) -γ + σ 2 (y)) + σ 2 (y) 2 ∂ 2 ϕ ∂x 2 x 2 + ∂ 2 ϕ ∂y 2 y 2 + 2 ∂ 2 ϕ ∂x∂y xy (t, x, y). (9)
Theorem 3. Let (x 0 , y 0 ) ∈ D. Let Assumptions 1, 2 and 3 hold. Then there exists a solution (x t , y t ) t≥0 to (2) staying in D almost surely.

Proof. Let us show that conditions (a) to (e) of Theorem 2 are fulfilled. Consider the sequence of sets (D n ) n≥1 defined by

D n = (1/(n + 1), n) × (1/(n + 1), 1 -1/(n + 1)). For any n ≥ 1, D n is open and D n ⊂ D n+1 . (a)
and (b) are satisfied with the limit D = R * + × (0, 1). According to Assumption 1, one can always find K n big enough such that max{|Φ(y) -α|; |κ(x) -γ|} ≤ K n for any (x, y) ∈ D n , and ensures the local Lipschitz condition (c). Now consider V of Definition 1 which is C 1,2,2 on D. Applying (9),

LV (x, y) = [κ(x) -κ(x)] (Φ(y) -α + σ 2 (y)) + [Φ(y) -Φ(ŷ)] (κ(x) -γ + σ 2 (y)) + κ(x) -κ(x) -xκ (x) + Φ(ŷ) -Φ(y) + yΦ (y) σ 2 (y)/2 . Since α = Φ(ŷ) and γ = κ(x), LV (x, y) = κ(x) -κ(x) -xκ (x) + Φ(y) -Φ(ŷ) + yΦ (y) σ 2 (y)/2 . ( 10
) Assumption 3 implies LV (x, y) ≤ max(σ 2 0 /2; 2K)V (x, y) + 2k for two positive constants K, k, checking condition (d). From Definition 1, inf x∈[0,+∞) V (x, y) = V 2 (y) + inf x∈[0,+∞) V 1 (x) = V 2 (y)
which implies that inf D\Dn V (x, y) ≥ inf{V 2 (y) : max{y, 1 -y} ≤ 1/n}, the latter going to infinity with n, recall (4). Similarly, inf y∈(0,1) V (x, y) goes to infinity as x goes to 0 or +∞. Condition (e) is then satisfied, which allows to apply Theorem 2.

Remark 3. Notice that ỹ < ŷ and thus x > x. Following Assumption 1, (10) at (x, ỹ) provides LV (x, ỹ) > 0. It is straightforward that (2) has no equilibrium point in D, nor on its boundary {0} × (0, 1) ∪ R * + × {0, 1}. If the point (0, 0) cancels (2), we highlight that LV (0, 0) < 0 and by continuity, it holds on a small region [0, ε) 2 . Recalling (4) implies that (x t , y t ) will diverge from (0, 0) almost surely if (x 0 , y 0 ) ∈ D.

A solution to (2) can be pictured as a trajectory continuously jumping from an orbit of (1) to another. Along this idea, V provides an estimate on trajectories, and can be related via Theorem 1 to the period T . Theorem 4. Let (x 0 , y 0 ) ∈ D, V 0 := V (x 0 , y 0 ), and (x t , y t ) t≥0 be a regular solution to [START_REF] Bahar | Stochastic delay LotkaVolterra model[END_REF]. We first introduce a constant 0 ≤ ρ ≤ V 0 , the set D(V 0 , ρ) := {(x, y) ∈ D : |V (x, y) -V 0 | ≤ ρ} ⊂ D and the stopping time τ ρ := inf{t > 0 : (x t , y t ) / ∈ D(V 0 , ρ)}. We then introduce two finite constants

R(V 0 , ρ) := max D(V 0 ,ρ) σ 2 (y) κ(x) -κ(x) -xκ (x) + yΦ (y) + Φ(y) -Φ(ŷ)
and

I(V 0 , ρ) := max D(V 0 ,ρ) σ 2 (y) (κ(x) -κ(x) + Φ(y) -Φ(ŷ)) 2 .
Then for all µ > 0

P [τ ρ > Θ(ρ, µ)] ≥ 1 - I(V 0 , ρ) µ 2 for Θ(ρ, µ) := 2 µ 2 + µ µ 2 + 2ρR(V 0 , ρ) + ρR(V 0 , ρ) (R(V 0 , ρ)σ) 2 . ( 11
)
Proof. Fix µ > 0. Now we define the F τρ -measurable set A µ = ω ∈ Ω : sup 0<t≤τρ M t (ω) ≤ µ where (M t ) t≥0 is a martingale defined by M t = 0 for t = 0 and for t > 0 by

M t = 1 √ t t 0 σ(y s ) (κ(x) -κ(x s ) + Φ(y s ) -Φ(ŷ)) dW (s) .
The process M is not right-continuous at t = 0 but still verifies E M 2 t ≤ I(V 0 , ρ) for all 0 < t ≤ τ ρ . The property holds by replacing M t by its càdlàg representation. Doob's martingale inequality can then be applied: P [A µ ] ≥ 1 -I(V 0 , ρ)/µ 2 . At last, using Itô's formula, we have from [START_REF] Goodwin | A growth cycle, Socialism, capitalism and economic growth[END_REF]:

|V (x t , y t ) -V 0 | ≤ 1 2 t 0 σ 2 (y s ) κ(x) -κ(x s ) -x s κ (x s ) + y s Φ (y s ) + Φ(y s ) -Φ(ŷ) ds + t 0 σ(y s ) (κ(x) -κ(x s ) + Φ(y s ) -Φ(ŷ)) dW s so that on {(t, ω) ∈ R + × Ω : (t, ω) ∈ A µ × [0, τ ρ (ω)]} |V (x t , y t ) -V 0 | ≤ 1 2 R(V 0 , ρ)t + µ √ t =: S(t) almost surely. Also, |e(t, ω)| ≤ ρ on that set. Put in another way, τ ρ > S -1 (ρ) =: Θ(ρ) on A µ .
According to Bayes rule,

P [τ ρ > Θ(ρ)] ≥ P τ ρ > Θ(ρ) A µ P [A µ ] ≥ P [A µ ] ≥ 1 - I(V 0 , ρ) µ 2
We now introduce the main result of the paper. We provide the following tailor-made definition for the cycling behavior of (2). Definition 2. Let (x * , y * ) ∈ E ⊆ R 2 and (x 0 , y 0 ) ∈ E\{(x * , y * )}. Let (x t , y t ) be a stochastic process starting at (x 0 , y 0 ) staying in E almost surely. We then introduce (ρ t ) t≥0 the angle between [x t -x * , y t -y * ] and [x 0 -x * , y 0 -y * ] . Let S := inf{t > 0 : |ρ t | ≥ 2π or (x t , y t ) = (x * , y * )} be a stopping time (a stochastic period). Then, the process (x t , y t ) is said to orbit stochastically around (x * , y * ) in E if S < +∞ almost surely.

Theorem 5. Let (x 0 , y 0 ) ∈ D\{(x, ỹ)} and (x t , y t ) a solution to (2) starting at (x 0 , y 0 ). Then (x t , y t ) orbits stochastically around (x, ỹ) in D. More precisely the system (2) produces clockwise orbits inside D. The angle ρ t is only defined if (x t , y t ) = (x, ỹ). This can be ensured by either proving that (x t , y t ) = (x, ỹ) for all t ≥ 0 almost surely, or by defining S as in Definition 2. See also Remark 4 The proof of Theorem 5 is removed to Section 4.

Proof of Theorem 5 4.1 Preliminary definitions and results

Recall that the probability space is given by (Ω, F, P) with the filtration generated only by W is completed with null sets. Our proof, although unwieldy, allows us to describe precisely the possible trajectories of solutions of (2). it consists in defining subregions (R i ) i of the domain D, see Definition 4 below illustrated by Fig. 1, and prove that the process exits from them in finite time by the appropriate frontier. According to Theorem 3 any regular solution of (2) is a Markov process. We then repeatedly change the initial condition of the system, as equivalent of a time translation and use Definition 5 herefater. We obtain recurrence properties via Theorem 3.9 in [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF]. Since it is repeatedly used hereafter, we provide here a version suited to our context. Theorem 6. Let (x t , y t ) t≥0 be a regular solution of (2) in D, starting at (x 0 , y 0 ) ∈ U , for some U ⊂ D. Let ϕ(t, x, y) ∈ C 1,2,2 (R + × U ) verifying ϕ(t, x, y) ≥ 0 for all (t, x, y) ∈ U and Lϕ(s, x, y) ≤ -ϕ(s) where ϕ(s) ≥ 0 and lim t t 0 ϕ(s)ds = +∞. Then (x t , y t ) leaves the region U in finite time almost surely. Definition 3. Let f be defined by f : x ∈ R + → f (x) := Φ -1 (αγ + κ(x)) as a concave decreasing function. For a solution (x t , y t ) to (2), we define θ t := y t /x t the finite variation process verifying dθ t = θ t (κ(x) -γ + α -Φ(y)) dt = θ t (Φ(f (x)) -Φ(y)) dt. Additionally, let θ := ỹ/x. Definition 4. We define eight sets (R i ) i=1,...,8 such that 8 i=1 R i = (ỹ, x) and

8 i=1 R i = D, by                          R 1 := {(x, y) ∈ D : y ≥ ỹ and θ t ≤ θ} R 2 := {(x, y) ∈ D : f (x) ≤ y ≤ ỹ} R 3 := {(x, y) ∈ D : y ≤ f (x) and x ≥ x} R 4 := {(x, y) ∈ D : x ≤ x and θ ≤ θ} R 5 := {(x, y) ∈ D : y ≤ ỹ and θ t ≥ θ} R 6 := {(x, y) ∈ D : ỹ ≤ y ≤ f (x)} R 7 := {(x, y) ∈ D : y ≥ f (x) and x t ≤ x} R 8 := {(x, y) ∈ D : y ≥ f (x) and x t ≤ x}.
Definition 5. Let (x t , y t ) be a solution to (2) starting at (x 0 , y 0 ) = (x, y) ∈ D. For any i ∈ {1, . . . , 8}, we define the stopping times τ i (x, y) := inf{t ≥ 0 : (x t , y t ) ∈ R i }. Remark 4. It seems rather clear that the point (x, ỹ) is not reached in finite time with a positive probability. In the following, the fact that LV (x, y) > ε for some small ε > 0 in a neighborhood of (x, ỹ) implies that (x, ỹ) is not a limit to almost every path of a solution to (2), recall Remark 3.

To ease the reading of the proof of Theorem 5 which follows from the following Propositions 1 to 10, we divide it in four quadrants around (x, ỹ). We first prove that the process cycles, even in infinite time, for some particular starting points.

Proposition 1. If (x 0 , y 0 ) ∈ R 1 , then P [τ 8 (x 0 , y 0 ) ≤ τ 7 (x 0 , y 0 )] = 0. If (x 0 , y 0 ) ∈ R 5 , then P [τ 4 (x 0 , y 0 ) ≤ τ 3 (x 0 , y 0 )] = 0.
Proof. This is a direct consequence of the absence of Brownian motion in θ. Take (x 0 , y 0 ) ∈ R 1 .

Then on [0, τ 3 (x 0 , y 0 )], the process θ is non increasing almost surely, meaning that R 8 cannot be reached without first crossing region R 7 . The other side is identical.

Remark 5. Proposition 1 holds even if τ i = +∞, for any i involved. It also implies that if (x 0 , y 0 ) ∈ R 1 ∪ R 5 , then τ i (x 0 , y 0 ) ≤ τ j (x 0 , y 0 ) almost surely for j ∈ {mod(i + 1, 8)}.

Eastern quadrant

We ought to prove that for (x 0 , y 0 ) ∈ R 1 , the process reaches R 3 in finite time almost surely.

Proposition 2. If (x 0 , y 0 ) ∈ R 1 then P [τ 2 (x 0 , y 0 ) < +∞] = 1. Proof. Let ϕ : y ∈ [0, 1] → √ y. Then ϕ(y) ≥ ϕ(ỹ) > 0 for any y such that (x, y) ∈ R 1 .
Moreover,

Lϕ(y) = ϕ(y) 2 κ(x) -γ + 3 4 σ 2 (y) ≤ - σ 2 (ỹ)ϕ(y) 8 ≤ - σ 2 (ỹ)h(ỹ) 8 < 0 .
Theorem 6 stipulates that (x t , y t ) leaves R 1 in finite time almost surely which is only possible via R 2 according to Proposition 1. Reaching the boundary is prevented by Theorem 3.

Proposition 3. If (x 0 , y 0 ) ∈ R 2 ∪ R 3 then P [τ 1 (x 0 , y 0 ) ∧ τ 4 (x 0 , y 0 ) < +∞] = 1.
Proof. We follow the proof of Proposition 2 with ϕ :

x ∈ R + → √ x. Proposition 4. If (x 0 , y 0 ) ∈ R 1 ∪ R 2 then P [τ 3 (x 0 , y 0 ) < +∞] = 1.
Proof.

Step 1. Let (υ n ) n≥0 be a sequence of stopping times defined by υ 0 = 0 and

υ n := inf{t ≥ υ n-1 : y t = ỹ or (x t , y t ) ∈ R 3 }, n ≥ 1.
By construction if (x υn , y υn ) ∈ R 3 for some n ≥ 1, then υ k = υ n for all k > n. Following Propositions 1, 2 and 3, υ n < +∞ for all n ≥ 1 almost surely, and {τ 3 (x, y) = +∞} ⊂ n≥1 {y υn = ỹ}. We prove in step 2 that this implies lim t→∞ θ t (ω) = 0, for P -a.e. ω ∈ {τ 3 (x, y) = +∞} .

Providing that (12) holds we immediately get P [τ 3 (x 0 , y 0 ) = +∞] ≤ P [lim n x υn = +∞] = 0.

Step 2. If ω ∈ {τ 3 (x, y) = +∞}, then for all n ≥ 1, y υn = ỹ and according to Proposition 3, (x t , y t ) does not converge to the set R 2 ∩ R 3 . Since θ t is a positive decreasing process for (x t , y t ) ∈ R 1 ∪ R 2 , Doob's martingale convergence theorem implies that θ t converges pathwise in L ∞ ([0, θ)). Assume now that θ t does not converge to 0 with t on E ⊂ {τ 3 (x 0 , y 0 ) = +∞}. Then for any ε > 0, and for almost every ω ∈ E

lim t t 0 1 {κ(xs(ω))-γ+α-Φ(ys(ω))<-ε} ds = C ε (ω) < +∞ . ( 13
)
If the integral (13) explodes to +∞ for some ε > 0 on some non null subset F ⊂ E, then for almost every ω ∈ F ,

L log θ t (ω) = (κ(x t (ω)) -γ + α -Φ(y t (ω))) < -ε1 {κ(xt(ω))-γ+α-Φ(yt(ω))<-ε}
and lim t↑∞ log θ t (ω) = -∞ for almost every ω ∈ F , implying that θ t converges to 0 on F , a contradiction with F ⊂ E, so that (13) holds on E. We then consider the random time t ε,n , being the first time such that tε,n 0

1 {κ(xs)-γ+α-Φ(ys)<-ε} ds ≥ C ε - 1 n , (14)
and k n the smallest index such that υ kn ≥ t ,n . Note that t ε,n is not a F-stopping time and k n is not F-adapted since they depend on C ε which is F ∞ -measurable. [START_REF] Hsu | A remark on the period of the periodic solution in the Lotka-Volterra system[END_REF] implies that there exists a random time s n ∈ (υ kn , υ kn + 1/n) such that -ε < κ(x sn ) -γ + α -Φ(y sn ) < 0, otherwise we would have a contradiction of ( 13) on a subset of E:

υ kn +1/n 0 1 {κ(xs)-γ+α-Φ(ys)<-ε} ds ≥ C ε .
This implies that lim n (s n -υ kn )(ω) = 0 for almost every ω ∈ E, and (y t ) t≥0 being a continuous process lim n y sn (ω) = ỹ, for P -a.e. ω ∈ D ⊂ {τ 3 (x 0 , y 0 ) = +∞} . This is impossible for ε > 0 small enough since θ t is strictly decreasing and thus E is a null set. ( 12) holds.

Southern quadrant

We show that starting from R 2 ∩ R 3 , (x t , y t ) reaches R 5 in finite time almost surely.

Proposition 5. If (x 0 , y 0 ) ∈ R 2 ∩ R 3 then P [τ 4 (x 0 , y 0 ) < +∞] = 1.

Proof.

Step 1. We consider ϕ t := ϕ(x t , y t ) with ϕ : (x, y) ∈ D\{(x, y)} → (y t -ỹ)/(x tx), and aim to prove that the process F t := F (ϕ(x t , y t )) with

F : ϕ ∈ (-π/2, π/2) → tan tan -1 (ϕ) + tan -1 (c) is a supermartingale on R 1 ∪ R 2 ∪ R 3 , for c ∈ (0, θ-1 ). Notice that it is bounded in R 1 ∪ R 2 ∪ R 3 . Applying Itô to ϕ first gives dϕ t = dy t x t - x - y t - ỹ (x t -x) 2 dx t + σ 2 (y t ) (x t -x) 2 y t - ỹ x t - x x 2 t -x t y t dt .
Then, noticing that F t = (ϕ t + c)(1 -ϕ t c), we obtain

dF t = 1 + c 2 (1 -ϕ t c) 2 dϕ t + c 1 -ϕ t c d ϕ t . It is clear that -(y -ỹ)(Φ(y) -α + σ 2 (y)) ≤ 0 for all y ∈ [0, 1). Now notice that for (x, y) ∈ R 1 , we have (x -x)(κ(x) -γ + σ 2 (y)) < 0 so that (x -x) 4 σ 2 (y) (1 -ϕc) 2 1 + c 2 LF ≤ (y -ỹ)(x -x)x 2 -xy(x -x) 2 + 1 ỹ/x -ϕ (ỹx -yx) 2 = (x -x) x 2 (y -ỹ) -xy(x -x) + x(ỹx -yx) = (x -x) 2 xx y x - ỹ x < 0 . Now on R 2 ∪ R 3 , x < x implies that (κ(x) -γ) < 0, so that (x -x) 4 σ 2 (y) (1 -ϕc) 2 1 + c 2 LF ≤ (y -ỹ)(x -x)x 2 -yx(x -x) 2 + 1 ỹ/x -ϕ (ỹx -yx) 2 = (x -x) x 2 (y -ỹ) -yx(x -x) + x(ỹx -yx) = (x -x) 2 x (y -ỹ) < 0 .
Denoting τ 1,4 := τ 1 (x 0 , y 0 ) ∧ τ 4 (x 0 , y 0 ), we conclude that F t∧τ 1,4 is a supermartingale for t ≥ 0. Using optional sampling theorem, assisted by Proposition 3, τ 1,4 < +∞ almost surely and

F 0 ≥ E F τ 1,4 = 1 c P [τ 4 (x, y) < τ 1 (x, y)] + cP [τ 1 (x 0 , y 0 ) < τ 4 (x 0 , y 0 )] Since M := max{F (ϕ(x, y)) : (x, y) ∈ R 2 ∩ R 3 } < c then P [τ 4 (x 0 , y 0 ) < τ 1 (x 0 , y 0 )] ≥ c(c -M ) c 2 + 1 > 0 ∀(x 0 , y 0 ) ∈ R 2 ∩ R 3 .
Step 2. According to Proposition 3, τ 1,4 < +∞ almost surely for any (x 0 , y 0 ) ∈ R 2 ∩ R 3 , and according to Proposition 4, τ 3 (x 0 , y 0 ) < +∞ P-a.s. for all (x 0 , y 0 ) ∈ R 1 . Taking (x 0 , y 0 ) ∈ R 2 ∩R 3 , we define the sequence (τ n 1,4 , τ n 3 ) n≥0 with τ 0 3 = 0 and

τ n 1,4 := inf{t ≥ τ n 3 : (x t , y t ) ∈ R 1 ∪ R 4 } τ n+1 3 := inf{t ≥ τ n 1,4 : (x t , y t ) ∈ (R 2 ∩ R 3 ) ∪ R 4 }
, for all n ≥ 1 .

We then have {τ 4 (x 0 , y 0

) = +∞} ⊂ ∩ n≥1 {x τ n 1,4 > x} for any (x 0 , y 0 ) ∈ R 2 ∩ R 3 . The sequence ({x τ n 1,4 > x}) n≥1
is decreasing in the sense of inclusion, so that

P [τ 4 (x, y) = +∞] = lim n P x τ n 1,4 > x . ( 15
)
Using Baye's rule,

P x τ n 1,4 > x ≤ n k=1 P x τ k 1,4 > x|x τ k-1 1,4 > x ≤ n k=1 P x τ k 1,4 > x|x τ k 3 > x .
Using step 1 of the present proof and the Markov property of (x t , y t ),

P x τ n 1,4 > x ≤ n k=1 P τ 1 (x τ k 3 , y τ k 3 ) < τ 4 (x τ k 3 , y τ k 3 ) ≤ n k=1 1 - c(c -M ) c 2 + 1 .
Plugging this inequality into (15) concludes the proof.

Remark 6. Notice that by choosing c properly in the above proof, it is possible to be arbitrarily close to R 5 in finite time. The device is used later in Proposition 9.

Proposition 6. If (x 0 , y 0 ) ∈ R 3 ∩ R 4 then P [τ 5 (x 0 , y 0 ) < +∞] = 1.
Proof.

Step 1. We claim that τ 2,5 := τ 2 (x 0 , y 0 ) ∧ τ 5 (x 0 , y 0 ) < +∞ almost surely. Consider the function ϕ : (x, y) ∈ D → √ x t∧υ 0 . The process ϕ t := ϕ(x t , y t ) is a positive supermartingale on

R 2 ∪ R 3 ∪ R 4 : Lϕ(x, y) = ϕ(x, y) 2 (Φ(y) -α + σ 2 (y) 2 ≤ - σ 2 (y)ϕ(x, y) 4 . ( 16
)
According to Doob's martingale convergence theorem, ϕ t converges point-wise with t. Let ε > 0 and define R ε := 4 i=2 R i ∩ {x ≥ ε}. Then ϕ t ≥ √ ε on R ε , and similarly to Proposition 3, we use Theorem 6 to assert that (x t , y t ) leaves R ε in finite time almost surely. This being true for any ε > 0, lim t ϕ t (ω) = 0 for almost every ω ∈ {τ 2,5 (ω) = +∞}. In R 5 , this is only possible if lim t y t (ω) = 0 also, implying that lim t (x t (ω), y t (ω)) = (0, 0) on this set. This being improbable, τ 2,5 < +∞ almost surely.

Step 2. By denoting τ 0 4 = 0, we then define the sequence (τ n 2,5 , τ n 4 ) n≥0 by

τ n 4 := inf{t ≥ τ n-1 2,5
:

x t = x or (x t , y t ) ∈ R 5 } τ n 2,5 := inf{t ≥ τ n 4 : (x t , y t ) ∈ R 2 ∪ R 5 }
, for all n ≥ 1 .

If (x τ 0 2,5 , y τ 0 2,5 ) ∈ R 2 , then, according to Proposition 5, the process reaches back R 4 in finite time. Using step 1, we have that P [τ n 4 < +∞] = P τ n 2,5 < +∞ = 1. By construction and Proposition 5, for n ≥ 1

{(x τ n 2,5 , y τ n 2,5 ) ∈ R 2 } ⊂ {x τ n 4 = x} = {(x τ n-1 2,5
, y τ n-1

2,5 ) ∈ R 2 } = {x τ n-1 2,5 > x}. (17)
Therefore, {τ 5 (x 0 , y 0 ) = +∞} = n≥0 {(x τ n 2,5 , y τ n 2,5 ) ∈ R 2 } and the sequence of sets

{(x τ n 2,5 , y τ n 2,5 ) ∈ R 2 } n≥0
is decreasing in the sense of inclusion. Altogether we get

P [τ 5 (x 0 , y 0 ) = +∞] = lim n P x τ n 2,5 > x . (18)
Now using Bayes formula and ( 17), we finally obtain for every n ≥ 1

P x τ n 2,5 > x ≤ n k=1 P x τ k 2,5 > x|x τ k-1 2,5 > x = n k=1 P x τ k 2,5 > x|x τ k 4 = x (19)
Putting ( 18) and ( 19) together, P [τ 5 (x 0 , y 0 ) = +∞] > 0 implies that

lim n P x τ n 2,5 > x|x τ n 4 = x = 1 . (20)
Step 3. Let ϕ : (t, x) ∈ R 2 + → √ x exp( 1 8 σ 2 (ỹ)t). According to [START_REF] Kiernan | Stochastic stability in macro models[END_REF] the process ϕ t := ϕ(t, x t ) is a supermartingale on [τ n 4 , τ n 2,5 ]. Fixing t > 0 and applying optional sampling theorem, we obtain

E ϕ(t ∧ τ n 2,5 , x t∧τ n 2,5 ) -ϕ(t ∧ τ n 4 , x t∧τ n 4 )|x t∧τ n 4 = x ≤ 0 . Since max(τ n 4 , τ n 2,5
) < +∞ almost surely, we apply Fatou's lemma and obtain

E exp 1 8 σ 2 (ỹ)[τ n 2,5 -τ n 4 ] x τ n 2,5 1
x τ n 2,5 <x + 1 x τ n 2,5 >x x τ n 4 = x ≤ √ x . (21)
Since x τ n 2,5 1

x τ n 2,5 <x ≥ 0 and x τ n 2,5 1 x τ n 2,5 >x ≥ √ x1 x τ n 2,5
>x for all n ≥ 1, ( 21) implies

E exp 1 8 σ 2 (ỹ)[τ n 2,5 -τ n 4 ] 1 x τ n 2,5 >x x τ n 4 = x ≤ 1 , leading to

Northern quadrant

Finally we prove that if (x 0 , y 0 ) ∈ R 7 , then the process reaches R 1 in finite time almost surely. One can notice that proofs are very similar to those of Subsections 4.2 and 4.3.

Proposition 8. If (x 0 , y 0 ) ∈ R 6 ∪ R 7 then P [τ 5 ∧ τ 8 (x 0 , y 0 ) < +∞] = 1.
Proof. Define the sequence of regions

{B n } n∈N through B n = R 6 ∪R 7 ∩{y < 1-k/n}∩{x > k/n}
where k > 0 is sufficiently small to have (x 0 , y 0 ) ∈ B 1 . Applying Itô to ϕ : (x, y) ∈ D → √

xx t we find that for all (x, y)

∈ B n Lϕ(x, y) = - 1 2ϕ(x, y) x Φ(y) -α + σ 2 (y) + 1 4 x 2 x -x σ 2 (y) ≤ - x 2 σ 2 (y) 8(x -x) 3/2 ≤ - k 2 σ 2 (1 -k/n) 8 √ n(nx -k) 3/2 < 0
while Lϕ(x, y) ≤ 0 in R 6 ∪R 7 . Doob's supermartingale convergence theorem implies the existence of the pointwise limit ϕ ∞ := lim t ϕ(x t∧τ 5,8 , y t∧τ 5,8 ) almost surely, where we use the notation τ 5,8 := τ 5 (x 0 , y 0 ) ∧ τ 8 (x 0 , y 0 ). In addition, Theorem 6 guarantees that every set B n is exited in finite time almost surely. Consequently if ω ∈ {τ 5,8 = +∞}, we have that either lim t x t (ω) = 0 or lim t y t (ω) = 1, a contradiction in either way.

Proposition 9. If (x 0 , y 0 ) ∈ R 6 ∩ R 7 then P [τ 8 (x 0 , y 0 ) < +∞] = 1.
Proof. The proof is identical to the one of Proposition 5, with small modifications. Here

ϕ t := ϕ(x t , y t ) with ϕ : (x, y) ∈ D\{(x, ỹ)} → (y -ỹ)/(x -x) and F : (x, y) ∈ D\{(x, ỹ)} → tan tan -1 (ϕ(x, y)) + tan -1 (c) . The process F t := F (x t , y t ) is a supermartingale on R 5 ∪ R 6 ∪ R 7 if
(x -x) 2 (1 -R t c) 2 1 + c 2 LF t ≤ y(x -x)[κ(x) -γ] -x(y -ỹ)[Φ(y) -α] + σ 2 (y) [y(x -x) -x(y -ỹ) + x (y -x(1/c -x + x))] ≤ y(x -x)[κ(x) -γ] -x(y -ỹ)[Φ(y) -α] -xxσ 2 (y)(1/c -θ) ≤ M -m(1/c -θ) ≤ 0 .
We then reproduce step 2 of the proof of Proposition 5, using Propositions 7 and 8 instead of Propositions 2 and 3.

Proposition 10. If (x 0 , y 0 ) ∈ R 7 ∩ R 8 then P [τ 1 (x 0 , y 0 ) < +∞] = 1.
Proof. We follow Proposition 6 with the minor following modifications.

1 We consider τ 1,6 := τ 1 (x 0 , y 0 )∧τ 6 (x 0 , y 0 ) the exit time of R 7 ∪R 8 . The process ϕ t := ϕ(x t , y t ) with ϕ : (x, y) ∈ D → x -2 t verifies

Lh t = -2h t Φ(y t ) -α + 3 2 σ 2 (y t ) < -εh t < -εh( θ) < 0
for some ε > 0. Indeed Φ(y) -α + σ 2 (y) ≥ 0 and is null only if y = ỹ, whereas σ 2 (y) = 0 only if y = 1. Applying Theorem 6 to R 7 ∪ R 8 , τ 1,6 < +∞ almost surely.

Step 2. If (x τ 1,6 , y τ 1,6 ) ∈ R 6 , then the process reaches R 8 in finite time almost surely according to Proposition 9. We define the sequence (τ n 1,6 , τ n 8 ) n≥0 with τ 0 8 := 0 and

τ n 1,6 := inf{t ≥ τ n 8 : (x t , y t ) ∈ R 6 ∪ R 1 } τ n+1 8 := inf{t > τ n 1,6 : (x t , y t ) ∈ (R 7 ∩ R 8 ) ∪ R 1 }
, for all n ≥ 0 .

Proceeding as in step 2 Proposition 6, we obtain that P [τ 1 (x 0 , y 0 ) = +∞] > 0 implies that

lim n P x τ n 1,6 < x|x τ n 8 = x = 1 . ( 24
)
Step 3. Define m := inf{2(Φ(y)-α)+3σ 2 (y) : y ∈ [ỹ, 1)}, which is strictly positive according to step 1. Consider the new process ϕ t := ϕ(x t , y t ) with ϕ : (x, y) ∈ D → exp(-mt)x 2 t . It is a positive submartingale on [0, τ 0 1,6 ], and similarly to step 3 of Proposition 6, we can obtain

x2 ≤ E x 2 τ n 1,6 e -m(τ n 1,6 -τ n 8 ) |x τ n 8 = x ≤ x2 E e -m(τ n 1,6 -τ n 8 ) 1 x τ n 1,6 <x |x τ n 8 = x + θ-2 E e -m(τ n 1,6 -τ n 8 ) 1 x τ n 1,6 ≥x |x τ n 8 = x .
Assuming [START_REF] Nguyen Huu | Dynamics of a stochastic Lotka-Volterra model perturbed by white noise[END_REF], we have

0 ≤ E (1 -e -m(τ n 1,6 -τ n 8 ) )1 x τ n 1,6 <x |x τ n 8 = x ≤ (1/ λ -1)(1 -P x τ n 1,6 < x|x τ n 8 = x ) n -→ 0 .
We then proceed exactly as in step 3 of Proposition 6 to finish the proof.

Example

In this section we assume that investment follows Say's law and Philips curve is provided by [START_REF] Grasselli | An analysis of the Keen model for credit expansion, asset price bubbles and financial fragility[END_REF]15].

Assumption 4. We let κ : x ∈ R + → (1 -x)/ν and Φ : y ∈ [0, 1) → Φ(y) := φ 1 (1-y) 2 + φ 0 .

Assumption 1 holds under Assumption 4. The unique non-hyperbolic equlibrium point in

D = R * + × (0, 1) is given by (x, ŷ) = 1 -νγ, 1 -φ 1 /(α -φ 0 ) . Functions of Definition 1 are given by V (x, y) = V 1 (x) + V 2 (y) with V 1 (x) = 1 ν x -x log x x + 1 , V 2 (y) = φ 1 log 1-ŷ 1-y + ŷ 1-ŷ log ŷ y + 1 y-y 2 -1 ŷ-ŷ 2 . ( 25
)
Although period T given by Theorem 1 is not explicit here, numerical computations allow to approximate it with a linear function of V 0 , see first part of Fig. 2. Following Remark 1, T does not converge to 0 with solutions of (1) concentrating to (x, ŷ). A local phase portrait with values of T is provided in second part of Fig. If we implicitly assume that the perturbation of the average growth rate α of the productivity is due to the flow of workers coming in and out of the fraction y t employed at time t, Assumption 5 conveniently expresses that this perturbation decreases with the employment rate since higher employment implies lower perturbation on the constant average rate α t . Other models can of course be considered.

Assumption 5 together with Assumption 4, and comparing with [START_REF] Reichenbach | Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model[END_REF], satisfy Assumption 3. Indeed for all y ∈ (0, 1),

σ 2 (y)Φ (y) = 2σ 2 0 φ 1 (1 -y) ≤ 2σ 2 0 V 2 (y) -φ 1 1 1 -ŷ ŷ log(ŷ) - 1 ŷ + log(1 -ŷ) (26)
and along with the sub-linearity of the log function,

-κ(x) -xκ (x) = 2x -1 ν ≤ 2 1 - x (V 1 (x) + x -x log(x)) . (27)
In line with Assumptions 1 and 3 the vertical asymptote at y = 1 implies that σ 2 (y)Φ(y) ≤ K 0 V 2 (y) + k 0 for some K 0 , k 0 ∈ R 2 + . Under Assumption 3 and following ( 26) and ( 27), K 0 = 0 and k 0 = σ 2 0 (φ 1 + φ 0 ). Assumption 5 also implies that (1 -ỹ) 2 is the root of a quadratic polynomial σ 2 0 (1 -ỹ) 4 -(α + φ 0 )(1 -ỹ) 2 + φ 1 = 0 . The latter shall have a unique root in (0, 1) to satisfy Assumption 2.

The following example of condition is sufficient. Assumption 6. We assume φ 1 ≤ (α + φ 0 )/2 and σ 0 ≤ max{(α + φ 0 )/(2 √ φ 1 ), (α + φ 0 -φ 1 )}.

We are now able to claim the existence of 2 .

K, k ∈ R 2 + such that |R(V 0 , ρ)| ≤ K(V 0 + ρ) + k, where R is defined in Proposition 4. A direct application provides R(V 0 , ρ) := max D(V 0 ,ρ) σ 2 0 (1 -y) 2 2x - x ν + 2φ 1 y (1 -y) 3 + φ 1 (1 -y) 2 - φ 1 (1 -ŷ)
Using [START_REF] Thygesen | A survey of Lyapunov techniques for stochastic differential equations[END_REF], this estimate becomes |R(V 0 , ρ)| ≤ K(V 0 +ρ)+k where K := (2σ 2 0 )/(1-x) and k is an explicitly calculable constant. Following the same procedure with [START_REF] Simmonds | A first look at perturbation theory[END_REF], I(V 0 , ρ) ≤ K(V 0 + ρ) 2 + k with the same K and k = k. Now choosing µ = (ρ -(K(V 0 + ρ) + k)θ/2)/ √ θ for some θ ≥ 0, so that Θ(ρ) = θ, Proposition 4 provides

P [τ ρ > θ] ≥ 1 - (K(V 0 + ρ) 2 + k )θ 1 2 (K(V 0 + ρ) + k)θ -ρ 2 .
Theorem 5 is a straightly observable phenomenon with simulations, see Fig. 3. Under the assumptions of this section, the system has been simulated using XPPAUT with a fourth order Runge-Kutta scheme for the deterministic part, and an Euler scheme for the Brownian part. Fig. 3 illustrates the effect of the volatility level σ 0 on trajectories of the system, as for the economic quantity P t := a t y t N t . Apart from specific subregions of D as R 1 or R 5 where Corollary 3.2 in [START_REF] Khasminskii | Stochastic stability of differential equations[END_REF] can provide an estimate for the expectation of the exit time, a bound for the expected period E [S] seems out of reach. Numerical simulations have nevertheless always provide reasonnable finite periods of stochastic orbits of (2). We thus expect that E [S] is finite for a wide range of values of (x 0 , y 0 ) ∈ D. Let us start with (x 0 , y 0 ) ∈ R 1 ∩R 8 and reformulate S of Definition 2 as the time the process crosses the line y = θx for the second time. This is equivalent to take (x 0 , y 0 ) ∈ R 4 ∩ R 5 . Resorting to numerical methods, we have simulated the system 2000 times for 100 different starting points in R 1 ∩ R 8 and recorded the position at the time when this line is crossed the second time, that is the positions after a full loop. Fig. 4 contains such examination for an array of values of σ 0 . The expected time E [S] to complete a full-loop is also illustrated. As observed, there seems to be a stable attractive fixed point to y 0 → E [y S ] for sufficiently large values of σ 0 . If the starting point is picked too close to (x, ŷ), the expected crossing value after one loop is further away from it. On the other hand, if the one starts extremely far away from (x, ŷ), say with y 0 < 0.25, then the expected value after on loop is higher. This implies that after many loops, the expectation converges, and so does E [S] with the number of loops around (x, ỹ). Assuming that E [S] < +∞ for enough initial points, Theorem 6 can be used with V at points (0, 0) and (x, ỹ) to prove the following conjecture.

Concluding remarks

This contribution attempts to draw the attention of dynamical system analysis onto macroeconomic models. Before looking into complex models of finance and crises, e.g. [START_REF] Costa-Lima | Destabilizing a stable crisis: employment persistence and government intervention in macroeconomics[END_REF][START_REF] Grasselli | An analysis of the Keen model for credit expansion, asset price bubbles and financial fragility[END_REF]15], we focus here on a Brownian perturbation added into a non-linear version of the Lotka-Volterra system used in Economics, the Goodwin model. To begin with, we recall the usual results for the deterministic planar oscillator: we provide the constant Entropy function and describe the period of the closed orbits drawned by the system. We then provide sufficient conditions for the stochastically perturbed system to stay in the meaningful domain D which is a a bounded subset of R 2 + for the y-component. The entropy function is actually of great use for the last result, additionally to prior estimates on variations of the system.

We finally prove what seems a fundamental and staightforward property of the system, namely that a solution (x t , y t ) rotates with perturbations around a unique point (x, ỹ). The definition of stochastic orbits provided here conventienly suits the intuition of how the deterministic concept can be extended. However it has clearly not the ambition to be a definitive concept and further investigations might confirm its usefulness or its precarity. The proof exploits the concept of reccurent domains in an intensive manner.

We expect that economists seek interest in (2), as other perturbed macroeconomic systems (e.g. [START_REF] Kiernan | Stochastic stability in macro models[END_REF][START_REF] Neamtu | The study of some stochastic macroeconomic models[END_REF]), for the possibility to adjust the model to observed past data (e.g. [START_REF] Arató | A famous nonlinear stochastic equation (Lotka-Volterra model with diffusion)[END_REF] and [START_REF] Harvie | Testing Goodwin: growth cycles in ten OECD countries[END_REF][START_REF] Mohun | Goodwin cycles and the US economy[END_REF]) and find a possible synthetic explanation for perturbations of business cycles (see [START_REF] Evans | Productivity shocks and real business cycles[END_REF][START_REF] Hansen | Indivisible labor and the business cycle[END_REF]).

Figure 1 :

 1 Figure 1: Covering of D := R * + ) × (0, 1) by (R i ) i=1...8 . Since f (0) < 1 and lim y↑1 Φ(y) = +∞, the graph illustrates the general case.

  we chose c ∈ (0, ( θ + M/m) -1 ) where (m, M ) are two positive constants given by m := min [x,x]×[ŷ,ỹ] xxσ 2 (y) and M := max [x,x]×[ŷ,ỹ] y(xx) [κ(x) -γ] -x(yŷ) [Φ(y) -α] . The justification is the following. The domain S c := D\{ θ ≤ ϕ(x, y) ≤ 1/c} contains the area of interest R 5 ∪ R 6 ∪ R 7 . Using Proposition 5, we can prove that F t is a supermartingale on S c \[x, x] × [ỹ, ŷ]. On [x, x] × [ỹ, ŷ],

Figure 2 :

 2 Figure 2: Up: values of T as a function of V . Down: Contour lines with values of T in a subset of D. Parameters set at (α, γ, ν, φ 0 , φ 1 ) = (0.025, 0.055, 3, 0.040064, 0.000064). Equilibrium point at (x, ŷ) = (0.8350, 0.80).

Figure 3 :

 3 Figure 3: Left column : phase diagram (x, y) of subsample paths of trajectories for (2) with different values of volatility σ 0 , starting from the green star and stopping at the red start. Right column: evolution of output P t over time for the subsample path.

Conjecture 1 .

 1 Consider the function S : y ∈ (0, ỹ) → E [y S ] ∈ (0, ỹ) such that (x t , y t ) is a solution to (2) with (x 0 , y 0 ) = (y/ θ, y), and S is the finite stopping time defined by Theorem 5. Then S has at least one fixed point in (0, ỹ).

Figure 4 :

 4 Figure 4: Expected values of employment y after one full loop y T (left), and expected elapsed time T (right). Computation performed in MATLAB, with 2000 simulations for every value single one of the 100 initial values taken along the line y = θx.

  where F 1 and F 2 are the two restrictions of F on R + and R -respectively.

								Notice that if t ∈ [0, T 1 ],
	then u t := Φ(y t ) + α ∈ [0, Φ(ȳ + α)]. Thus, F 1 (u t ) is a strictly increasing function of t taking its
	values in [0, V (x, ŷ)]. Getting back to x = e z for G, we have for z ≥ z
	G(z) :=	x	x∧e z	κ(s) -γ s	ds +	e z x∧e z	κ(s) -γ s	ds

Acknowledgment

Both authors want to thank Matheus Grasselli for leading ideas and presentation suggestion. Remaining errors are authors responsibility.

Let's assume that P [τ 5 (x 0 , y 0 ) = +∞] > 0, so that (20) holds. According to [START_REF] Mohun | Goodwin cycles and the US economy[END_REF], we get

Markov inequality then leads to the following convergence for any ε > 0:

Now Bayes rules with [START_REF] Neamtu | The study of some stochastic macroeconomic models[END_REF] provides

Therefore on this set, the continuous mapping theorem asserts that (x, y) at consecutive stopping times converge in probability. By continuity, this implies lim n y τ n 4 (ω) = ỹ and lim n x τ n 2,5 (ω) = x for P -a.e. ω ∈ {τ 5 (x 0 , y 0 ) = +∞}. By the Markov property of (x t , y t ), lim t→∞ (x t , y t )(ω) = (x, ỹ) for almost every ω ∈ {τ 5 (x 0 , y 0 ) = +∞} . We conclude that P [τ 5 (x 0 , y 0 ) = +∞] = 0.

Western Quadrant

Theorem 6 then states that (x t , y t ) exits R ε in finite time almost surely. Since that θ is non-decreasing on this set, and recalling Theorem 2, it is only possible via R 7 and P [τ 7 (x 0 , y 0 ) < +∞] = 1. This holds for any ε > 0.

Step 2. Assume now that (x 0 , y 0 ) ∈ (R 5 ∪ R 6 )\R ε . According to step 1, {τ 7 (x 0 , y 0 ) = +∞} ⊂ {x t ≥ x, ∀t ≥ 0} and thus {τ 7 (x 0 , y 0 ) = +∞} ⊂ {θ t ≤ f (x)/x, ∀t ≥ 0}. Because θ t is non decreasing and according to Doob's martingale convergence theorem, θ t converges to θ 0 ∈ L ∞ ([ θ, f (ω)/ω]) on {τ 7 (x 0 , y 0 ) = +∞}. This implies that (x t , y t ) converges with t to R 6 ∩ R 7 on {τ 7 (x 0 , y 0 ) = +∞}. Since σ(y) > σ(f (0)), this convergence is improbable.


Pétergáspár

Zoltán Szabó
email: szabo.zoltan@sztaki.mta.hu

Oliver Sename
email: olivier.sename@gipsa-lab.grenoble-inp.fr

Luc Dugard
email: luc.dugard@gipsa-lab.grenoble-inp.fr

PRINCIPLES OF THE ACTIVE COLLISION MITIGATION SYSTEM FOR VEHICLES INVOLVED IN LIGHT IMPACT

Keywords: collision mitigation, yaw control, bicycle model, integrated control

The paper presents a collision mitigation method in which the aim is to stabilize the motion of the vehicle after a light impact has occurred. A yaw rate servo control based on the bicycle model is applied for control purposes. The collision is assumed to have been detected by a passive safety system, e.g., airbag sensors. The control system is activated by the monitoring signal. After the collision has occurred the motion of the vehicle and the operation of the control system are illustrated through CarSim simulation examples. Two control solutions are analyzed: in the first case active steering is used while in the second case steering is combined with differential braking. With the knowledge of certain vehicle parameters and the impact force and direction, post impact vehicle states (yaw rate, velocities, roll rate) can be analyzed. A sensitivity analysis is carried out to determine the validity range of the vehicle model in the presence of uncertainties and different kinds of impact forces. Several software packages are used for the design and analysis of the controlled system. The control design is performed by using the Matlab/Simulink software and the verification of the designed controller is performed by using the CarSim software.

INTRODUCTION

In the past decades vehicle safety has become one of the key issues in the automotive industry. Together with the effective organization of the traffic flow, the engineers' goal is to decrease the number and seriousness of vehicle accidents. In the developed countries with the grown motorization rate road capacity is nearly at it's maximum, hence the number of conflict situations among vehicles increased. Accordingly vehicles participating in traffic need to meet constantly higher safety standards. Safety features can be rated to two major categories: passive safety equipment (seat belts, safety cabin, airbags, etc.) mitigate the severity of accidents already occurred, while active safety tools are responsible for the prevention of dangerous traffic situations (ABS,ESP, ACC, etc.). Statistics show that in the developed countries the number of fatal accidents decreased in the past few decade despite the dramatic growth of the traffic. The main factor contributing in this result besides the advanced traffic engineering is the higher safety standards of the vehicles.

However to approach the goal of a road traffic without fatal accidents defined by government agencies, various type of hazards must be eliminated. Several statistics [START_REF] Langwieder | Retrospective Safety Analysis of Car Collisions Resulting in Serious Injuries[END_REF] shown that a vehicle involved in a light impact may experience a severe secondary crash, and 1/3 of all accidents with severe injuries are due to multiple events. Thus the task of controlling a vehicle in an accident is an important issue studied by relatively few researchers.

One of the first study on collision control was reported by Chan [START_REF] Chan | Feasibility Analysis of Steering Control as a Driver-Assistance Function in Collision Situations[END_REF], in which a post-impact steering control was investigated. The controller used information about the vehicle position in the lane and the heading angle, which can be realized with technically complex and expensive tools (marker sensors, computer vision, etc...). In addition, steering control alone without braking is probably not capable to deal with a severe collision with bigger impact angles.

Bosch, a German company, released a prototype of a Secondary Collision Mitigation (SCM) system. Here the active and passive safety systems of the car are combined. The airbag sensors detect the collision event by measuring the vehicle's acceleration (passive), and after the detection phase automatic braking is applied (active). Accordingly the kinetic energy and the distance traveled after the collision is reduced for the subject vehicle, thus the risk and/or the severity of a secondary accident is mitigated.

One may think that an electronic stability control (ESC) is capable of stabilizing a vehicle after an impact, but research conducted by Thor [START_REF] Thor | The Efficiency of Electronic Stability Control after Light Collisions[END_REF] has shown the limitations of such systems. Since the disturbance and the control source of such control system are both from the tire force variations, an exogenous impact induced vehicle motion is likely to be beyond the operation range of the ESC.

A four-DOF collision vehicle model was proposed by Jianbo Lu et al. [START_REF] Zhou | Collision model for vehicle motion prediction after light impacts[END_REF] for collision control. With the knowledge of certain vehicle parameters and the impact force and direction (or the bullet vehicle's velocity and heading angle), the post impact vehicle states (yaw rate, velocities, roll rate) were determined. The results were compared with the results calculated with the three-DOF momentum-conservationbased model and was validated by the calculations of CarSim. One of the major achievement of this four-DOF model was the capability to consider tire forces and roll rate variations. This model was the base for the collision detection and control developed later with a sliding surface method and model predictive supervisory control [START_REF] Zhou | Vehicle Stabilization in Response to Exogenous Impulsive Disturbances to the Vehicle Body[END_REF] [START_REF] Zhou | Vehicle stabilisation in response to exogenous impulsive disturbances to the vehicle body[END_REF].

The motivation of this is paper is to propose the design of a control method dealing with the stabilization of the vehicle after a light collision has occurred. The control goal is to track the desired yaw rate set by the driver with the use of active steering and differential braking. The design method is then validated with the use of simulation software Carsim, where control strategies using different actuators are compared to each other along with the uncontrolled collision case.

The paper is organized as follows: in Section 2 a vehicle model used for collision control is introduced. The description of the collision is discussed in Section 3. In Section 4 a yaw rate tracking linear quadratic design method is presented. The simulation results are discussed in Section 5. Finally, conclusions are presented in Section 6.

VEHICLE MODEL FOR COLLISION CONTROL

For modeling of the vehicle motion a commonly used simplification, the so called bicycle model is used. This single track bicycle model with 3-DOF as shown in Fig. 1 is unable to capture certain aspects of the vehicle motion, such as the roll and the pitch dynamics. In this work these motions are not included, since the main task of the controller is to maintain the vehicle's directional stability, thus the modeling of the yawing motion is sufficient. In later work the model can be modified to incorporate other aspects of the vehicle dynamics in order to avoid such treats as a turnover.

The left and right wheels at the back and at the front of the vehicle are represented by one wheel in the bicycle model. The major assumption of this model is that the wheels velocity vector is in the same direction as the wheels orientation angle, which is only true for small velocities. The state variables are the yaw rateψ & and the vehicle body side slip angle β. The vehicle velocity v is considered as a changing parameter in the model. The control input may vary depending on the control strategy.

In the state equation below two controller inputs are considered: the steering angle of the front tires δ(considered equal for both sides of the vehicle) and the moment applied on the vehicle vertical axis M f generated by differential braking of the rear wheels. The impulse of the collision is formalized as a disturbance acting on the input, where F y is the lateral force applied at the center of gravity of the vehicle, M z is the impact moment applied at the vehicle vertical axis.

l c l c x           +           +             + - - - - + - = 0 1 1 0 0 1 1 1 1 1 2 1 2 1 1 2 2 1 1 2 2 2 2 2 2 1 1 & (1)
where

[ ] T x β ψ& = , [ ] T f M u δ = , [ ] T z y M F d = , max min δ δ δ ≤ ≤ , max min f f M M M ≤ ≤ .
The steering angle δ of the front tires is constrained by the construction of the steering system, while the moment applied on the vehicle M f is constrained by the principle of limiting the friction. This maximum moment can be calculated with the following equation:

d F M M zr f f µ = - = min max
, where µ is the friction coefficient, d is the half length of the rear axle and F zr is the normal load of the rear axle given by: b a

a mg F zr + = (2)
The moment applied at the vehicle center is given by differential braking of the rear wheel (see Fig. 2). The relation between this moment and the braking force is given as:

b F F M R xr L xr f ) ( , , - = (3)
where a and b are the distances of the front and rear axle from the center of gravity (see Fig. 1), m is the vehicle mass, g is the gravity constant, F xr,L and F xr,R is the longitudinal wheel force.

Fig. 2 Rear axle geometry

The tire side force constants c 1 and c 2 are identified at constant speed. The vehicle velocity v is a dynamic parameter considered to be constant (v=104 km/h). Note, that this vehicle model is nonlinear with the quadratic parameter v and the tire side slip forces c 1 and c 2 , thus for a global solution for the above equation a gain scheduling or LPV controller must be designed.

Measuring vehicle state β is rather difficult and expensive, thus it is practical to estimate it. Vehicle side slip estimation has been introduced by several authors. The side slip angle can be estimated by using GPS measurements in conjunction with other vehicle sensors as shown by Bevly et. al. or can be estimated using observers [START_REF] Bevly | Integrating INS Sensors with GPS Measurements for Continuous Estimation of Vehicle Sideslip, Roll, and Tire Cornering Stiffness[END_REF][START_REF] Bevly | Estimation of vehicle sideslip, tire force and wheel cornering stiffness[END_REF]. In this paper for the estimation of the state variables an observer was designed.

COLLISION DESCRIPTION

There are several types of collisions which can occur on the roads with different severity. In this paper, some assumptions need to be made in order to design an effective post impact control strategy. First of all, after the collision impact the steering and brake system must function properly for at least a short period of time. This assumption probably means that the velocity differential between the vehicles is less than 40-55 km/h for rear-end collisions.

The resultant impulse vector of the collision is assumed to have a specific point of application on the target vehicle's periphery. This application point together with the force vector amplitude and angle can be calculated by the measures of the airbag sensor. Fig. 3 The scheme of the collision impact Typical time duration of a collision is around 0.1-0.2second. Hereinafter the time duration assumed to be 0.15s. The impact force is assumed to build up in a spike form. The collision scene is shown in Fig. 3, where the bullet vehicle is hitting the target vehicle which is controlled after the impact. Note that this rear end collision is with a certain angle, thus the impact force has components on both the x and y axis.

YAW RATE SERVO CONTROLLER DESIGN

Control objectives can differ depending on the collision event, the traffic situation and technical facilities. Stopping the vehicle after the impact as soon as possible may be a logical control objective. However, this can still result in dangerous situations like a suddenly stopping vehicle on motorway. Thus, in this paper the control goal is to maintain the target vehicle directional stability.

For this purpose a yaw rate tracking controller was designed with a servo structure (see Fig. 4). In this scheme the state feedback is combined with an output feedback with integrating property. The vehicles statesψ & and β were used for state feedback and the system output ψ& is tracked. The reference signal is the calculated yaw rate, which is set by the driver with the use of the steering wheel:

)) tan( ) (tan( ) cos( 2 1 δ δ β ψ - + = = b a v r r & (4)
where δ 1 and δ 2 are the front and rear wheels cornering angle.

Fig. 4 The scheme of the yaw rate tracking control

Then a new state is defined as follows:

∫ - = t r dt z 0 ) ( ψ ψ & & (5)
Let the augmented state vector be defined as follows:

[ ] T z x x = ~
, where x ˆ is the estimated state vector. The state equation is modified containing the above error signal:

d mv J r u mv c J J l c x mv c c mv l c l c J l c l c Jv l c l c x                 +           +                 +                 - + - - - - + - = 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 1 0 1 0 ~11 1 2 1 2 1 1 2 2 1 1 2 2 2 2 2 2 1 1 & (6)
The input signal is then as follows:

) ( ) ( ) ( 2 , 1 t z K t x K t u i + - = (9)
where K T is the gain of the states, while K i is the gain of the augmented state z defined as an error signal. The control goal is to follow the desired yaw rate by the intervention of the steering angle and the brake torque. The design of control gains is based on different robust control design methods such as LQ or Hinf.

SIMULATION RESULTS

The post impact design method has been validated with simulation performed in CarSim. The impact of the collision was given as exterior force acting on the vehicle. The vehicle speed was set to constant 104 km/h before the collision, and after the impact no driver intervention was assumed. In the simulation a full size SUV was used with the parameters given in Table 1.

The impact of the collision was simulated by applying a force vector on the vehicle with longitudinal (F x ) and lateral (F y ) components. The performances of the designed controllers are shown in Fig. 5, where the yaw rate tracking ability of the vehicle is analyzed. 1.745 m µ 0.7 -Table 1 Vehicle data

CONCLUSIONS

A yaw-rate servo control method has been introduced to avoid stability loss in the case of a light impact collision. The simulation results have shown that by applying both differential braking and active steering the vehicle's yaw rate set by the driver can be tracked much faster than by applying solely the break or the steering input or without any control action. The vehicle's directional stability can be preserved with the proposed method without significant deviation in the lateral position of the vehicle or without the spinning out of the lane.

Future work must consider the problem of collision detection and validation to avoid any control action taken based on false sensor information or sensor error. A sensitivity analysis must be carried out to describe the margins of such post impact control systems and to describe collision situations where the control action may be ineffective or harmful. A three dimensional vehicle model should also be applied to capture roll and pitch dynamics of the vehicle in order to avoid a possible rollover.

Fig. 1

 1 Fig. 1 Single track bicycle model

Fig. 5

 5 Fig. 5 Post impact yaw rate tracking with different methods

ACKNOWLEDGEMENT

This work was supported by the TÁMOP-4.2.2.A-11/1/KONV-2012-0012: Basic research for the development of hybrid and electric vehicles. and the PICS-CROTALE projects.


As our world is becoming ever more digitalized companies have more and more data to process. This steep increase of the unstructured data available is asking for a shift of perspective from the traditional database approach to an efficient distributed computing platform specifically designed for handling petabytes of unstructured information available in jobs such as: personalized advertising, advanced data mining or classification.

Modeling and Control of

MapReduce Systems BEREKMERI, Mihaly, GIPSA-lab, mihaly.berekmeri@gipsa-lab.grenoble-inp.fr BOUCHENAK, Sara, LIG, sara.bouchenak@imag.fr BOGDAN, Robu, GIPSA-lab, bogdan.robu@gipsa-lab.fr MARCHAND, Nicolas, GIPSA-lab, nicolas.marchand@gipsa-lab.fr Grenoble Images Parole Signal Automatique UMR CNRS 5216 -Grenoble Campus 38400 Saint Martin d'Hères -FRANCE Systems based on the MapReduce programming model are emerging as a central tool for deploying jobs that process large datasets in parallel. However the configuration of MapReduce systems is a complex process and at the moments it's left up to the user. These ad-hoc configuration methods make it difficult for small companies to take advantage of the growth of cloud computing solutions that provide resources as a service. Furthermore, the definition of SLAs becomes a complicated process for the user and the service provider as well. We propose a control theoretical approach to solving these problems. This implies the development of a general model that captures the dynamics of MapReduce systems. Finally, we intend to provide novel control methods that ease the configuration process and guarantee service level objectives such as constraints on system performance (execution times) and dependability (latency, availability) while optimizing resource consumption.

I. MapReduce Dynamic Model

•

V. Our goals

Cloud Computing is next milestone of IT evolution. It incorporates the long standing dream of providing computing as a service over the internet. Cloud computing can offer many computing forms as a service such as: hardware, platform, software, storage. Infrastructure as a service is one of the most wide spread service models at the moment and it offers computers as a service. Platform as a service models offer computing frameworks such as programming models and languages, operating systems. Software as a service provides an environment for running end user applications in the cloud.

Modeling issues: the models are specific to certain job metric, such as job finish time and don't consider latency, availability metrics.  Most of the models don't account for any kind of faults.  Randomness of such systems is not incorporated into current models.  Ad-hoc configuration of MR framework poses many challenges. SLA problems: a general framework for defining the terms of the Service Level Agreements, for MapReduce jobs, is needed.  It is very difficult for the user to check that the constraints of the service level agreements are kept, difficult to differentiate between providers.  Abusive jobs are becoming increasingly difficult to handle in virtualized environments where multiple jobs use the same cluster. Control Insertion: novel control laws are needed that can enforce the Service Level Agreements, both performance and dependability metrics.  More energy efficient control laws are desired.

IV. Current problems and development perspectives

III. MapReduce

Cloud computing, using the pay as you go model, offers an enticing new approach to deploying such task. The dominant new perspective emerging, in large scale data processing, is the programming model called MapReduce.

Scalability and elasticity are a few of the key aspects of Cloud computing. Seemingly resources scale up infinitely on demand. In the background, cloud computing takes care of such non trivial problems as data security and availability, multitenancy issues, virtualization, hardware and software maintenance.

•

  Transfer the problems into the control theory framework • The model has to ease the definition of service level objectives • Account for uncertainties • Provide guarantees in terms of performance, dependability and cost • Optimize the system configuration • Minimize deployment costs • Handle multitenancy • Test it on real life systems, such as AmazonEc2, Grid500 MapReduce is a programming paradigm especially designed for large scale unstructured data processing. Its success lies in its simplicity, scalability and fault-tolerance.The runtime environments which build upon the MapReduce framework automatically take care of: data partitioning replication, data consistency load balancing and fault tolerance task distribution and scheduling Only two functions have to implemented: the Map function and the Reduce function. The Map function takes an input set of (key,value) pair and outputs an intermediate (key,value) pair. The MapReduce framework automatically groups all the values associated with the same keys and forwards it to a Reduce function. The Reduce function processes these values and usually gives as output a reduced set of values. Finally, from a user perspective, the MapReduce framework hides all the messy overhead of parallel computing and lets us focus on the task at hand.

  MapReduce is a programming model especially designed for large scale unstructured data processing. Its success lies in its simplicity, scalability and fault-tolerance. The runtime environments, which build upon the MapReduce framework, automatically take care of:  data partitioning  replication, data consistency  load balancing and fault tolerance  task distribution and scheduling Only two functions have to be implemented: the Map function and the Reduce function. The Map function takes an input set of (key,value) pair and outputs an intermediate (key,value) pair. The MapReduce framework automatically groups all the values associated with the same keys and forwards it to a Reduce function. The Reduce function processes these values and usually gives as output a reduced set of values. Finally, from a user perspective, the MapReduce framework hides all the messy overhead of parallel computing and lets us focus on the task at hand.