Hadi Hachem

Hengbo Cui

Reizo Kato

Pere Alemany

Enric Canadell

Olivier Jeannin

Marc Fourmigué

Dominique Lorcy

Mixed-Valence Conductors from Ni Bis(diselenolene) Complexes with a Thiazoline Backbone

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

■ INTRODUCTION

Nickel and gold bis(dithiolene) complexes have focused a great deal of attention for affording highly conducting systems, either as single-component 1 or as multi-component molecular materials. 2,[START_REF]A solution of PPh4Br (200 mg) dissolved in 10 mL. of CH2Cl2/CH3CN (3/7) was introduced in both compartments as supporting electrolyte. The nickel complex[END_REF] In order to generate the partially or fully oxidized metal bis(dithiolene) complex, the corresponding monoanionic species [M(dt)2] -1 (M = Ni, Au) is generally used in electrocrystallization experiments, and this complex should be at least soluble in a convenient solvent to be oxidized. It should be stressed here that upon electro-oxidation, the large majority As the chemical approaches to generate the monoanionic Au and Ni complexes are essentially the same, authors generally investigated in parallel both monoanionic metal complexes with the same ligand. 4,5,6,7,8 Within this frame, we have particularly developed one family of ligands (Chart 1c), namely the N-alkyl-1,3-thiazoline-2-thione-4,5-dithiolate (Rthiazdt, R = alkyl group) [START_REF] Eid | Influence of the thiazole backbone on the structural, redox, and optical properties of dithiolene and diselenolene complexes[END_REF] and investigated their corresponding Ni and Au complexes.

Chart 1.

Our chemical approach concerning these complexes relies on the initial preparation of the reduced species such as the d 8 dianionic Ni 2+ complexes, which are further oxidized in air to the monoanionic open-shell complexes, or for the gold complexes, the direct synthesis of the monoanionic d 8 Au 3+ species. Then through electrochemical oxidation, the neutral species are most often isolated as single component molecular conductors, either with nickel as closed shell species [Ni(R-thiazdt)2] 0 or with gold as radical complexes [Au(R-thiazdt)2] • . Such neutral Ni bis(dithiolene) complexes exhibit, given their closed shell nature, semi conducting behavior for [Ni(R-thiazdt)] (R = Me, Et, Pr). [START_REF] Filatre-Furcate | Radical or not radical: compared structures of metal (M = Ni, Au) bisdithiolene complexes with a thiazole backbone[END_REF][START_REF] Hachem | Single component conductors based on closed-shell Ni and Pt bis(dithiolene) complexes: metallization under high pressure[END_REF] We previously showed, however, that [Ni(Et-thiazdt) 2] 0 (R = Et) turns metallic under pressure in the low temperature range when the pressure reaches 4.4 GPa and in the whole temperature range for pressures above 6.9 GPa. [START_REF] Hachem | Single component conductors based on closed-shell Ni and Pt bis(dithiolene) complexes: metallization under high pressure[END_REF] In the gold series, we have shown that the presence of a substituent smaller than the ethyl one on the ligand allowed us to go from a Mott insulator material for [Au(Et-thiazdt)2] • , [START_REF] Tenn | A single-component molecular metal based on a thiazole dithiolate gold complex[END_REF] to a metallic system for [Au(Me-thiazdt)2] • . [START_REF] Gal | Stable metallic state of a neutral radical single-component conductor at ambient pressure[END_REF] Altogether, the use of the R-thiazdt ligand, either with nickel or gold, systematically led to the neutral complexes, rather than mixed-valence salts, upon electrocrystallization of the monoanionic precursors.

Besides the changes of the R substituent, we also investigated the replacement of sulfur atoms for selenium ones in the metallacycles using the R-thiazds ligand (N-alkyl-1,3thiazoline-2-thione-4,5-diselenolate) (Chart 1f). Such substitution parallels that reported earlier with the dsit ligand (Chart 1d), the diselenolene analog of dmit. It should be stressed at this point that very few [M(dsit)2] diselenolene complexes have been reported and structurally characterized to date. They are limited to salts of the monoanionic [Ni(dsit)2] 1-nickel complex, 14 and one single mixed-valence conducting palladium salt, namely (Me4N)[Pd(dsit)2]2. [START_REF] Faulmann | crystal structure and electrical properties of two new palladium complexes of selenium and sulfur ligands[END_REF] Besides, one should mention the dsise analogs 16 (Chart 1e) with only one mention of a mixed-valence conducting salt (Bu4N)[Ni(dsise)2]2. [START_REF] Olk | Synthese und Eigenschaften von molekularen Leitern auf der Basis von dmit-isologen Chelaten Kristall-und Molekülstruktur des Tetramethylammonium-bis-[bis-(1,3-dithiol-2-selon-4,5 diselenolato) nickelat[END_REF] We have shown that the oxidation of [Ni(Et-thiazds)2] -1 salts afforded the neutral [Ni(Et-thiazds)2] 0 , which exhibits an essentially 1D electronic structure with large band dispersion and small HOMO-LUMO gap and a semi-conducting behavior, with σRT = 8 × 10 -3 S cm -1 and Eact = 0.12 eV. [START_REF] Hachem | Introducing Selenium in Single-Component Molecular Conductors Based on Nickel Bis(dithiolene) Complexes[END_REF] By analogy with our recent results on gold diselenolene complexes with the less sterically demanding R = Me analog [Au(Me-thiazds)2] -1 , [START_REF] Yzambart | Anisotropic chemical pressure effects in single-component molecular metals based on radical dithiolene and diselenolene gold complexes[END_REF][START_REF] Gal | Mixed-valence gold bis(diselenolene) complex turning metallic under pressure[END_REF] we investigate here the oxidation of the corresponding monoanionic N-methyl substituted nickel diselenolene analogue, which is to say [Ni(Methiazds)2] -1 (Chart 1f), in order to see whether this complex during electrocrystallization would generate the unknown single-component conductor [Ni(Me-thiazds)2] 0 or a rare mixed-valence salt. Herein, we describe the synthesis of three monoanionic nickel complexes of the Me-thiazds diselenolene ligand with different counterions, [C] [Ni(Me-thiazds)] with C = Bu4N + , Et4N + and Ph4P + and the outcome of their electrocrystallization, providing rare examples of mixed-valence salts of nickel diselenolene complexes, further blessed with high conductivities.

■ RESULTS AND DISCUSSION

Syntheses. The monoanionic nickel diselenolene complexes were prepared starting from the cyanoethyl-protected diselenolene proligand 1 (Scheme 1). [START_REF] Eid | Influence of the thiazole backbone on the structural, redox, and optical properties of dithiolene and diselenolene complexes[END_REF] The proligand 1 was deprotected through the addition of sodium methanolate and then, after successive additions of NiCl2. 6H2O and Ph4PCl or Et4NBr or Bu4NCl, the dianionic species of the nickel complexes The [Ph4P][Ni(Me-thiazds)2] salt crystallizes in the monoclinic system, space group P21/n, with both ions in general position. As shown in Figure 1, the diselenolene ligands are disordered on two positions with a 92:8 distribution. Due to the presence of the unsymmetrical Me-thiazds ligand, the cis and trans isomers can be formed. However, as the Ni complex is localized in a general position in the unit cell, this crystallographic disorder is compatible only with a trans configuration of the complex. This disorder is reminiscent to the one observed for the analogous In the solid state, the monoanionic complexes in [Ph4P] [Ni(Methiazds)] interact only along the a axis (S•••S: 3.62 Å), and are well separated by Ph4P + cations associated into inversioncentered pairs in a so-called phenyl embrace motif (Figure S1). The magnetic susceptibility of this complex (Figure S2) is well fitted with a Curie-Weiss law, giving a Curie constant of 0.404, and a  value of -0.76 K, with a g = 2.077 value deduced from EPR experiments.

gold complex, [Ph4P][Au(Me-thiazds)2]. 20
The redox behavior of the three novel salts has been analyzed by cyclic voltammetry in CH2Cl2 using NBu4PF6 as supporting electrolyte. Their UV-vis-NIR absorption properties were measured in CH2Cl2 at room temperature. For the three salts, the redox system corresponding to the oxidation of the radical anion to the neutral state is not fully reversible, but characterized by a strong adsorption phenomenon occurring at the electrode (Figure 2a). Contrariwise, the redox system corresponding to the reduction into the dianionic species is fully reversible. This trend of deposition of the oxidized species at the electrode upon oxidation is different to the one observed for the N-Et analogues presumably due to a decrease in solubility of the N-Me oxidized species. The counterion does not have a significant influence on the redox potentials (Table 1). Table 1. Redox potentials (E in V vs SCE) in CH2Cl2, with 0.1M NBu4PF6 as a supporting electrolyte, absorption maxima max (nm) and molar extinction coefficients  (M -1 cm -1 ) for the We also performed spectro-electrochemical experiments as they give access to the absorption properties of the neutral species generated at the electrode. The UV-vis-NIR spectroscopic investigations carried out on freshly prepared dichloromethane solution of the monoanionic diselenolene complexes allowed us to determine the absorption maxima (max) of the lowest energy NIR electronic transitions for the monoanionic species (Figure 2b). If we compare the spectra obtained for the three monoanionic species, either as a Ph4P + salt, Bu4N + or Et4N + one, there is no influence of the counterion on the wavelength of the low energy absorption band. If

NIR absorptions of the [C][Ni(Me-thiazds)2] salts (C = Ph4P + , Et4N + , Bu4N + ) E -2/-1 Epa/pc -1/0 max () (nm, M -1 cm -1 ) Ref [Et4N][
we compare now these diselenolene complexes with the dithiolene analogue, [PPh4][Ni(Methiazdt)2] (max = 1278 nm), [START_REF] Hachem | Single component conductors based on closed-shell Ni and Pt bis(dithiolene) complexes: metallization under high pressure[END_REF] it can be noticed that the replacement of the sulfur atoms by selenium ones induces an hypsochromic shift of this NIR absorption band of about 90 nm. As can be seen in Figure 2b, upon gradual oxidation, an overall decrease of the NIR absorption band was observed resulting from the oxidation of the monoanionic species but concomitantly no spectral signature of the neutral species appeared (Figure 2b). Instead, upon oxidation of the monoanionic species, a strong deposit occurred on the platinum grid, attributed to the formation of the highly insoluble oxidized species.

In order to crystallize and identify the nature of the oxidized species, electrocrystallization experiments were conducted with the three monoanionic complexes. We found that the choice of the electrolyte used in the electrocrystallization cell is particularly important to favor a given phase. Moreover, high quality crystals were obtained only when the counterion of the supporting electrolyte was the same as the counterion of the complex to be The highest occupied and lowest unoccupied levels of one trimer are a set of six levels built from the HOMO and LUMO of the Ni(Me-thiazds)2 monomers and delocalized within the trimer. Three of these levels are doubly filled in the neutral species. Since the unit cell of the layer contains two trimers, the band structure contains six pairs of bands (Figure 5). These bands exhibit a quite poor energy dispersion, especially those marked with a red arrow, which are Transport and magnetic measurements confirmed these assumptions. The material shows a semiconducting behavior, with a room temperature conductivity of 4 × 10 -4 S.cm -1 , and an activation energy of 203 meV (Figure 6). This semiconducting behavior persists at higher pressures (up to 11.9 GPa), with an increase of the RT conductivity by 3 orders of magnitude (σRT = 0.25 S.cm -1 ), while the activation energy is divided by two down to 68 meV at 11.9 GPa. This electronic localization is also confirmed by the temperature dependence of the magnetic susceptibility (Figure S3). It shows a behavior that is well fitted to a Curie-Weiss law, giving a Curie constant of 0.409, and a  value of +0.92 K. The calculated g value amounts to 2.088, which is very close to what was observed for the monoanionic precursor [PPh4][Ni(Me-thiazds)2] (g = 2.077). This shows that, due to the strong trimerization, each unit acts as a single radical moiety, with very weak interactions between the S = ½ trimeric species.

T (K) P (GPa)
1 bar 0.8 1.7 2.9 5.3 9.3 11.9

higher pressures in the Diamond Anvil Cell (DAC) were unsuccessful as the gold contacts crystals appear to react with the crystals. Hückel band structure calculations (Figure S4) as those used above for the 1:3 Ph4P + salt in Figure 5 show that the Fermi level indeed cuts a band with a substantial energy dispersion of ~0.4 eV. Such a dispersion (calculated with the same computational settings) 2b,24 for this type of dimerized metal bis(dithiolene) type layers is usually associated with a metallic behavior, thus confirming our initial expectation. At this point we decided to turn to DFT calculations that by considering electronic interactions explicitly will allow a more in-depth exploration of this apparent contradiction. The calculated band structure for [NEt4][Ni(Me-thiazds)2]2 assuming double occupation of the levels is reported in Figure 9a. Because of the strong cation disorder, we removed the cations from the calculation replacing them by a uniform background charge keeping the neutrality of the unit cell. This procedure has been tested in numerous cases and shown to provide band structures almost undistinguishable from those using the full unit cell. [START_REF] Alemany | Structural and electronic control of the metal to insulator transition and local orderings in the theta-(BEDT-TTF)2X organic conductors[END_REF][START_REF] Abhervé | Conducting chiral nickel(II) bis(dithiolene) complexes: structural and electron transport modulation with the charge and the number of stereogenic centres[END_REF][START_REF] Canadell | Concerning the Band Structure of D[M(dmit)2]2 (D = TTF, Cs, NMe4; M = Ni, Pd) Molecular Conductors and Superconductors: Role of the HOMO and LUMO[END_REF] Because of the very short intra-dimer Se•••Se contacts and the Ni-over-Se type dimerization (see Figure 3b), the splitting of the HOMO and LUMO levels in a dimeric unit is not the typical one found in many metal bis(dithiolene) dimerized salts (see Figure 9b). 2b,3b,27 The bonding/antibonding combinations of the HOMOs are the main components of the lower/upper levels of the set of four dimer orbitals; the bonding/antibonding combinations of the LUMOs occur in between them. Such ordering is essentially kept in the layer according to the band structure of Figure 9a which corresponds to a metallic system. The bonding combinations of both the HOMO and LUMO levels are filled providing a strong stability to the dimers and the electron transferred from the cation partially fills the antibonding combination of the LUMOs. This level ordering is unexpected since in the absence of any calculation and with one electron transferred to the dimer, one would expect that the bonding combination of the LUMOs would house the electron. This level ordering is still another consequence of the so called two-band behavior. 2b,24,27 In order to find a plausible explanation for the observed activated conductivity of this salt we carried out spin-polarized DFT calculations to see if states with electrons localized on the dimers more stable than the metallic one could be found. This proved to be a difficult task since our trials invariably converged to the delocalized metallic state. After several unsuccessful attempts, we considered the possibility of some inadequacy in our computational approach, either because of the use of the uniform background charge or from some deficiency in the computational settings that was exaggerating the Se•••Se interactions, thus favoring the metallic state. In order to test the latter we carried out calculations with exactly the same computational details for a single-molecule conductor based on a very similar complex, [Au(Et-thiazds)2]. [START_REF] Yzambart | Anisotropic chemical pressure effects in single-component molecular metals based on radical dithiolene and diselenolene gold complexes[END_REF] We were able to reproduce the subtle energy difference between metallic and antiferromagnetic localized states of the previous calculations carried out with a different plane wave based computational approach, [START_REF] Yzambart | Anisotropic chemical pressure effects in single-component molecular metals based on radical dithiolene and diselenolene gold complexes[END_REF] so that the problems found for the present case seem not to arise from an inadequate choice of our computational method.

As mentioned above, the use of a uniform background of charge for molecular conductors where counterions exhibit disorder has been tested in numerous cases, and shown to provide band structures almost undistinguishable from those using the full unit cell with the disordered counterion frozen in the majority position. [START_REF] Alemany | Structural and electronic control of the metal to insulator transition and local orderings in the theta-(BEDT-TTF)2X organic conductors[END_REF][START_REF] Abhervé | Conducting chiral nickel(II) bis(dithiolene) complexes: structural and electron transport modulation with the charge and the number of stereogenic centres[END_REF][START_REF] Alemany | Electronic structure and anion ordering in (TMTSF)2ClO4 and (TMTSF)2NO3: A first-principles study[END_REF] Thus, we suspected that the difficulty in converging toward localized states was most likely unrelated to technical problems with the computations. For this reason, we devised a different strategy to search for possible localized states by rigidly shifting the dimers along different directions of the lattice so as to decrease the Se•••Se interactions selectively. As shown in the band structure of Figure 9a, the band widths along the chain direction (-Y) are substantially larger than those along the interchain direction (-X), specially for the partially filled band. Thus, we decided to explore the behavior of a single chain of dimers along the b axis. We used a 2a×2b×2c cell so as to have well separated chains. We found that an antiferromagnetic (AF) state lying very slightly below the metallic state (4 meV/dimer) can be obtained (Figure S5). The ferromagnetic state is considerably higher in energy than the metallic state (40 meV/dimer). The stability of this AF state increases when the dimers are rigidly shifted along the chain, decreasing the Se•••Se contacts between them. However, as soon as the inter-chain interactions are switched on, only the metallic state can be obtained. It is thus clear that the chain of dimers has two almost degenerate states: metallic and AF. The two states are so close in energy that even if the interchain interactions are weak, they slightly stabilize the metallic state which becomes the more stable alternative when the total spin is S = 0. However, we suspected that if we allowed antiferromagnetic interactions along the inter-chain direction maybe the localized AF state would be the more stable state for the layer, as it was for the isolated chain. A calculation using a 2a×2b×1c cell confirmed this expectation and the fully AF state could be located and was found to be 1 meV/dimer more stable than the metallic state. Of course, this very small energy difference should not be taken at its right face and the only meaningful conclusion of this computational exercise is that [Et4N] [Ni(Me-thiazds)2]2 has two possible ground states, metallic and localized, which are very close in energy. Why then the present system, although quite conductive as suggested by the calculations, chooses to exhibit a semiconducting behavior? The simplest possibility would rely on a slight predominance of electronic repulsions over delocalization effects. In that case, a very small pressurization should stabilize the metallic state albeit we were not able to verify it experimentally. However, a plausible answer can be attained after consideration of the results for the Bu4N + salt.

The mixed-valence salt with the Bu4N + cation crystallizes in the triclinic system, space group P1 ̅ . The overall structure and organization are very close to that of the Et4N + salt, with strong dimerization of the complex units. For instance, the shorter inter-dimer Se•••Se contacts along the stacking b axis are now 3.44 Å and those between the stacks now go down to 3.37 Å.

These values are very similar to those quoted above for the Et4N + salt, at 3.46 and 3.39 Å respectively. As mentioned above, the strong disorder of the Bu4N + moiety in the crystal structure could not be solved, even when the data collection was performed at low temperatures (150 K). For this reason, the cation was squeezed out of the crystal structure. This issue poses a hurdle in assessing the true stoichiometry between the complexes and cations, a crucial aspect concerning the overall band filling of this salt. Considering the unit cell volume of the Et4N + salt (971 Å 3 for 47 non-hydrogen atoms), the Bu4N + salt with 55 non-hydrogen atoms should have, with a 1:2 stoichiometry, a cell volume around 1133 Å 3 , instead of the observed 1016 Å 3 volume. This corresponds to a [Bu4N]x[Ni(Me-thiazds)2]2 salt with an x value of 0.66. Analysis of the SQUEEZE procedure during the structural refinement of [Bu4N]x[Ni(Me-thiazds)2]2 to represent the disordered Bu4N + cation gives a very similar value (x = 0.70). As expected from the structural discussion, the calculated DFT band structure is practically identical to that of the Et4N + salt (Figure S6) and the only noticeable effect of the partial occupation of the cation sites is a displacement of the corresponding Fermi level.

The temperature dependence of the resistivity at ambient pressure shows a highly conducting but activated behavior, with RT = 1.25 S cm -1 and Eact = 19 meV, comparable to those found in the stoichiometric Et4N + salt. Resistivity measurements under pressure could be performed and showed an increase in conductivity and a decrease in activation energy between 1 bar and 4.7 GPa (Figure 10). Further increase in pressure up to 9.6 GPa showed an essentially semi-metallic behavior over the whole temperature range, with a RT = 16 S cm -1 and an activation energy divided by two down to 9 meV. With these results in mind, we can understand the origin of the activated conductivity of the two salts. The energy difference between the localized and metallic states of the NEt4 + salt is extremely small. Because of the non-stoichiometry, the filling of the (LU-LU) ab bands in [NBu4]x[Ni(Me-thiazds)2]2 decreases thus going out of the half-filled situation which is known to favor the localization. Thus, a better conductivity or even a metallic behavior is expected for the NBu4 + salt. A simple chemical argument illustrates this point. The activated conductivity of a localized system essentially arises from the electronic repulsion (U) created when an electron jumps from one site (dimer) to the next one. When x ≈ 2/3, around one third of the dimers do not have any electron in the (LU-LU) ab level. Consequently, when one electron jumps to one of these empty levels, no electron repulsion is created and the jump is not associated with a strong energy penalty (activation energy). Whereas all jumps are activated in the NEt4 + salt, a substantial number of them are non-activated in the NBu4 + salt. So to speak, the nonstoichiometry should introduce a kind of metallicity in the system resulting with a much better conductivity than for the NEt4 + salt. However, our conductivity measurements do not agree with such expectation. Only under a substantial pressurization a semimetallic state is reached.

There are two basic mechanisms for the occurrence of an electron localization in layers such as those found in the present salts. First, an intrinsic mechanism where the electronic repulsions drive the localization of electrons (Mott-type mechanism). [START_REF] Mott | Metal-Insulator Transitions[END_REF] Second, an extrinsic disorder induced localization (Anderson-type mechanism). [START_REF] Anderson | Absence of Diffusion in Certain Random Lattices[END_REF][START_REF] Hayes | Defects and Defect Processes in Non -metallic Solids[END_REF] Although both mechanisms must be simultaneously operative in these salts, the severe disorder in the cation sites and the absence of a clear preference for a localized state in the DFT calculations strongly advocate for the dominance of the cation disorder as the more plausible origin for the activated nature of the conductivity. We believe that the two salts would be metallic in agreement with the occurrence of very short inter-dimer Se…Se contacts if it were not for the severe cationic disorder.

■ CONCLUSIONS

In added to the proligand 1 (200 mg, 0.5 mmol). After complete dissolution, the solution was stirred at room temperature for 30 mn. Then a solution of NiCl2,6H2O in MeOH (5 mL) was added, followed 6 hours later by the addition of C + Br -(0.5 mmol, 210 mg of PPh4Br and 161 mg of NBu4Br) . After stirring for 15 h, the formed precipitate was filtered and recrystallized under an air atmosphere from CH2Cl2/MeOH 20/80 to afford the monoanion complex as dark crystals.

[PPh4] [Ni(Me-thiazds)

R1 = ||Fo| -|Fc||/|Fo|. b wR2 = [w(Fo 2 -Fc 2 ) 2 ]/[w(Fo2) 2 ] 1/2 .
Transport measurements. Single crystals were mounted in the DAC by using the same technique as reported earlier. [START_REF] Cui | Metallization of the Single Component Molecular Semiconductor [Ni(ptdt)2] under Very High Pressure[END_REF] The sample was encapsulated with a mixture of epoxy and alumina. The diamond culet size was 0.7 mm. Electrical contacts were obtained by attaching four 10 μm gold wires with gold paint, and the four-probe DC method was used for all measurements. The contacts were made on the corners of the elongated plates. The stacking axis of the trimers/dimers in the three mixed-valence salts is located within this plane, as illustrated in Fig. S7 Band structure calculations. The first-principles calculations for the solid were carried out using a numeric atomic orbitals density functional theory (DFT) approach [START_REF] Hohenberg | Inhomogeneous Electron Gas[END_REF][START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Effects[END_REF] developed for efficient calculations in large systems and implemented in the SIESTA code. 39 We used the generalized gradient approximation (GGA) to DFT and, in particular, the functional of Perdew, Burke, and Ernzerhof. [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] To study the relative energies of states with localized electrons, spin polarized band calculations for appropriate supercells have been undertaken. All calculations included a Hubbard correction term Ueff = 6.0 eV for the S 3p and Se 4p states. [START_REF] Dudarev | Electronenergy-loss spectra and the structural stability of nickel oxide: An LSDA+U study[END_REF] In previous work [START_REF] Kiyota | Electronic engineering of a tetrathiafulvalene charge-transfer salt via reduced symmetry induced by combined substituents[END_REF] we have found that this U term on the chalcogen atoms is needed for appropriately describing the electronic structure of molecular conductors were accurate experimental information on the bandwidth and charge transfer is available. Only the valence electrons are considered in the calculation, with the core being replaced by normconserving scalar relativistic pseudopotentials [START_REF] Troullier | Efficient pseudopotentials for plane-wave calculations[END_REF] factorized in the Kleinman-Bylander form. [START_REF] Kleinman | Efficacious Form for Model Pseudopotentials[END_REF] We have used a split-valence double- basis set including polarization orbitals with an energy shift of 10 meV for all atoms. [START_REF] Artacho | Linear-Scaling ab-initio Calculations for Large and Complex Systems[END_REF] The energy cutoff of the real space integration mesh was 350

Ry. The Brillouin zone was sampled using grids 46 of (10×10×5) k-points for the calculations using a single cell; those for larger unit cells were modified appropriately. The experimental crystal structures were used for the computations.

The tight-binding band structure calculations were of the extended Hückel type [START_REF] Whangbo | The Band Structure of the Tetracyanoplatinate Chain[END_REF] and used a modified Wolfsberg-Helmholtz formula [START_REF] Ammeter | Counterintuitive Orbital Mixing in Semiempirical and ab Initio Molecular Orbital Calculations[END_REF]

Accession Codes For Table of content only

Synopsis

Dimeric or trimeric associations are observed in highly conducting mixed-valence nickel bis(diselenolene) complexes, depending one the nature of the counter ion (Ph4P + , Et4N + , Bu4N + ) and the stoichiometry of the salt

  of monoanionic gold complexes [Au(dt)2] -1 gives the open-shell species [Au(dt)2] • , 1 i.e. the neutral radical complexes, leading to single component molecular conductors, while the monoanionic nickel complexes afford mainly multi-component, mixed-valence conductors (C + )[Ni(dt)2]n (n = 2,3) as the prototypical [Ni(dmit)2] salts (Chart 1a) but rarely the neutral diamagnetic species [Ni(dt)2] 0 .

  Scheme 1. Synthetic route toward the monoanionic, radical, Ni bis(diselenolene)complex, [Et4N][Ni(Me-thiazds)2] , [Bu4N][Ni(Me-thiazds)2] and [Ph4P][Ni(Me-thiazds)2].

Figure 1 .

 1 Figure 1. Molecular view of the anion radical species [Ni(Me-thiazds)2] -1 in [Ph4P][Ni(Methiazds)2]. The two disordered moieties with a 92:8 distribution are highlighted with light grey and black bonds, respectively.

Figure 2 .

 2 Figure 2. (a) Cyclic voltammetry and (b) UV-vis-NIR monitoring of the electrochemical oxidation of [PPh4][Ni(Me-thiazds)2] in CH2Cl2, with 0.2M NBu4PF6 as a supporting electrolyte.

18 Scheme 2

 182 Scheme 2 Electrocrystallization conditions and outcomes.

Figure 3 .Table 2 .

 32 Figure 3. Molecular view of (a) the trimeric motif in [Ph4P][Ni(Me-thiazds)2]3 and (b) the dimeric motif in [Et4N][Ni(Me-thiazds)2]2.

Figure 4 .

 4 Figure 4. (a) Projection of the unit cell of [Ph4P][Ni(Me-thiazds)2]3 along b, (b) detail of the anionic layer showing interactions between trimers in the (bc) plane.

  those half-filled by the transferred electrons. Such band structure clearly shows that the trimers interact only weakly, thus suggesting this salt to behave as a Mott insulator with one localized (but delocalized within the trimer) S = ½ radical per trimer.

Figure 5 .

 5 Figure 5. Calculated extended Hückel band structure for the anionic layer of [PPh4][Ni(Methiazds)2]3 where Γ = (0, 0), Y = (b*/2, 0), Z = (0, c*/2), M = (b*/2, c*/2). The red arrow points out the bands that would be occupied by the two transferred electrons.

Figure 6 .

 6 Figure 6. (a) Temperature dependence of the resistivity of [PPh4][Ni(Me-thiazds)2]3 at different pressures between 1 bar and 11.9 GPa.

Figure 7

 7 Figure 7 (a) Projection view of the unit cell of [NEt4][Ni(Me-thiazds)2]2 along b showing in light and dark grey atoms the disorder model on the Et4N + cation, (b) End-on view of the anionic layer. Hydrogen atoms have been omitted for clarity.

Figure 8 .

 8 Figure 8. Temperature dependence of the resistivity of [NEt4][Ni(Me-thiazds)2]2 at ambient pressure

Figure 9 .

 9 Figure 9. (a) Calculated DFT band structure for [NEt4][Ni(Me-thiazds)2]2 assuming that all levels are doubly filled. The Fermi level is represented by a dashed line and Γ, X, Y, M, and Z refer to Γ = (0, 0, 0), X = (1/2, 0, 0), Y = (0, 1/2, 0), M = (1/2, 1/2, 0) and Z= (0, 0, 1/2) in units of the triclinic Brillouin zone. (b) Schematic diagram showing the splitting of the Ni(Methiazds)2 HOMO and LUMO levels in the [Ni(Me-thiazds)2]2 dimers of Figure 3b. The "b" and "ab" labels refer to bonding and antibonding combinations, respectively.

Figure 10 .

 10 Figure 10. Temperature dependence of resistivity of [NBu4]x[Ni(Me-thiazds)2]2 at different pressures.

  for [nBu4N]0.66[Ni(Me-thiazds)2]2. Daphne Oil 7373 was used as the pressure transmitting medium. The pressure was determined by the shift in the ruby fluorescence R1 lines at room temperature.

  to calculate the non-diagonal H values. All valence electrons were considered in the calculations and the basis set consisted of Slater-type orbitals of double- quality for Ni 3d and of single- quality for Ni 3s and 3p, C 2s and 2p, S 3s and 3p, Se 4s and 4p and H 1s. The ionization potentials, contraction coefficients and exponents were taken from previous work. 49 pubs.acs.org/doi/10.1021/acs.inorgchem.XXXX.

Figures

  Figures S1-S7 (PDF): Additional crystal structure figure, Temperature dependence of the magnetic susceptibility curves, calculated extended Hückel band structure, Calculated DFT band structures, crystal faces indexation.

  contrast with nickel bis(R-thiazdt) dithiolene complexes, which afford single-component neutral species, nickel bis(Me-thiazds) diselenolene complexes lead to rare multi-component, mixed-valence salts. Depending on the cation, salts with different stoichiometries were obtained. The PPh4 + salt contains anionic layers with trimeric units associated through apical Ni-Se bonds leaving the central nickel atom in an octahedral coordination; it exhibits semiconducting behavior with an activation energy Eact of 203 meV. This activated conductivity is kept under pressures up to ~12 GPa although the conductivity increases by three orders of magnitude and Eact decreases to 68 meV. The Et4N + and Bu4N + salts contain strongly dimerized Chemicals and materials from commercial sources were used without further purification. All the reactions were performed under an argon atmosphere. Melting points were measured on a Kofler hot-stage apparatus and are uncorrected. Mass spectra were recorded by the CRMPO, Rennes. Methanol, acetonitrile and dichloromethane were dried using an Inert pure solvent column device. CVs were carried out on a 10 -3 M solution of complex in CH2Cl2 with NBu4PF6 0.1 M. Potentials were measured versus Saturated Calomel Electrode (SCE). The spectro-electrochemical setup was performed in CH2Cl2 with Bu4NPF6 0.2 M using a Pt grid as the working electrode, a Pt wire as the counter electrode and SCE reference electrode. A Shimatzu 3600 spectrophotometer was employed to record the UV-vis-NIR spectra. Proligand 1 and [NEt4][Ni(Me-thiazds)2] were prepared as previously reported.

	■ EXPERIMENTAL SECTION
	General.
	units with two intra-dimer Ni-Se bonds leading to a square-pyramidal nickel coordination. The
	cations in both salts are disordered and the stoichiometry is 1:2 for the Et4N + and 2/3:2 for the
	Bu4N + salts. The Et4N + compound exhibits a semiconducting behavior at ambient pressure, with
	a high conductivity (RT = 6 S cm -1 ) and very low Eact (12 meV). Based on a DFT study we
	suggest that the activated conductivity is mostly disorder-induced. The essentially isostructural
	NBu4 + salt [NBu4]x[Ni(Me-thiazds)2]2 is characterized by an non-integer stoichiometry (x ≈
	2/3) and exhibits a similar semi-conducting behavior, while turning semi-metallic under
	pressure. At variance with most other systems incorporating these R-thiazdt and R-thiazds
	ligands which provide the single-component neutral complexes, we demonstrate here that
	nickel complexes of the Me-thiazds diselenolene ligand are able to organize into highly
	conducting 2:1 mixed-valence systems with strongly dimerized complexes stabilized by short
	Ni•••Se intra-dimer bonds. This structural motif contrasts with the well-known [Pd(dmit)2] 2:1
	salts which exhibit a variety of ground states depending on the choice of cation, 32 where the
	dimeric moieties are associated through metal-metal Pd-Pd intradimer bonds. The observed
	sensitivity to pressure effects lets us infer that chemical pressure could also be used here to tune
	their properties, by replacing for example the Et4N + cation by smaller (Me4N + ; Me4P + ,
	Et3MeN + , Et3MeP + ) or larger ones (Et4P + , Et4As + ), as exemplified in the [Pd(dmit)2] series. 32

9

Syntheses.

[C][Ni(Me-thiazds)2]

. Under inert atmosphere, a solution of sodium methanolate in MeOH (1.5 mmol, prepared from 35 mg of Na in 10 mL of dry MeOH) was

Table 3 .

 3 Crystallographic data a,b

		[PPh4]	[PPh4]	[Et4N]	[nBu4N]0.66
		[Ni(Me-thiazds)2]	[Ni(Me-thiazds)2]3	[Ni(Me-thiazds)2]2	[Ni(Me-thiazds)2]2
	CCDC	2224750	2224751	2224748	2224749
	Formulae	C32H26N2NiPS4Se4	C48H38N6Ni3PS12Se12 C12H16N2.5NiS4Se4	C8H6N2NiS4Se4
	FW (g.mol -1 )	972.31	2238.18	698.06	632.94
	System	monoclinic	monoclinic	triclinic	triclinic
	Space group	P21/n	C2/c	𝑃1 ̅	𝑃1 ̅
	a (Å)	10.6095(8)	49.634(5)	6.4328(5)	6.4074(9)
	b (Å)	19.6820(14)	7.3788(7)	8.1900(3)	8.1303(11)
	c (Å)	16.5774(13)	20.312(2)	18.7194(13)	19.673(3)
	α (deg)	90	90	91.113(4)	82.976(5)
	β (deg)	99.828(3)	113.440(4)	99.767(6)	89.302(5)
	γ (deg)	90	90	91.548(5)	88.260(5)
	V (Å 3 )	3410.8(4)	6825.2(12)	971.29(11)	1016.6(3)
	T (K)	150(2)	150(2)	293(2)	150(2)
	Z	4	4	2	2
	Dcalc (g.cm -3 )	1.893	2.178	2.387	2.068
	µ (mm -1 )	5.157	7.649	8.923	8.513
	Total refls	48468	43482	12136	33238
	Abs corr	multiscan	multiscan	multiscan	multiscan
	Uniq refls (Rint)	7818(0.0893)	7591 (0.1099)	4561 (0.0463)	4654 (0.0765)
	Uniq refls (I > 2σ(I)) 6314	6251	3067	3820
	R1, wR2	0.0575, 0.124	0.0597, 0.1174	0.0669, 0.14	0.0449, 0.1019
	R1, wR2 (all data)	0.0773, 0.133	0.0786, 0.1225	0.1139, 0.1621	0.0575, 0.1081
	GOF	1.109	1.136	1.075	1.025
	a

■ ACKNOWLEDGMENTS This work was supported in part by Université de Rennes 1 (PhD grant to H. H.) and by the JSPS KAKENHI (Grant JP16H06346). The stay of H. H. to RIKEN was supported in part by RIKEN International Program Associate and by Rennes Métropole. We thank Takaaki Minamidate (RIKEN) and Thierry Guizouarn (Rennes) for the magnetic susceptibility (SQUID) measurements Work in Spain was supported by MINECO through Grants PGC2018-096955-B-C44 and PID2021-128217NB-I00, Generalitat de Catalunya (2017SGR1506 and 2017SGR1289). E. C. acknowledges support of the Spanish MINECO through the Severo Ochoa FUNFUTURE (CEX2019-000917-S) Excellence Centre distinction and P. A. from the Maria de Maeztu Units of Excellence Program (MDM-2017-0767). Their support is gratefully acknowledged.

Olivier Jeannin -Institut des Sciences Chimiques de Rennes, Université de Rennes, CNRS, UMR 6226, F-35000 Rennes, France; orcid.org/0000-0002-1411-5656

Notes

The authors declare no competing financial interest.


Zaid M Odibat
email: z.odibat@gmail.com

Nathalie Corson
email: nathalie.corson@univ-lehavre.fr

M A Aziz-Alaoui
email: aziz.alaoui@univ-lehavre.fr

Cyrille Bertelle
email: cyrille.bertelle@univ-lehavre.fr

Synchronization of chaotic fractional-order systems via linear control

Keywords: Chaos synchronization, Fractional-order system, Caputo fractional derivative, Mittag-Leffler function, Stability, Linear control

The chaotic dynamics of fractional-order systems has attracted much attention recently. Chaotic synchronization of fractional-order systems is further studied in this paper. We investigate the chaos synchronization of two identical systems via a suitable linear controller applied to the response system. Based on the stability results of linear fractional-order systems, sufficient conditions for chaos synchronization of these systems are given. Control laws are derived analytically to achieve synchronization of the chaotic fractionalorder Chen, Rössler and modified Chua systems. Numerical simulations are provided to verify the theoretical analysis.

Introduction

Chaos, as an important topic in nonlinear science, has been investigated and studied in mathematical and physical communities in the last few decades and the research efforts have devoted to chaos control and chaos synchronization problems in many dynamical systems. During the last two decades, synchronization in chaotic dynamical systems has attracted a great deal of attention [START_REF] Yamada | Stability theory of synchroized motion in coupled-oscillator systems[END_REF]; [START_REF] Pecora | Synchronization in chaotic systems[END_REF][START_REF] Ott | Controling chaos[END_REF][START_REF] Chen | From chaos to order: methodologies, persectives and applications[END_REF][START_REF] Derivière | Estimation of attractors and synchronization of generalized Lorenz systems[END_REF][START_REF] Aziz-Alaoui | Synchronization of chaos[END_REF]] since the seminal papers by Yamada and Fujisaka [1983] and [START_REF] Pecora | Synchronization in chaotic systems[END_REF]. A way to study synchronization is to use a controller to make the output of the slave (reponse) system copy in some manner the master (drive) system one.

On the other hand, the development of models based on fractional-order differential systems has recently gained popularity in the investigation of dynamical systems. Fractional derivatives provide an excellent instrument to describe memory and hereditary properties of various materials and processes. The main reason for using the integer-order models was the absence of solution methods for fractional differential equations. The advantages or the real objects of the fractional-order systems [START_REF] Petráš | A note on the fractional-order Chua's system[END_REF] are that we have more degrees of freedom in the model and that a "memory" is included in the model (fractional-order systems have an unlimited memory).

Recently, studying fractional-order systems has become an active research area. The chaotic dynamics of fractional-order systems began to attract much attention in recent years [START_REF] Podlubny | Fractional Differential Equations[END_REF]Hil-fer, 2000]. It has been shown that fractional-order systems, as generalizations of many well-known systems, can also behave chaotically, such as the fractional Duffing system [START_REF] Ge | Chaos in a fractional order modified Duffing system[END_REF], the fractional Chua system [START_REF] Petráš | A note on the fractional-order Chua's system[END_REF][START_REF] Hartley | Chaos in a fractional order Chua's system[END_REF], the fractional Rössler system [Li & Chen, 2004a], the fractional Chen system [START_REF] Li | Chaos in Chen's system with a fractional order[END_REF]Li & Chen, 2004b; [START_REF] Lu | A note on the fractionalorder Chen system[END_REF]], the fractional Lorenz system [START_REF] Grigorenko | Chaotic dynamics of the fractional Lorenz system[END_REF]], the fractional Arneodo system [START_REF] Arneodo | Asymptotic chaos[END_REF] and the fractional Lü system [START_REF] Deng | Chaos synchronization of the fractional Lü system[END_REF]. In [START_REF] Hartley | Chaos in a fractional order Chua's system[END_REF]Li & Chen, 2004a; [START_REF] Li | Chaos in Chen's system with a fractional order[END_REF]Li & Chen, 2004b] it has been shown that some fractional-order systems can produce chaotic attractors with order less than 3. Recent studies show that chaotic fractional-order systems can also be synchronized [START_REF] Lu | A note on the fractionalorder Chen system[END_REF] [START_REF] Peng | Synchronization of fractional order chaotic systems[END_REF]. In many literatures, synchronization among fractional-order systems is only investigated through numerical simulations. A simple method to synchronize chaotic fractional-order systems based on the stability criteria of linear differential systems, see [START_REF] Matignon | Stability results of fractional differential equations with applications to control processing[END_REF][START_REF] Deng | Stability analysis of linear fractional differential system with multiple time delays[END_REF], is presented in [Sheu et In this paper, we recall and present some stability results for linear fractional-order systems. Then, based on these results, we study the chaos synchronization of fractional-order systems. Using the driveresponse concept, a linear feedback control law is derived to achieve synchronization of the two chaotic fractional-order systems. The synchronization controllers are investigated theoretically and numerical simulations are then presented in the case of fractional-order Chen, Rössler and modified Chua systems.

Preliminaries and definitions

There are several definitions of a fractional derivative of order α > 0 [START_REF] Podlubny | Fractional Differential Equations[END_REF][START_REF] Caputo | Linear models of dissipation whose Q is almost frequency independent, Part II[END_REF].

The two most commonly used are the Riemann-Liouville and Caputo definitions. Each definition uses Riemann-Liouville fractional integration and derivatives of whole order. The difference between the two definitions appears in the order of evaluation. The Riemann-Liouville fractional integral operator of order α ≥ 0 of the function f (t) is defined as,

J α f (t) = 1 Γ(α) t 0 (t -τ ) α-1 f (τ )dτ, t > 0. ( 1
)
Some properties of the operator J α can be found, for example, in [START_REF] Podlubny | Fractional Differential Equations[END_REF][START_REF] Gorenflo | Fractional calculus: Integral and differential equations of fractional order[END_REF][START_REF] Oldham | The Fractional Calculus[END_REF]. We recall only the two following properties. For µ ≥ -1, α, β ≥ 0 and γ > -1, we have,

J α J β f (t) = J α+β f (t), J α t γ = Γ(γ + 1) Γ(α + γ + 1) t α+γ .
In this study, Caputo definition is used and the fractional derivative of f (t) is defined as, J m-α D m f (t),

D α f (t) = 1 Γ(m -α) t 0 f (m) (τ ) (t -τ ) α-m+1 dτ, (2) for m -1 < α ≤ m, m ∈ IN, t > 0.
Caputo definition has the advantage of dealing properly with initial value problems in which the initial conditions are given in terms of the field variables with their integer order which is the case in most physical processes. Fortunately, the Laplace transform of the Caputo fractional derivative satisfies,

L{D α f (t)} = s α L{f (t)} - m-1 k=0 f (k) (0 + )s α-1-k , (3) where m -1 < α ≤ m.
The Laplace transform of Caputo fractional derivative requires the knowledge of the (bounded) initial values of the function and of its integer derivatives of order k = 1, 2, • • • , m-1. It has been found that the solution of many fractional-order differential equations can be written in terms of Mittag-Leffler functions, which are relevant for their connection with fractional calculus. The one-parameter Mittag-Leffler function E α (z) is defined by [START_REF] Podlubny | Fractional Differential Equations[END_REF],

E α (z) = ∞ k=0 z k Γ(αk + 1) , α > 0, z ∈ C, (4)
which is an entire function. The Laplace transform for E α (z) is given by [START_REF] Podlubny | Fractional Differential Equations[END_REF],

L{E α (-λt α )} = s α-1 s α + λ . (5)
The Mittag-Leffler function provides a simple generalization of the exponential function. For studying the stability of fractional order systems, we need the final value theorem.

f (t) = lim s-→0 sF (s). (6)
The final value theorem gives information about the asymptotic behavior of f (t) (how f (t) behaves as t -→ +∞) directly from F (s). This result plays an important role in the derivation of the linear feedback control laws to achieve synchronization of the chaotic fractional-order systems.

Stability results of linear fractional systems

Stability of linear fractional systems, which is of main interest in control theory, has been thoroughly investigated where necessary and sufficient conditions have been derived, see [START_REF] Matignon | Stability results of fractional differential equations with applications to control processing[END_REF][START_REF] Deng | Stability analysis of linear fractional differential system with multiple time delays[END_REF] or references therein. In [START_REF] Matignon | Stability results of fractional differential equations with applications to control processing[END_REF], Matignon introduced the stability properties for some linear fractional-order systems. In [START_REF] Deng | Stability analysis of linear fractional differential system with multiple time delays[END_REF], Deng et al. studied the stability of n-dimensional linear fractional differential equations with time delays. In this section, we recall the main stability properties and then, we present a new property for a general class of fractional-order systems. For this object, we consider the following linear system of fractional differential equations,

d α dt α x(t) = Ax(t), x(0) = x 0 , (7)
where x ∈ IR n , the matrix

A ∈ IR n × IR n , α = [α 1 , α 2 , • • • , α n ] indicates the fractional orders, d α dt α = [ d α 1 dt α 1 , d α 2 dt α 2 , • • • , d αn dt αn ] and d α i
dt α i is the Caputo fractional derivative of order α i , where 0

< α i ≤ 1 for i = 1, 2, • • • , n. If α = α 1 = α 2 = • • • = α n ,
then the stability of the fractional-order system [START_REF] Petráš | A note on the fractional-order Chua's system[END_REF] has been studied in [START_REF] Matignon | Stability results of fractional differential equations with applications to control processing[END_REF], where necessary and sufficient conditions have been introduced. The following stability result was derived in [START_REF] Matignon | Stability results of fractional differential equations with applications to control processing[END_REF].

Theorem 2 The autonomous fractional-order system [START_REF] Petráš | A note on the fractional-order Chua's system[END_REF], when

α = α 1 = α 2 = • • • = α n , is asymptot- ically stable iff |arg(spec(A))| > απ/2.
In this case the components of the state decay towards 0 like t -α .

In case of α = 1, the stability occurs if no poles of the linear system [START_REF] Petráš | A note on the fractional-order Chua's system[END_REF] Theorem 3 Suppose that α i 's are rational numbers between 0 and 1,

for i = 1, 2, • • • , n. Let γ = 1/m
where m is the least common multiple of the denominators m i of α i 's, where

α i = k i /m i , k i , m i ∈ IN , i = 1, 2, • • • , n. Then system (7) is asymptotically sta- ble if all roots λ of the equation det diag(λ mα1 , λ mα2 , • • • , λ mαn ) -A = 0 satisfy |arg(λ)| > γπ/2. Now, if α 1 , α 2 , • • •, α n are
any positive numbers between 0 and 1, not necessary equals or rational numbers, then we have the following new result.

Theorem 4

The autonomous fractional-order system ( 7) is asymptotically stable if |arg(spec(A))| > π/2, that is a 11 , a 22 , • • • , a nn < 0, when A is an upper or lower triangular matrix.

Proof. For simplicity, we give the proof when A is an 3 × 3 lower triangular matrix. A similar way can be followed to prove the general theorem. Consider the fractional-order system,

       d α1 x dt α1 d α2 y dt α2 d α3 z dt α3        =    a 11 0 0 a 21 a 22 0 a 31 a 32 a 33       x y z    (8)
Applying Laplace transform to system (8), letting

X(s) = L((x(t)), Y (s) = L(y(t)), Z(s) = L(z(t)
) and using formula (3), we obtain,

                 s α1 X(s) -s α1-1 x(0) = a 11 X(s) s α2 Y (s) -s α2-1 y(0) = a 21 X(s) + a 22 Y (s) s α3 Z(s) -s α3-1 z(0) = a 31 X(s) + a 32 Y (s) + a 33 Z(s) (9)
It follows from Eqs. ( 9) that,

                                           X(s) = s α 1 -1 x(0) s α 1 -a11 Y (s) = s α 2 -1 y(0) s α 2 -a22 + a21s α 1 -1 x(0) (s α 1 -a11)(s α 2 -a22) Z(s) = s α 3 -1 z(0) s α 3 -a33 + a31s α 1 -1 x(0) (s α 1 -a11)(s α 3 -a33) + a32s α 2 -1 y(0) (s α 2 -a22)(s α 3 -a33) + a21a32s α 1 -1 x(0) (s α 1 -a11)(s α 2 -a22)(s α 3 -a33) (10)
In particular, using (5), we have x(t) = x(0) E α1 (a 11 t α1 ), where E α (t) is the Mittag-Leffler function. From the properties of this function, see [Gorenflo & Mainardi, 1997], we can conclude that x(t) -→ 0 as t -→ +∞ if a 11 < 0. In view of [START_REF] Ge | Chaos in a fractional order modified Duffing system[END_REF], all poles of sX(s), sY (s) and sZ(s) lie in the open left half plane if a 11 , a 22 , a 33 < 0. Hence, using the final value theorem we get,

       lim t-→∞ x(t) = lim s-→0 sX(s) = 0, lim t-→∞ y(t) = lim s-→0 sY (s) = 0, lim t-→∞ z(t) = lim s-→0 sZ(s) = 0. ( 11
)
As a result, the states x(t), y(t), z(t) decay towards 0, as t -→ +∞, if a 11 , a 22 , a 33 < 0. Therefore, in this case, system (8) is asymptotically stable.

Synchronization of fractionalorder systems

In this section, based on the stability results of fractional differential equations, we briefly discuss the is-sue of controlling fractional-order chaotic systems to realize synchronization with linear error feedback control. In order to observe synchronization behavior, we construct the master (drive) system and the slave (response) system as,

M aster                                    d α1 x m dt α1 = b 11 x m + b 12 y m + b 13 z m +g 1 (x m , y m , z m , t) d α2 y m dt α2 = b 21 x m + b 22 y m + b 23 z m +g 2 (x m , y m , z m , t) d α3 z m dt α3 = b 31 x m + b 32 y m + b 33 z m +g 3 (x m , y m , z m , t) (12)
Slave

                                 d α1 x s dt α1 = b 11 x s + b 12 y s + b 13 z s +g 1 (x m , y m , z m , t) + u 1 (t) d α2 y s dt α2 = b 21 x s + b 22 y s + b 23 z s +g 2 (x m , y m , z m , t) + u 2 (t) d α3 z s dt α3 = b 31 x s + b 32 y s + b 33 z s +g 3 (x m , y m , z m , t) + u 3 (t) (13)
where d α i dt α i is the fractional differential operator in Caputo sense, 0 < α i ≤ 1, g i is a nonlinear function and b ij are real constants, for i, j = 1, 2, 3. Subscripts m and s stand for the master system and slave system, respectively, u(t) = [u 1 (t), u 2 (t), u 3 (t)] T is the linear controller to be designed such that these two chaotic systems synchronize. Defining the synchronization error as,

e 1 (t) = x s (t) -x m (t) e 2 (t) = y s (t) -y m (t) e 3 (t) = z s (t) -z m (t)
we get the error system, Our aim is to determine the controller u(t), given below, such that the drive system [START_REF] Li | Chaos and hyperchaos in fractional order Rössler equations[END_REF] and the response system [START_REF] Li | Chaos in Chen's system with a fractional order[END_REF] are synchronized ( e(t) -→ 0, as t -→ +∞).

                     d α1
u(t) =    u 1 (t) u 2 (t) u 3 (t)    =    a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33    •    e 1 (t) e 2 (t) e 3 (t)    (15)
where a ij are real constants. For this purpose, we design the controller u(t) = [u 1 (t), u 2 (t), u 3 (t)] T such that the solutions of the error system e i (t) go to 0, i = 1, 2, 3, as t goes to +∞. In this paper, three cases are studied to design the controller u(t).

Case 1. If α = α 1 = α 2 = α 3 then, using Theorem 2, the drive system (12) and the response system (13) are synchronized when |arg(spec(A u ))| > απ/2, where,

A u =    a 11 + b 11 a 12 + b 12 a 13 + b 13 a 21 + b 21 a 22 + b 22 a 23 + b 23 a 31 + b 31 a 32 + b 32 a 33 + b 33    (16)
In this case, we choose the controller u(t) such that |arg(λ)| > απ/2 for every eigenvalue λ of the matrix A u . Of course, if all eigenvalues of A u are negative real numbers then both systems achieve synchronization.

Case 2. If α i 's are rational numbers such that α i = k i /m i , γ = 1/m where m is the least common multiple of the denominators m i of α i 's, k i , m i ∈ IN , for i = 1, 2, 3, then, using Theorem 3, the drive system ( 12) and the response system (13) are synchronized when all roots λ of the characteristic equation det diag(λ mα1 , λ mα2 , λ mα3 ) -A u = 0 satisfy |arg(λ)| > γπ/2. In this case, we choose the controller u(t) such that all the roots of the equation det diag(λ r1 , λ r2 , λ r3 ) -A u = 0 satisfy |arg(λ)| > γπ/2, where r i = mk i /m i .

Case 3. In particular, if α 1 , α 2 , α 3 are any positive real numbers between 0 and 1 then, using Theorem 4, the drive system ( 12) and the response system (13) are synchronized when A u is an upper or lower triangular matrix and all eigenvalues of A u are negative real numbers. In this case we choose the controller u(t) such that A u becomes an upper or lower triangular matrix and all roots of the characteristic equation

(λ -a 11 -b 11 )(λ -a 22 -b 22 )(λ -a 33 -b 33 ) = 0 are negative real numbers.
To demonstrate these techniques, some examples of synchronization for two identical fractional-order chaotic systems are discussed in the following sections.

Synchronization of fractionalorder Chen systems

Here, we consider Chen system, which has been found by Chen and Ueta [1999] in 1999. Its fractional version reads as,

                     d α1 x dt α1 = a(y -x) d α2 y dt α2 = (c -a)x -xz + cy d α3 z dt α3 = xy -bz (17)
where [START_REF] Arneodo | Asymptotic chaos[END_REF] is the original integer order Chen system, which is chaotic, for example, when (a, b, c) = [START_REF] Oldham | The Fractional Calculus[END_REF][START_REF] Ott | Controling chaos[END_REF][START_REF] Wang | Extending synchronization scheme to chaotic fractional-order Chen systems[END_REF], see [START_REF] Chen | Yet another chaotic attractor[END_REF]. Simulations are performed to obtain chaotic behavior of the fractional-order Chen system for different fractional orders α. Simulation results demonstrate that chaos indeed exists in the fractional-order Chen system with order less than 3 (i.e. α 1 + α 2 + α 3 < 3). For example, chaotic attractors are found in [Yan & Li, 2007] when α = (0.95, 0.95, 0.95) and (a, b, c) = [START_REF] Oldham | The Fractional Calculus[END_REF][START_REF] Ott | Controling chaos[END_REF][START_REF] Wang | Extending synchronization scheme to chaotic fractional-order Chen systems[END_REF]. In [Li & Chen, 2004b] chaotic behaviors are found when α = (0.9, 0.9, 0.9) and (a, b, c) = [START_REF] Oldham | The Fractional Calculus[END_REF][START_REF] Ott | Controling chaos[END_REF][START_REF] Wang | Extending synchronization scheme to chaotic fractional-order Chen systems[END_REF]. Moreover, in [START_REF] Wang | Extending synchronization scheme to chaotic fractional-order Chen systems[END_REF], it is found that for the parameters α = (0.985, 0.99, 0.98) and (a, b, c) = (35, 3, 28) the fractional-order Chen system can display chaotic attractors. The chaotic attractors of fractional Chen system (17) for some fractional orders are shown in Figs. 1234. According to our approach, the drive and the response fractional-order Chen systems are described as,

α = (α 1 , α 2 , α 3 ) is subject to 0 < α 1 , α 2 , α 3 ≤ 1. When α = (1, 1, 1), system
      d α 1 xm dt α 1 d α 2 ym dt α 2 d α 3 zm dt α 3       =   -a a 0 c -a c 0 0 0 -b     xm ym zm   +   0 -xmzm xmym   (18)       d α 1 xs dt α 1 d α 2 ys dt α 2 d α 3 zs dt α 3       =   -a a 0 c -a c 0 0 0 -b     xs ys zs   +   0 -xmzm xmym   +   u1(t) u2(t) u3(t)   ( 19
)
where the controller u(t) = [u 1 (t), u 2 (t), u 3 (t)] T is defined in [START_REF] Lu | A note on the fractionalorder Chen system[END_REF]. Then the error system gives,

d α dt α e(t) = B u e(t), (20)
where e(t) = (e 1 (t), e 2 (t), e 3 (t)) and,

B u =   
                 a 12 = -a a 13 = 0 a 23 = 0 a 11 < a a 22 < -c a 33 < b (21)
or,

                 a 21 = a -c a 31 = 0 a 32 = 0 a 11 < a a 22 < -c a 33 < b (22)
Example 1 Taking α = (0.95, 0.95, 0.95), (a, b, c) = (35, 3, 28) and u(t) = (37, -33, 0), (2, -31, 0), (0, 0, -5) e(t), then the roots of the equation det λI -B u = 0 are,

λ 1 = -5, λ 2 = 1 2 -1 + √ 15i , λ 3 = 1 2 -1 - √ 15i ,
where i 2 = -1. Of course, all roots λ 1 , λ 2 , λ 3 have negative real parts, and so they lie in the region |arg(λ)| > (0.95)π/2. Therefore, the drive system (18) and the response system [START_REF] Li | Synchronization of fractional order chaotic systems[END_REF] are synchronized.

The error functions evolution, in this case, is shown in Fig. 5 (a).

Example 2 If we take α = (0.9, 0.9, 0.9), (a, b, c) = (35, 3, 28) and u(t) = (32, -35, 0), (7, -26, -20), (0, 30, 2) e(t), then the roots of the equation det λI -B u = 0 are,

λ 1 = -3, λ 2 = 1 2 1 + √ 2391 i , λ 3 = 1 2 1 - √ 2391 i .
Even the eigenvalues λ 2 and λ 3 have positive real parts, systems ( 18) and ( 19) are synchronized since |arg(λ 2,3 )| > (0.9)π/2. The error functions evolution, in this case, is shown in Fig. 5 (b).

Example 3 If we take α = (0.985, 0.99, 0.98), (a, b, c) = [START_REF] Oldham | The Fractional Calculus[END_REF][START_REF] Ott | Controling chaos[END_REF][START_REF] Wang | Extending synchronization scheme to chaotic fractional-order Chen systems[END_REF] and u(t) = (25, 0, 0), (7, -33, 5), (0, 0, 1) e(t), we have r 1 = 985, r 2 = 990, r 3 = 980 and m = 1000 and then the equation det diag(λ 985 , λ 990 , λ 980 ) -B u = 0 can be written as, (λ 985 + 10)(λ 990 + 5)(λ 980 + 2) = 0.

Using simple calculations, we can show that all roots of above equation lie in the region |arg(λ)| > π/2000. Therefore, both systems ( 18) and ( 19) are synchronized. Also, we can conclude that the systems are synchronized since the controller u(t) satisfies the control law [START_REF] Peng | Synchronization of fractional order chaotic systems[END_REF]. The error functions evolution, in this case, is shown in Fig. 5 (c).

Example 4 If we take α = (0.985, 0.99, 0.98), (a, b, c) = [START_REF] Oldham | The Fractional Calculus[END_REF][START_REF] Ott | Controling chaos[END_REF][START_REF] Wang | Extending synchronization scheme to chaotic fractional-order Chen systems[END_REF] and u(t) = (27, -35, 0), (8, -30, 0), (5, 0, 2.5) e(t), then u(t) satisfies the control law [START_REF] Zhou | Chaos control and synchronization in a fractional neuron network system[END_REF], and so both systems ( 18) and ( 19) are synchronized. The error functions evolution, in this case, is shown in Fig. 5 (d).

Figure 5 From Fig. 5, it is obvious that the components of the error system (20) decay towards zero as t goes to +∞. Therefore, we can numerically conclude that the designed linear controller can effectively control the chaotic fractional-order Chen system to achieve synchronization between the drive system (18) and the response system [START_REF] Li | Synchronization of fractional order chaotic systems[END_REF].

Synchronization of fractionalorder Rössler systems

Now, we consider the Rössler system, which has been found by [START_REF] Rössler | An equation for continuous chaos[END_REF] in 1976. Its fractional version reads as,

                     d α1 x dt α1 = -(y + z) d α2 y dt α2 = x + ay d α3 z dt α3 = z(x -c) + b (23)
where 23) is the original integer order Rössler system, which is chaotic, for example, when (a, b, c) = (0.2, 0.2, 5), see Fig 6 . Simulations are performed to obtain chaotic behavior of the fractional-order Rössler system and the results demonstrate that chaos indeed exists with order less than 3 (i.e. α 1 + α 2 + α 3 < 3). For example, chaotic attractors are found in [Li & Chen, 2004a] when α = (0.9, 0.9, 0.9) and (a, b, c) = (0.4, 0.2, 10). Also, in [START_REF] Peng | Synchronization of fractional order chaotic systems[END_REF] it is found that for the parameters α = (1, 1, 0.8) and (a, b, c) = (0.2, 0.2, 5) the fractional-order Rössler system can display chaotic behaviors. Moreover, we find that chaotic behaviors can be displayed for the fractional-order Rössler system when (a, b, c) = (0.2, 0.2, 5) and α = (0.95, 0.95, 0.95) or α = (0.9, 0.9, 0.9), see Figs. 7 and8.

α = (α 1 , α 2 , α 3 ) is subject to 0 < α 1 , α 2 , α 3 ≤ 1. When α = (1, 1, 1), system (

Figure 6

Figure 7 Figure 8 According to our approach, the drive and the response fractional-order Rössler systems are described as,

      d α 1 xm dt α 1 d α 2 ym dt α 2 d α 3 zm dt α 3       =   0 -1 -1 1 a 0 0 0 -c     xm ym zm   +   0 0 xmzm + b   (24)       d α 1 xs dt α 1 d α 2 ys dt α 2 d α 3 zs dt α 3       =   0 -1 -1 1 a 0 0 0 -c     xs ys zs   +   0 0 xmzm + b   +   u1(t) u2(t) u3(t)   (25)
where the controller u(t) = [u 1 (t), u 2 (t), u 3 (t)] T is defined in [START_REF] Lu | A note on the fractionalorder Chen system[END_REF]. Then the error system gives,

d α dt α e(t) = C u e(t), (26)
where,

C u =    a 11 a 12 -1 a 13 -1 a 21 + 1 a 22 + a a 23 a 31 a 32 a 33 -c    .
In view of case 1, if α 1 = α 2 = α 3 = α, then the drive system [START_REF] Deng | Stability analysis of linear fractional differential system with multiple time delays[END_REF] and the response system [START_REF] Sheu | Chaos in a new system with fractional order[END_REF] are synchronized if all roots of the equation det λI -C u = 0 lie in the region |arg(λ)| > απ/2, see example 5 below. Also, in view of case 2, if

α 1 = k 1 /m 1 , α 2 = k 2 /
m 2 and α 3 = k 3 /m 3 are rational numbers, then both systems are synchronized if all roots of the equation det diag(λ r1 , λ r2 , λ r3 ) -C u = 0 lie in the region |arg(λ)| > π/2m, where m is the least common multiple of {m 1 , m 2 , m 3 } and r i = mα i , see example 6 below. Moreover, in particular and according to case 3, if α 1 , α 2 , α 3 are any positive real numbers between 0 and 1, then synchronization of the drive system (24) and the response system ( 25) is sufficiently achieved under the control laws,

                 a 12 = -1 a 13 = 1 a 23 = 0 a 11 < 0 a 22 < -a a 33 < c (27) or,                  a 21 = -1 a 31 = 0 a 32 = 0 a 11 < 0 a 22 < -a a 33 < c (28)
Example 5 Taking α = (0.9, 0.9, 0.9), (a, b, c) = (0.4, 0.2, 10) and u(t) = (2, 9, 1), (-18, -2.4, 0), (0, 0, 7) e(t), then the roots of the equation det λI -C u = 0 are,

λ 1 = -3, λ 2 = √ 132 i, λ 3 = - √ 132 i.
Since |arg(λ 1,2,3 )| > (0.9)π/2, the drive system ( 24) and the response system (25) are synchronized. The error functions evolution, in this case, is shown in Fig. 9 (a).

Example 6 If we take α = (1, 1, 0.8), (a, b, c) = (0.2, 0.2, 5) and u(t) = (-2, -2, 1), (6, -1.2, 0), (0, 0, -2) e(t), we have r 1 = 10, r 2 = 10, r 3 = 8 and m = 10 and then the equation det diag(λ 10 , λ 10 , λ 8 )-C u = 0 can be written as,

(λ 8 + 2)(λ 20 + 3λ 10 + 20) = 0.
Simply, we can show that all roots of above equation lie in the region |arg(λ)| > π/20. Therefore, both systems ( 24) and ( 25) are synchronized. The error functions evolution, in this case, is shown in Fig. 9 (b).

From Fig. 9, for the given parameters, we can conclude that the components of the error system [START_REF] Yan | On chaos synchronization of fractional differential equations[END_REF] decay towards zero as t goes to +∞, and so the drive system (24) and the response system [START_REF] Sheu | Chaos in a new system with fractional order[END_REF] are synchronized.

Synchronization of fractionalorder cubic Chua systems

Now, we consider the Chua's equations [START_REF] Chua | Canonical realization of Chua's circuit family[END_REF][START_REF] Chua | Chaos synchronization in Chua's circuit[END_REF] with cubic nonlinearity (i.e. cubic Chua system), see for example [START_REF] Huang | Chua's equation with cubic nonlinearity[END_REF]. Its fractional version reads as,

                     d α1 x dt α1 = a(y + bx + cx 3 ) d α2 y dt α2 = x -y + z d α3 z dt α3 = -βy -γz (29)
where α = (α 1 , α 2 , α 3 ) is subject to 0 < α 1 , α 2 , α 3 ≤ 1. We fix the parameters as, (a, b, c) = (9.5, 0.15, -0.3) , (β, γ) = (14, 0.02), [START_REF] Zhu | Chaos and synchronization of the fractional-order Chua's system[END_REF] and we perform numerical simulations to obtain chaotic behaviors of the fractional-order cubic Chua system. First of all, the integer case α = (1, 1, 1) is studied and chaotic attractors are found as shown in Fig. 10. Then, in the fractional-order case, with α = (0.98, 0.98, 0.98) or α = (0.96, 0.96, 0.96), chaotic behaviors are also obtained as shown in Figs. 11 and 12.

Figure 10

Figure 11 Figure 12 According to our approach, the drive and the response systems for the fractional-order cubic Chua system are described as,

      d α 1 xm dt α 1 d α 2 ym dt α 2 d α 3 zm dt α 3       =   -ab a 0 1 -1 1 0 -β -γ     xm ym zm   +   -acx 3 m 0 0   (31)       d α 1 xs dt α 1 d α 2 ys dt α 2 d α 3 zs dt α 3       =   -ab a 0 1 -1 1 0 -β -γ     xs ys zs   +   -acx 3 m 0 0   +   u1(t) u2(t) u3(t)   (32)
Then, the error system gives,

d α dt α e(t) = D u e(t), (33)
where,

D u =    a 11 -ab a 12 + a a 13 a 21 + 1 a 22 -1 a 23 + 1 a 31 a 32 -β a 33 -γ    .
Similar to the previous systems, in view of case 1, if α 1 = α 2 = α 3 = α, then the drive system (31) and the response system [START_REF] Yu | The synchronization of fractional-order Rössler hyperchaotic systems[END_REF] are synchronized if all roots of the equation det λI -D u = 0 lie in the region |arg(λ)| > απ/2. Also, in view of case 2, if α 1 = k 1 /m 1 , α 2 = k 2 /m 2 and α 3 = k 3 /m 3 are rational numbers, then both systems are synchronized if all roots of the equation det diag(λ r1 , λ r2 , λ r3 ) -D u = 0 lie in the region |arg(λ)| > π/2m, where m is the least common multiple of {m 1 , m 2 , m 3 } and r i = mα i . Moreover, in particular and according to case 3, as in examples 7 and 8, if α 1 , α 2 , α 3 are any positive real numbers between 0 and 1, then synchronization of the drive system (31) and the response system [START_REF] Yu | The synchronization of fractional-order Rössler hyperchaotic systems[END_REF] is sufficiently achieved under the control laws,

                 a 12 = -a a 13 = 0 a 23 = -1 a 11 < ab a 22 < 1 a 33 < γ (34) or,                  a 21 = -1 a 31 = 0 a 32 = β a 11 < ab a 22 < 1 a 33 < γ (35)
Example 7 Taking α = (0.98, 0.98, 0.98), the same parameters as given in Eq.

(30) and u(t) = (-2.5, -9.5, 0), (2, -3, -1), (5, 3, -2) e(t), then u(t) satisfies the control law [START_REF] Gorenflo | Fractional calculus: Integral and differential equations of fractional order[END_REF], and so both systems [START_REF] Wu | Chaos in the fractional order unified system and its synchronization[END_REF] and ( 32) are synchronized. The error functions evolution, in this case, is shown in Fig. 13 (a).

Example 8 If we take α = (0.96, 0.96, 0.96), the same parameters as given in Eq. ( 30) and u(t) = (-4, 3, 5), (-1, 0.2, 1), (0, 14, -3) e(t), then u(t) satisfies the control law [START_REF] Oldham | The Fractional Calculus[END_REF], and so both systems [START_REF] Wu | Chaos in the fractional order unified system and its synchronization[END_REF] and ( 32) are synchronized. The error functions evolution, in this case, is shown in Fig. 13 (b).

Figure 13

From Fig. 13, for the given parameters, we can conclude that the components of the error system [START_REF] Caputo | Linear models of dissipation whose Q is almost frequency independent, Part II[END_REF] decay towards zero as t -→ +∞, and so the drive system (31) and the response system (32) are synchronized.

Conclusion

In this paper, we present theoretical results for drive-response synchronization between fractionalorder systems, based on stability results of linear fractional-order systems. Suitable sufficient conditions for achieving synchronization of two identical fractional-order systems via a suitable linear controller applied to the response system are derived. We briefly discuss the issue of controlling fractional-order chaotic Chen, Rössler and modified Chua systems to realize synchronization with linear error feedback control. For this purpose, we design the controller such that the components of the error system decay towards zero as t goes to +∞. Numerical simulations are given to verify the effectiveness of the proposed synchronization scheme and to show that some typical chaotic systems can be synchronized. We emphasize that this scheme of of synchronization can be extended to other chaotic fractional-order systems. 17), when α = (0.9, 0.9, 0.9) and (a, b, c) = [START_REF] Oldham | The Fractional Calculus[END_REF][START_REF] Ott | Controling chaos[END_REF][START_REF] Wang | Extending synchronization scheme to chaotic fractional-order Chen systems[END_REF]. Figure 5: Synchronization of the fractional-order Chen system (17). It is obvious that the evolution of the error functions e1(t), e2(t), e3(t) → 0 as t → +∞. 23), when α = (0.9, 0.9, 0.9) and (a, b, c) = (0.2, 0.2, 5). Figure 9: Synchronization of the fractional-order Rössler system [START_REF] Matignon | Stability results of fractional differential equations with applications to control processing[END_REF]. It is obvious that the evolution of the error functions e1(t), e2(t), e3(t) → 0 as t → +∞. Figure 13: Synchronization of the fractional-order Chua cubic system [START_REF] Li | Chaos synchronization of the Chua system with a fractional order[END_REF]. It is obvious that the evolution of the error functions e1(t), e2(t), e3(t) → 0 as t → +∞.

  e 1 dt α1 = b 11 e 1 + b 12 e 2 + b 13 e 3 + u 1 (t) d α2 e 2 dt α2 = b 21 e 1 + b 22 e 2 + b 23 e 3 + u 2 (t) d α3 e 3 dt α3 = b 31 e 1 + b 32 e 2 + b 33 e 3 + u 3 (t)[START_REF] Li | Chaos in the fractional order Chen system and its control[END_REF]

Figure 1 Figure 2 Figure 3 Figure 4

 1234 Figure 1 Figure 2 Figure 3 Figure 4

Figure 1 :

 1 Figure 1: Chaotic attractor of the Chen system (17), when α = (1, 1, 1) and (a, b, c) = (35, 3, 28).

Figure 2 :

 2 Figure 2: Chaotic attractor of the Chen system (17), when α = (0.9, 0.9, 0.9) and (a, b, c) =[START_REF] Oldham | The Fractional Calculus[END_REF][START_REF] Ott | Controling chaos[END_REF][START_REF] Wang | Extending synchronization scheme to chaotic fractional-order Chen systems[END_REF].

Figure 3 :

 3 Figure 3: Chaotic attractor of the Chen system (17), when α = (0.85, 0.85, 0.85) and (a, b, c) = (35, 3, 28).

Figure 4 :

 4 Figure 4: Chaotic attractor of the Chen system (17), when α = (0.985, 0.99, 0.98) and (a, b, c) = (35, 3, 28).

Figure 6 :

 6 Figure 6: Chaotic attractor of the Rössler system (23), when α = (1, 1, 1) and (a, b, c) = (0.2, 0.2, 5).

Figure 7 :

 7 Figure 7: Chaotic attractor of the Rössler system (23), when α = (0.95, 0.95, 0.95) and (a, b, c) = (0.2, 0.2, 5).

Figure 8 :

 8 Figure 8: Chaotic attractor of the Rössler system[START_REF] Matignon | Stability results of fractional differential equations with applications to control processing[END_REF], when α = (0.9, 0.9, 0.9) and (a, b, c) = (0.2, 0.2, 5).

Figure 10 :

 10 Figure 10: Chaotic attractor of the Chua cubic system[START_REF] Li | Chaos synchronization of the Chua system with a fractional order[END_REF], when α = (1, 1, 1) and the parameters given in Eq.[START_REF] Zhu | Chaos and synchronization of the fractional-order Chua's system[END_REF].

Figure 11 :

 11 Figure 11: Chaotic attractor of the fractional Chua cubic system (29), when α = (0.98, 0.98, 0.98) and the parameters given in Eq. (30).

Figure 12 :

 12 Figure12: Chaotic attractor of the fractional Chua cubic system[START_REF] Li | Chaos synchronization of the Chua system with a fractional order[END_REF], when α = (0.96, 0.96, 0.96) and the parameters given in Eq.[START_REF] Zhu | Chaos and synchronization of the fractional-order Chua's system[END_REF].

  ; Deng & Li, 2005; Li et al., 2003; Li & Zhou, 2005; Zhou et al., 2008;

  al., 2007; Yan & Li, 2007; Li & Yan, 2007]. Numerical algorithms for chaos synchronization of fractional-order systems based on Laplace transform theory are presented in [Wang et al., 2006; Li et al., 2006; Zhu et al., in press; Wu et al., 2008; Yu & Li, 2008].

  lie in the closed right half plane of the Laplace plane which agrees with well-known results for ordinary linear differential systems. If α 1 , α 2 , • • •, α

n are rational positive numbers, then we have the following result, which is introduced in

[START_REF] Deng | Stability analysis of linear fractional differential system with multiple time delays[END_REF]

.

  In view of case 1, if α 1 = α 2 = α 3 = α, then the drive system[START_REF] Deng | Chaos synchronization of the fractional Lü system[END_REF] and the response system[START_REF] Li | Synchronization of fractional order chaotic systems[END_REF] are synchronized if all roots of the equation det λI -B u = 0 lie in the region |arg(λ)| > απ/2, see examples 1 and 2 below. Also, in view of case 2, ifα 1 = k 1 /m 1 , α 2 = k 2 /m 2 and α 3 = k 3 /m 3 are rational numbers, then both systems are synchronized if all roots of the equation det diag(λ r1 , λ r2 , λ r3 ) -B u = 0 lie in the region |arg(λ)| > π/2m, where m is the least common multiple of {m 1 , m 2 , m 3 } and r i = mα i , see example 3 below. Moreover, in particular and according to case 3, as in examples 3 and 4 below , if α 1 , α 2 , α 3 are any positive real numbers between 0 and 1, then synchronization of the drive system (18) and the response system[START_REF] Li | Synchronization of fractional order chaotic systems[END_REF] is sufficiently achieved under the control laws,

	a 12 + a a 32 a 21 + c -a a 22 + c a 11 -a a 31	a 13 a 33 -b a 23	   .