Alessandro Torcini

Dynamical phases of the Hindmarsh-Rose neuronal model: Studies of the transition from bursting to spiking chaos

The dynamical phases of the Hindmarsh-Rose neuronal model are analyzed in detail by varying the external current I. For increasing current values, the model exhibits a peculiar cascade of nonchaotic and chaotic period-adding bifurcations leading the system from the silent regime to a chaotic state dominated by bursting events. At higher I-values, this phase is substituted by a regime of continuous chaotic spiking and finally via an inverse period doubling cascade the system returns to silence. The analysis is focused on the transition between the two chaotic phases displayed by the model: one dominated by spiking dynamics and the other by bursts. At the transition an abrupt shrinking of the attractor size associated with a sharp peak in the maximal Lyapunov exponent is observable. However, the transition appears to be continuous and smoothed out over a finite current interval, where bursts and spikes coexist. The beginning of the transition (from the bursting side) is signaled from a structural modification in the interspike interval return map. This change in the map shape is associated with the disappearance of the family of solutions responsible for the onset of the bursting chaos. The successive passage from bursting to spiking chaos is associated with a progressive pruning of unstable long-lasting bursts.

Localization and equipartition of energy in the b-FPU chain: chaotic breathers

Abstract The evolution towards equipartition in the β-FPU chain is studied considering as initial condition the highest frequency mode. Above an analytically derived energy threshold, this zone-boundary mode is shown to be modulationally unstable and to give rise to a striking localization process. The spontaneously created excitations have strong similarity with moving exact breathers solutions. But they have a finite lifetime and their dynamics is chaotic. These chaotic breathers are able to collect very efficiently the energy in the chain. Therefore their size grows in time and they can transport a very large quantity of energy. These features can be explained analyzing the dynamics of perturbed exact breathers of the FPU chain. In particular, a close connection between the Lyapunov spectrum of the chaotic breathers and the Floquet spectrum of the exact ones has been found. The emergence of chaotic breathers is convincingly explained by the absorption of high frequency phonons whereas a breathers metastability is for the first time identified. The lifetime of the chaotic breather is related to the time necessary for the system to reach equipartition. The equipartition time turns out to be dependent on the system energy density e only. Moreover, such time diverges as e−2 in the limit e → 0 and vanishes as e −1 4 for e → ∞.

Modulated amplitude waves and defect formation in the one-dimensional complex Ginzburg-Landau equation

The transition from phase chaos to defect chaos in the complex Ginzburg–Landau equation (CGLE) is related to saddle-node bifurcations of modulated amplitude waves (MAWs). First, the spatial period P of MAWs is shown to be limited by a maximum PSN which depends on the CGLE coefficients; MAW-like structures with period larger than PSN evolve to defects. Second, slowly evolving near-MAWs with average phase gradients ν ≈ 0 and various periods occur naturally in phase chaotic states of the CGLE. As a measure for these periods, we study the distributions of spacings p between neighbouring peaks of the phase gradient. A systematic comparison of p and PSN as a function of coefficients of the CGLE shows that defects are generated at locations where p becomes larger than PSN. In other words, MAWs with period PSN represent “critical nuclei” for the formation of defects in phase chaos and may trigger the transition to defect chaos. Since rare events where p becomes sufficiently large to lead to defect formation may only occur after a long transient, the coefficients where the transition to defect chaos seems to occur depend on system size and integration time. We conjecture that in the regime where the maximum period PSN has diverged, phase chaos persists in the thermodynamic limit.

Modulated Amplitude Waves and the Transition from Phase to Defect Chaos

The mechanism for transitions from phase to defect chaos in the one-dimensional complex Ginzburg-Landau equation (CGLE) is presented. We describe periodic coherent structures of the CGLE, called modulated amplitude waves (MAWs). MAWs of various periods P occur in phase chaotic states. A bifurcation study of the MAWs reveals that for sufficiently large period, pairs of MAWs cease to exist via a saddle-node bifurcation. For periods beyond this bifurcation, incoherent near-MAW structures evolve towards defects. This leads to our main result: the transition from phase to defect chaos takes place when the periods of MAWs in phase chaos are driven beyond their saddle-node bifurcation.

Proton dynamics in supercooled water by molecular dynamics simulations and quasielastic neutron scattering

A detailed study of the single‐particle dynamics of liquid water in normal and supercooled regime has been carried out by comparing molecular dynamics (MD) simulation results with now available high resolution quasielastic neutron scattering (QENS) data. Simulation runs have been performed at 264, 280, 292, and 305 K, using the extended simple point charge model, well suited for reproducing single‐particle properties of H2O. The microscopic dynamics has been probed over a wide range of times and distances. The MD results indicate that a substantial coupling between translational and rotational dynamics exists already at about 1 ps. The decay of the translational dynamic correlations has been phenomenologically analyzed in terms of three exponential components, and the agreement between the parameters thus obtained from experimental and simulation derived datasets is quite satisfactory. Both QENS and MD data can not be described with sufficient accuracy by simple diffusion models over the entire range of e...

Molecular dynamics results for stretched water

Detailed computer simulation results of several static and dynamical properties of water, obtained by using a realistic potential model proposed by Jorgensen et al., in the supercooled region, at densities well below the coexistence curve, are reported. We have analyzed the structural properties by evaluating the volume distributions of Voronoi polyhedra as well as angular and radial distributions of molecular clusters. In particular, the homogeneity of the system has carefully been checked. The investigated dynamical properties mainly concern the density and temperature dependence of the diffusion coefficient. The results are compared both with previous simulations, performed with different models, and with experimental findings. Some differences stress the fact that the conclusions drawn on the physical process underlying density and temperature behavior, can strongly be influenced by the use of different potential models.

The Hamiltonian Mean Field Model: From Dynamics to Statistical Mechanics and Back

The thermodynamics and the dynamics of particle systems with infiniterange coupling display several unusual and new features with respect to systems with short-range interactions. The Hamiltonian Mean Field (HMF) model represents a paradigmatic example of this class of systems. The present study addresses both attractive and repulsive interactions, with a particular emphasis on the description of clustering phenomena from a thermodynamical as well as from a dynamical point of view. The observed clustering transition can be first or second order, in the usual thermodynamical sense. In the former case, ensemble inequivalence naturally arises close to the transition, i.e. canonical and microcanonical ensembles give different results. In particular, in the microcanonical ensemble negative specific heat regimes and temperature jumps are observed. Moreover, having access to dynamics one can study non-equilibrium processes. Among them, the most striking is the emergence of coherent structures in the repulsive model, whose formation and dynamics can be studied either by using the tools of statistical mechanics or as a manifestation of the solutions of an associated Vlasov equation. The chaotic character of the HMF model has been also analyzed in terms of its Lyapunov spectrum.

Equilibrium and dynamical properties of two-dimensional N -body systems with long-range attractive interactions

A system of N classical particles in a 2D periodic cell interacting via a longrange attractive potential is studied numerically and theoretically. For low energy density U a collapsed phase is identified, while in the high energy limit the particles are homogeneously distributed. A phase transition from the collapsed to the homogeneous state occurs at critical energy Uc. A theoretical analysis within the canonical ensemble identifies such a transition as first order. But microcanonical simulations reveal a negative specific heat regime near Uc. This suggests that the transition belongs to the universality class previously identified by Hertel and Thirring (Ann. of Physics, 63, 520 (1970)) for gravitational lattice gas models. The dynamical behaviour of the system is strongly affected by this transition : below Uc anomalous diffusion is observed, while for U > Uc the motion of the particles is almost ballistic. In e-mail: torcini@fi.infn.it, web : http://torcini.de.unifi.it/∼torcini e-mail: antoni@sebigbos.mpipks-dresden.mpg.de 1 the collapsed phase, finite N -effects act like a ”deterministic” noise source of variance O(1/N), that restores normal diffusion on a time scale that diverges with N . As a consequence, the asymptotic diffusion coefficient will also diverge algebraically with N and superdiffusion will be observable at any time in the limit N → ∞. A Lyapunov analysis reveals that for U > Uc the maximal exponent λ decreases proportionally to N−1/3 and vanishes in the mean-field limit. For sufficiently small energy, in spite of a clear non ergodicity of the system, a common scaling law λ ∝ U1/2 is observed for any initial conditions. In the intermediate energy range, where anomalous diffusion is observed, a strong intermittency is found. This intermittent behaviour is related to two different dynamical mechanisms of chaotization. PACS Numbers: 05.40.+j, 05.45.+b, 05.70.Fh, 64.60.Cn Typeset using REVTEX 2

Periodic orbits in coupled Hénon maps: Lyapunov and multifractal analysis

A powerful algorithm is implemented in a 1-d lattice of Henon maps to extract orbits which are periodic both in space and time. The method automatically yields a suitable symbolic encoding of the dynamics. The arrangement of periodic orbits allows us to elucidate the spatially chaotic structure of the invariant measure. A new family of specific Lyapunov exponents is defined, which estimate the growth rate of spatially inhomogeneous perturbations. The specific exponents are shown to be related to the comoving Lyapunov exponents. Finally, the zeta-function formalism is implemented to analyze the scaling structure of the invariant measure both in space and time.

Transition to stochastic synchronization in spatially extended systems

Spatially extended dynamical systems, namely coupled map lattices, driven by additive spatio-temporal noise are shown to exhibit stochastic synchronization. In analogy with low-dimensional systems, synchronization can be achieved only if the maximum Lyapunov exponent becomes negative for sufficiently large noise amplitude. Moreover, noise can suppress also the nonlinear mechanism of information propagation, which may be present in the spatially extended system. An example of phase transition is observed when both the linear and the nonlinear mechanisms of information production disappear at the same critical value of the noise amplitude. The corresponding critical properties cannot be estimated numerically with great accuracy, but some general argument suggests that they could be ascribed to the Kardar-Parisi-Zhang universality class. Conversely, when the nonlinear mechanism prevails on the linear one, another type of phase transition to stochastic synchronization occurs. This one is shown to belong to the universality class of directed percolation.