Boundary crisis and suppression of Fermi acceleration in a dissipative two-dimensional non-integrable time-dependent billiard

Some dynamical properties for a dissipative time-dependent oval-shaped billiard are studied. The system is described in terms of a four-dimensional nonlinear mapping. Dissipation is introduced via inelastic collisions of the particle with the boundary, thus implying that the particle has a fractional loss of energy upon collision. The dissipation causes profound modifications in the dynamics of the particle as well as in the phase space of the non-dissipative system. In particular, inelastic collisions can be assumed as an efficient mechanism to suppress Fermi acceleration of the particle. The dissipation also creates attractors in the system, including chaotic. We show that a slightly modification of the intensity of the damping coefficient yields a drastic and sudden destruction of the chaotic attractor, thus leading the system to experience a boundary crisis. We have characterized such a boundary crisis via a collision of the chaotic attractor with its own basin of attraction and confirmed that inelastic collisions do indeed suppress Fermi acceleration in two-dimensional time-dependent billiards.     

Dynamic modeling and simulation of a real world billiard

Gravitational billiards provide an experimentally accessible arena for testing formulations of nonlinear dynamics. We present a mathematical model that captures the essential dynamics required for describing the motion of a realistic billiard for arbitrary boundaries. Simulations of the model are applied to parabolic, wedge and hyperbolic billiards that are driven sinusoidally. Direct comparisons are made between the model?s predictions and previously published experimental data. It is shown that the data can be successfully modeled with a simple set of parameters without an assumption of exotic energy dependence.     

Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems

A rich variety of dynamical scenarios has been shown to occur when a fixed point of a non-smooth map undergoes a border-collision. This paper concerns a closely related class of discontinuity-induced bifurcations, those involving equilibria of n-dimensional piecewise-smooth flows. Specifically, transitions are studied which occur when a boundary equilibrium, that is one lying within a discontinuity manifold, is perturbed. It is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios. Conditions to classify among the possible simplest scenarios are given for piecewise-smooth continuous, Filippov and impacting systems. Also, we investigate the possible birth of other attractors (e.g. limit cycles) at a boundary-equilibrium bifurcation.     

Implications of heterogeneous inputs and connectivity on the synchronization in excitable networks

We study the combined implications of connectivity and heterogeneous inputs on the synchronization features of a one-dimensional chain of diffusively coupled FitzHugh Nagumo (FHN) systems. The uncoupled systems are triggered into a regime of chaotic firing by periodic parametric forces modeling external stimuli. Due to the parameter dispersion involved in randomly distributed amplitudes and/or phases of the forces the units are nonidentical and the firing events on the chain of uncoupled units will be asynchronous leading to a distribution of the spiking times. Interest is focused on mutually synchronized spikings arising through the coupling where the connectivity of the network may range from nearest-neighbor interaction to global interactions. From our studies we conclude that increasing the interaction radius does not necessarily entail better spike synchrony and the coupling strength plays a more important role than connectivity. It is found that for driving with random amplitudes together with random phases a critical interaction radius exists beyond which firing becomes suppressed if the coupling between the units is too strong. In such cases of ‘firing death’ the units perform only small-amplitude oscillations which are mutually synchronous. The optimal coupling for spike synchrony is of intermediate strength and altering the connectivity does not really matter for the degree of spike synchrony. Distinct to that, when all the phases are equal and only the amplitudes of the forces are randomly distributed enhanced spike synchrony is achieved for sufficiently strong coupling regardless of the interaction radius.     

The saturation bifurcation in coupled oscillators

We examine examples of weakly nonlinear systems whose steady states undergo a bifurcation with increasing forcing, such that a forced subsystem abruptly ceases to absorb additional energy, instead diverting it into an initially quiescent, unforced subsystem. We derive and numerically verify analytical predictions for the existence and behavior of such saturated states for a class of oscillator pairs. We also examine related phenomena, including zero-frequency response to periodic forcing, Hopf bifurcations to quasiperiodicity, and bifurcations to periodic behavior with multiple frequencies.     

Geometrical properties of local dynamics in Hamiltonian systems: The Generalized Alignment Index (GALI) method

We investigate the detailed dynamics of multi-dimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion, the number of deviation vectors, their linear (in)dependence and the spectrum of Lyapunov exponents. The different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics, leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on N-dimensional tori. More specifically we introduce the Generalized Alignment Index of order k () as the volume of a generalized parallelepiped, whose edges are k initially linearly independent unit deviation vectors with respect to the orbit studied whose magnitude is normalized to unity at every time step. We show analytically and verify numerically on particular examples of N-degree-of-freedom Hamiltonian systems that, for chaotic orbits, tends exponentially to zero with exponents that involve the values of several Lyapunov exponents. In the case of regular orbits, fluctuates around non-zero values for 2≤k≤N and goes to zero for N<k≤2N following power laws that depend on the dimension of the torus and the number m of deviation vectors initially tangent to the torus: ∝t^{−2(k−N)+m} if 0≤m<k−N, and ∝t^−(k−N) if m≥k−N. The is a generalization of the Smaller Alignment Index (SALI) (). However, provides significantly more detailed information on the local dynamics, allows for a faster and clearer distinction between order and chaos than SALI and works even in cases where the SALI method is inconclusive.     

Stability of nonstationary states of classical,many-body dynamical systems

We summarize recent arguments which show that for a broad class of classical, many-body dynamical model systems with short-range interactions (such as coupled maps, cellular automata, or partial differential equations), collectively chaotic states—nonstationary states wherein some Fourier amplitude varies chaotically in time—cannot occur generically. While chaos occurs ubiquitously on alocal level in such systems, the macroscopic state of the system typically remains periodic or stationary. This implies that the dimensionD of chaotic (strange) attractors must diverge with the linear sizeL of the system likeD(L/C)^d ind space dimensions, where (<) is the spatial coherence length. We also summarize recent work which demonstrates that in spatially isotropic systems that have short-range interactions and evolve (like coupled maps) in discrete time, periodic states are never stable under generic conditions. In spatially anisotropic systems, however, short-range interactions that exploit the anisotropy and so allow for the stabilization of periodic states do exist.     

Deterministic chaos and natural phenomena

The natural time seriesf ₀ F ₂, F10.7, and AE are analyzed and low-dimensional attractors are found, characterized by the correlation dimension and the lower bound of the Kolmogorov entropy. Sources of noise in natural time series are discussed and the concept of extended systems is introduced and used to explain why the number of data required to calculate the correlation dimension of natural time series is higher than that reported by other authors.     

Nonequilibrium potentials near instabilities

The existence of polynomial approximations for nonequilibrium potentials determined by a master equation near an instability of arbitrary codimension with diagonalizable linear part is studied. It is shown that the approximations exist, provided some relations are satisfied between the coefficients of the master equation.     

Control of pattern formation by time-delay feedback with global and local contributions

We consider the suppression of spatiotemporal chaos in the complex Ginzburg-Landau equation by a combined global and local time-delay feedback. Feedback terms are implemented as a control scheme, i.e., they are proportional to the difference between the time-delayed state of the system and its current state. We perform a linear stability analysis of uniform oscillations with respect to space-dependent perturbations and compare with numerical simulations. Similarly, for the fixed-point solution that corresponds to amplitude death in the spatially extended system, a linear stability analysis with respect to space-dependent perturbations is performed and complemented by numerical simulations.     

Synchronization of chaotic fractional-order systems via active sliding mode controller

In this paper, we propose a controller based on active sliding mode theory to synchronize chaotic fractional-order systems in master-slave structure. Master and slave systems may be identical or different. Based on stability theorems in the fractional calculus, analysis of stability is performed for the proposed method. Finally, three numerical simulations (synchronizing fractional-order Lü-Lü systems, synchronizing fractional order Chen-Chen systems and synchronizing fractional-order Lü-Chen systems) are presented to show the effectiveness of the proposed controller. The simulations are implemented using two different numerical methods to solve the fractional differential equations.     

Deciphering interactions of complex systems that do not satisfy detailed balance

We present an extension of a novel method for understanding complex systems, which has been applied to non-equilibrium systems both in and out of detailed balance. For the non-detailed balance case, in which non-zero currents are a principal indicator of complexity, there has been an incomplete understanding of the distance function in the observable representation embedding. To deal with this we construct a new transition matrix by accounting for this current and compute the eigenvalues and eigenvectors. From these, we define a metric whose distance provides a useful measure of the relation among variables. Use of this method provides insights into long-range correlation, and chaotic properties. As an example we show that these distances can be used to control chaos in a simple dynamical system.     

Some remarks on the abundance of stable periodic orbits inside homoclinic lobes

We consider a family F_? of area-preserving maps (APMs) with a hyperbolic point H_? whose invariant manifolds form a figure-eight and we study the abundance of elliptic periodic orbits visiting homoclinic lobes (EPL), a domain typically dominated by chaotic behavior. To this end, we use the Chirikov separatrix map (SM) as a model of the return to a fundamental domain containing lobes. We obtain an explicit estimate, valid for families F_? with central symmetry and close to an integrable limit, of the relative measure of the set of parameters ? for which F_? has EPL trajectories. To get this estimate we look for EPL of the SM with the lowest possible period. The analytical results are complemented with quantitative numerical studies of the following families F_? of APMs:

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The SM family, and we compare our analytical results with the numerical estimates.

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The standard map (STM) family, and we show how the results referring to the SM model apply to the EPL visiting the lobes that the invariant manifolds of the STM hyperbolic fixed point form.

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The conservative Hénon map family, and we estimate the number of a particular type of symmetrical EPL related to the separatrices of the 4-periodic resonant islands.

The results obtained can be seen as the quantitative analogs to those in Simó and Treschev (2008) [9], although here we deal with the a priori stable situation instead.     

Measures with infinite Lyapunov exponents for the periodic Lorentz gas

We study invariant measures for the periodic Lorentz gas which are supported on the set of points with infinite Lyapunov exponents. We construct examples of such measures which are measures of maximal entropy and ones which are not.     

A new method for finding cycles by semilinear control

In the paper two new results that prove typically real and local univalent properties of certain class of rational functions in the unit disc $\mathbb{D}$ are presented. They provide a theoretical justification of a new method for finding periodic solutions of the discrete vector dynamical systems. The main goal of the paper is to propose and investigate the two-delays generalization of the scheme introduced by Ö. Morgül. The analysis shows that the coefficients in the proposed method can be chosen in a way that improves the admissible range of parameters. The method is elementary and easy to implement. Furthermore, numerical experiments with several well known maps show effectiveness of the control scheme.     

Chaotic attractors in incommensurate fractional order systems

In this paper, based on the stability theorems in fractional differential equations, a necessary condition is given to check the existence of 1-scroll, 2-scroll or multi-scroll chaotic attractors in a fractional order system. This condition is proposed for incommensurate order systems in general, but in the special case it converts to the condition given in the previous works for the commensurate fractional order systems. Though the presented condition is only a necessary (and not sufficient) condition for the existence of chaos it can be used as a powerful tool to distinguish for what parameters and orders of a given fractional order system, chaotic attractors can not be observed and for what parameters and orders, the system may generate chaos. It can also be used as a tool to confirm or reject results of a numerical simulation. Some of the numerical results reported in the previous literature are confirmed by this tool.     

Synchronization dynamics of chemically coupled cells with activator–inhibitor pathways

Systems of interacting cells containing an activator–inhibitor pathway, regulating naturally in their inner parts their end-product concentrations through a sequence of biochemical reactions with feedback-loops: an end-product inhibition of the first substrate, and an autocatalytic activation of the end-product through an allosteric enzyme-mediated reaction are investigated. The individual cells are considered to be identical and are described by nonlinear differential equations recently proposed following the concerted transition model. The chemical and electrical coupling types, realized by exchange of metabolites across concentration of the cells are used in order to analyze the onset of phase and complete synchronization in the biochemical system. It is found that depending on the coupling nature and the range of coupling strength, cells enter into different synchronization regimes going from low-quality to high-quality synchronization. The synchronization manifold's stability is analyzed. The results are supported by numerical simulations using indicators such as the conditional Lyapunov exponents and the rate of change of the Lyapunov function. The results indicate that the system cannot completely synchronize under the single action of the chemical coupling. The combined effect of both chemical and electrical couplings is found to be of capital importance in the onset of complete synchronization and high quality synchronization.     

Study of bifurcations in a model Hartree-Fock calculation

The Hartree-Fock procedure is used to study the behaviour of the ground state of a system ofM spinless electrons distributed overN equivalent and equidistant sites (M ⩽N) as a function of the strength of the mutual repulsion between the electrons. Below a critical strength, all initial configurations are seen, after repeated iterations, to converge to a unique solution. Above this critical strength, in addition to the initial configurations which lead to a unique solution, there exist configurations which on repeated iterations give rise to stable two-period solutions. Although the number of independent stable two-period solutions depends on the coupling strength, for no value of the coupling are stable solutions of periodicity higher than two seen.