Global periodic structure of integrable hyperbolic map

Integrable hyperbolic mappings are constructed within a scheme presented by Suris. The Cosh map is a singular map, of which fixed point is unstable. The global behavior of periodic orbits of the Sinh map is investigated referring to the Poincaré–Birkhoff resonance condition. Close to the fixed point, the periodicity is indeed determined from the Poincaré–Birkhoff resonance condition. Increasing the distance from the fixed point, the orbit is affected by the nonlinear effect and the average periodicity varies globally. The Fourier transformation of the individual orbits determines overall spectrum of global variation of the periodicity.     

Statistics of Poincaré recurrences in local and global approaches

The basic statistical characteristics of the Poincaré recurrence sequence are obtained numerically for the logistic map in the chaotic regime. The mean values, variance and recurrence distribution density are calculated and their dependence on the return region size is analyzed. It is verified that the Afraimovich–Pesin dimension may be evaluated by the Kolmogorov–Sinai entropy. The peculiarities of the influence of noise on the recurrence statistics are studied in local and global approaches. It is shown that the obtained numerical data are in complete agreement with the theoretical results. It is demonstrated that the Poincaré recurrence theory can be applied to diagnose effects of stochastic resonance and chaos synchronization and to calculate the fractal dimension of a chaotic attractor.     

Horseshoe chaos in a class of simple Hopfield neural networks

A class of new simple Hopfield neural networks is revisited. To confirm the chaotic behavior in these Hopfield neural networks demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a rigorous verification of existence of horseshoe chaos by virtue of topological horseshoes theory and estimates of topological entropy in the derived Poincaré maps.     

Flip bifurcations of an SIR epidemic model with birth pulse and pulse vaccination

In this paper, we study an SIR epidemic model with birth pulse and pulse vaccination. We present a new constructor method of Poincaré maps. Using this method, we construct a Poincaré map. However, for this Poincaré map, we can’t directly use the bifurcation theorem to discuss the existence of flip bifurcations. We use a new method to investigate the existence of flip bifurcations. We establish that the system undergoes flip bifurcation when the maximum birth rate passes some critical values. Furthermore, some numerical simulations are given to illustrate our results.     

Suppressing chaos of a complex Duffing's system using a random phase

The effect of random phase for a complex Duffing's system is investigated. We show as the intensity of random noise properly increases the chaotic dynamical behavior will be suppressed by the criterion of top Lyapunov exponent, which is computed based on the Khasminskii's formulation and the extension of Wedig's algorithm for linear stochastic systems. Also Poincaré map analysis, phase plot and the time evolution are carried out to confirm the obtained results of Lyapunov exponent on dynamical behavior including the stability, bifurcation and chaos. Thus excellent agreement between these results is found.     

Chaos in a nonautonomous eco-epidemiological model with delay

In this paper, we propose and analyze a nonautonomous predator-prey model with disease in prey, and a discrete time delay for the incubation period in disease transmission. Employing the theory of differential inequalities, we find sufficient conditions for the permanence of the system. Further, we use Lyapunov’s functional method to obtain sufficient conditions for global asymptotic stability of the system. We observe that the permanence of the system is unaffected due to presence of incubation delay. However, incubation delay affects the global stability of the positive periodic solution of the system. To reinforce the analytical results and to get more insight into the system’s behavior, we perform some numerical simulations of the autonomous and nonautonomous systems with and without time delay. We observe that for the gradual increase in the magnitude of incubation delay, the autonomous system develops limit cycle oscillation through a Hopf-bifurcation while the corresponding nonautonomous system shows chaotic dynamics through quasi-periodic oscillations. We apply basic tools of non-linear dynamics such as Poincaré section and maximum Lyapunov exponent to confirm the chaotic behavior of the system.     

Bifurcation and chaos of an axially accelerating viscoelastic beam

This paper investigates bifurcation and chaos of an axially accelerating viscoelastic beam. The Kelvin–Voigt model is adopted to constitute the material of the beam. Lagrangian strain is used to account for the beam's geometric nonlinearity. The nonlinear partial–differential equation governing transverse motion of the beam is derived from the Newton second law. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. By use of the Poincaré map, the dynamical behavior is identified based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented in the case that the mean axial speed, the amplitude of speed fluctuation and the dynamic viscoelasticity is respectively varied while other parameters are fixed. The Lyapunov exponent is calculated to identify chaos. From numerical simulations, it is indicated that the periodic, quasi-periodic and chaotic motions occur in the transverse vibrations of the axially accelerating viscoelastic beam.     

Alexander polynomials and Poincaré series of sets of ideals

Earlier the authors considered and, in some cases, computed Poincaré series for two sorts of multi-index filtrations on the ring of germs of functions on a complex (normal) surface singularity (in particular, on the complex plane). A filtration of the first class was defined by a curve (with several branches) on the surface singularity. A filtration of the second class (called divisorial) was defined by a set of components of the exceptional divisor of a modification of the surface singularity. Here we define and compute in some cases the Poincaré series corresponding to a set of ideals in the ring of germs of functions on a surface singularity. For the complex plane, this notion unites the two classes of filtrations described above.     

Chaos in singular systems

In this paper we consider singular systems of differential equations and we show that, under right conditions, the Poincaré map associated to those systems, and not just a suitable iterate, behaves chaotically. We use the notion of exponential dichotomies to prove the existence of a transverse homoclinic orbit of our system and after use the shadow lemma to show that the Poincaré map associated to its topologically conjugate to the Bernouilli shift on a set of two symbols. Entrata in Redazione il 3 aprile 1997 e, in versione riveduta, il 30 ottobre 1997.     

Classical perturbation theory and resonances in some rigid body systems

We consider the system of a rigid body in a weak gravitational field on the zero level set of the area integral and study its Poincaré sets in integrable and nonintegrable cases. For the integrable cases of Kovalevskaya and Goryachev–Chaplygin we investigate the structure of the Poincaré sets analytically and for nonintegrable cases we study these sets by means of symbolic calculations. Based on these results, we also prove the existence of periodic solutions in the perturbed nonintegrable system. The Chaplygin integrable case of Kirchhoff’s equations is also briefly considered, for which it is shown that its Poincaré sets are similar to the ones of the Kovalevskaya case.     

Dynamics of a generalized Gause-type predator–prey model with a seasonal functional response

We extend a previous Gause-type predator–prey model to include a general monotonic and bounded seasonally varying functional response. The model exhibits rich dynamical behaviour not encountered when the functional response is not seasonally forced. A theoretical analysis is performed on the model to investigate the global stability of the boundary equilibria and the existence of periodic solutions. It is shown that, under certain well-defined conditions, the Poincaré map of the model undergoes a Hopf bifurcation leading to the appearance of a quasi-periodic solution. Numerical results are given for the Poincaré sections and bifurcation diagrams for Holling-types II and III functional responses, using the amplitude of seasonal variation as bifurcation parameter. The model shows a rich variety of behaviour, including period doubling, quasi-periodicity, chaos, transient chaos, and windows of periodicity.     

Hopf bifurcation in an hexagonal governor system with a spring

The complex dynamical behaviors of the hexagonal governor system with a spring are studied in this paper. We go deeper investigating the stability of the equilibrium points in the hexagonal governor system with a spring. These systems have a rich variety of nonlinear behaviors, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. By studying numerical simulations, it is possible to provide reliable theory and effective numerical method for other systems.     

On computing Poincaré map by Hénon method

The trajectory of the autonomous chaotic system deviates from the original path leading to a deformation in its attractor while calculating Poincaré map using the method presented by Hénon [Hénon M. Physica D 1982;5:412]. Also, the Poincaré map obtained is found to be the Poincaré map of deformed attractor instead of the original attractor. In order to overcome these drawbacks, this method is slightly modified by introducing an important change in the existing algorithm. Then it is shown that the modified Hénon method calculates the Poincaré map of the original attractor and it does not affect the system dynamics (attractor). The modified method is illustrated by means of the Lorenz and Chua systems.     

Steady-state chaotic responses of a class of linear vibration absorbers

The steady state behavior of nonautonomous systems of two coupled nonlinear oscillators with small internal damping is analyzed by numerical integration of the motion equations, by varying the frequency of the periodical external excitation. A variety of periodic, quasi-periodic and chaotic oscillations are detected, whose properties are examined by means of Poincaré mappings, Lyapunov exponents spectra and fractal dimension measurements.     

Chaos control in passive walking dynamics of a compass-gait model

The compass-gait walker is a two-degree-of-freedom biped that can walk passively and steadily down an incline without any actuation. The mathematical model of the walking dynamics is represented by an impulsive hybrid nonlinear model. It is capable of displaying cyclic motions and chaos. In this paper, we propose a new approach to controlling chaos cropped up from the passive dynamic walking of the compass-gait model. The proposed technique is to linearize the nonlinear model around a desired passive hybrid limit cycle. Then, we show that the nonlinear model is transformed to an impulsive hybrid linear model with a controlled jump. Basing on the linearized model, we derive an analytical expression of a constrained controlled Poincaré map. We present a method for the numerical simulation of this constrained map where bifurcation diagrams are plotted. Relying on these diagrams, we show that the linear model is fairly close to the nonlinear one. Using the linearized controlled Poincaré map, we design a state feedback controller in order to stabilize the fixed point of the Poincaré map. We show that this controller is very efficient for the control of chaos for the original nonlinear model.     

Fractal dimension of a random invariant set

In recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche's techniques to this case, which can be overcome by careful use of the Poincaré recurrence theorem. We prove that under the same conditions as in Debussche's paper and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2d Navier–Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations.     

Global Symplectic Polynomial Approximation of Area-Preserving Maps

Different methods are proposed and tested for transforming a nonlinear differential system, and more particularly a Hamiltonian one, into a map without having to integrate the whole orbit as in the well known Poincaré map technique . We construct a synthetic map, based on a global fitting, which satisfies the symplectic condition. Taking the Standard Map as a model problem we point our attention on methods suitable for comparing the model map and its synthetic counterpart. We test the agreement of the fitting on finer scales through the visual representation, the computation of the rotation number and the measure of the local distribution of the Lyapunov characteristic exponents. Comparing these results with those obtained by Froeschlé and Petit using a method based on Taylor interpolation, we show that the symplectic character is a crucial condition for the recovering of the finest details of a dynamical system. On the other hand the global character of our method makes the generalization to any system of differential equations difficult.     

Arbitrary waveform generator biologically inspired

This work shows and analyzes a system that produces arbitrary waveforms, which is a simplification, based on spatial discretization, of the BVAM model proposed by Barrio et al. in 1999 [1] to model the biological pattern formation. Since the analytical treatment of non-linear terms of this system is often prohibitive, its dynamic has been analyzed using a discrete equivalent system defined by a Poincaré map. In this analysis, the bifurcation diagrams and the Lyapunov exponent are the tools used to identify the different operating regimes of the system and to provide evidence of the periodicity and randomness of the generated waveforms. Also, it is shown that the analyzed system presents the period doubling phenomenon, the values of its bifurcation points are related by the Feigenbaum constant and they converge to the onset of chaos. It is shown that, the analyzed system can be electronically implemented using operational amplifiers to produce arbitrary waveforms when varying a single control parameter. The functionality and behavior of the ideal electronic implementation of the analyzed system is shown by the simulations obtained from the MatLab–Simulink™ toolbox. Finally, some problems related to a real electronic implementation are discussed. This paper gives a brief overview of how ideas from biology can be used to design new systems that produce arbitrary waveforms.     

Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane

We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.