Narrow Results

The dynamics of the soliton in a self-attractive Bose–Einstein condensate under the gravity are investigated. First, we apply the inverse scattering method, which gives rise to equation of motion for the center-of-mass coordinate of the soliton. We analyze the amplitude-frequency characteristic for nonlinear resonance. Applying the Krylov–Bogoliubov method for the small parameters the dynamics of soliton on the phase plane are considered. Hamiltonian chaos under the action of the gravity on the Poincaré map are studied.

This paper considers an autonomous nonlinear system of differential equations derived in [Leipnik, 1979]. A criterion for the existence of closed orbits in similar systems is presented. Numerical results are made rigorous by the use of interval analytic techniques in establishing the existence of a periodic solution which is not asymptotically stable. The limitations of the method of locating orbits are considered when a promising candidate for a closed orbit is shown not to intersect itself.

This paper studies the chaotification problem of driving a continuous-time system to a chaotic state by using an impulsive control input. The controller is designed to ensure the controlled orbit be bounded and, meanwhile, have positive Lyapunov exponents. This is proved to be not only possible but also implementable near a stable limit cycle of the given system. Two numerical examples are given to illustrate the effectiveness of the proposed chaotification method.

In this paper we present a simplified model of a three-body problem. Place three parallel lines in the plane. Place one mass on each of the lines and let their positions evolve according to Newton's inverse square law of gravitation. We prove the KAM theory applies to our model and simulations are presented. We argue that this model provides an ideal, accessible entry point into the beautiful mathematics involved in the study of the three-body problem.

The dynamical behavior of two coupled parametrically excited van der Pol oscillators is investigated in this paper. A special road to chaos is explored in detail. Period-doubling bifurcation associated with one of the frequencies of the system may be observed, the other frequency of the coupled oscillators plays a role in the evolution. It is found that one of the frequencies of the system contributes to the cascade of period-doubling bifurcations associated with the other frequency, which leads to a generalized modulated chaos.

This paper studies a foodweb model concerning hare–lynx-vegetation, and presents a computer assisted proof for existence of horseshoe for Poincaré map derived from this model.

In this paper we present a relationship between the algebraic notion of proper system, the geometric notion of contact point and the dynamic notion of Poincaré map for piecewise linear differential systems. This allows to present sufficient conditions (which are also necessary under additional hypotheses) for the existence of Poincaré maps in piecewise linear differential systems. Moreover, an adequate parametrization of the Poincaré maps make such maps invariant under linear transformations.

The homoclinic bifurcation and chaos in a system of weakly coupled simple pendulum and harmonic oscillator subject to light dampings and weakly external and (or) parametric excitation of bounded noise is studied. The random Melnikov process is derived and mean-square criteria is used to determine the threshold amplitude of the bounded noise for the onset of chaos in the system. The threshold amplitude is also determined by vanishing the numerically calculated maximal Lyapunov exponent. The threshold amplitudes are further confirmed by using the Poincaré maps, which indicate the path from periodic motion to chaos or from random motion to random chaos in the system as the amplitude of bounded noise increases.

In this paper we revisit the well-known Chua's circuit and give a discussion on entropy of this circuit. We present a formula for the topological entropy of a Chua's circuit in terms of the Poincaré map derived from the ordinary differential equations of this Chua's circuit by computer simulation arguments.

This paper presents, in a tutorial manner, nonlinear phenomena such as bifurcations and chaotic behavior in DC–DC switching converters. Our purpose is to present the different modeling approaches, the main results found in the last years and some possible practical applications. A comparison of the different models is given and their accuracy in predicting nonlinear behavior is discussed. A general Poincaré map is considered to model any multiple configuration of DC–DC switching converters and its Jacobian matrix is derived for stability analysis. More emphasis is done in the discrete-time approach as it gives more accurate prediction of bifurcations. The results are reproduced for different examples of DC–DC switching converters studied in the literature. Some methods of controlling bifurcations are applied to stabilize Unstable Periodic Orbits (UPOs) embedded in the dynamics of the system. Statistical analysis of these systems working in the chaotic regime is discussed. An extensive list of references is included.

Due to strong contraction in the outer regions of Chua's circuit with piecewise-linear dynamics, the system can be described by an approximate one-dimensional map. We confirm by simulation and experiment that the initial slope of this map is defined by .

The transition from periodicity to chaos in a DC-DC Buck power converter is studied in this paper. The converter is controlled through a direct Pulse Width Modulation (PWM) in order to regulate the error dynamics at zero. Results show robustness with low output error and a fixed switching frequency. Furthermore, some rich dynamics appear as the constant associated with the first-order error dynamics decreases. Finally, a transition from periodicity to chaos is observed. This paper describes this transition and the bifurcations in the converter. Chaos appears in the system with a stretching and folding mechanism. It can be observed in the one-dimensional Poincaré map of the inductor current. This Poincaré map converges to a tent map with the variation of the system parameter k_s.

In this paper, we demonstrate chaos in low dimensional cellular neural networks for some weight matrices. To verify chaoticity of the dynamics in these cellular neural networks, we consider a cross-section properly chosen for the attractors obtained and study the dynamics of the corresponding Poincaré maps, and rigorously verify the existence of horseshoe in the manner of computer-assisted proof arguments.

We obtain numerically a horseshoe in a Poincaré map derived from a cellular neural network described by four-dimensional autonomous ordinary differential equations. Contrary to the horseshoe numerically found in the Hodgkin–Huxley model, which showed evidence that the Poincaré map derived from the Hodgkin–Huxley model has just one expanding direction on some invariant subset, the horseshoe obtained in this paper proves that the Poincaré map derived from the neural network have two expanding directions on some invariant subset.

In this paper, we study a new class of simple three-neuron chaotic cellular neural networks with very simple connection matrices. To study the chaotic behavior in these cellular neural networks demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a rigorous verification of the existence of horseshoe chaos using topological horseshoe theory and the estimate of topological entropy in derived Poincaré maps.

The bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p₁ and one hyperbolic saddle p₂ are investigated, where p₁ is assumed to undergo transcritical bifurcation. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic loop connecting p₁ (resp. p₂) and the coexistence of one homoclinic loop and one periodic orbit are established. Secondly, we analyze bifurcations of heteroclinic loop accompanied by transcritical bifurcation, namely, nonhyperbolic equilibrium p₁ splits into two hyperbolic saddles and , a heteroclinic loop connecting and p₂, homoclinic loop with (resp. p₂) and heteroclinic orbit joining and (resp. and p₂; p₂ and ) are found. The results achieved here can be extended to higher dimensional systems.

In this paper, chaos in the smooth Chua's equation is revisited. To confirm the chaotic behavior in the smooth Chua's equation demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a computer assisted verification of existence of horseshoe chaos by virtue of topological horseshoes theory.

In this paper, we use the method of displacement functions to study the existence, stability and bifurcation of periodic solutions of scalar periodic impulsive differential equations. We obtain some new and interesting results on saddle-node bifurcation and double-period bifurcation of periodic solution.

In this paper, the general perturbation of piecewise Hamiltonian systems on the plane is considered. When the unperturbed system has a family of periodic orbits, similar to the perturbations of smooth system, an expression of the first order Melnikov function is derived, which can be used to study the number of limit cycles bifurcated from the periodic orbits. As applications, the number of bifurcated limit cycles of several concrete piecewise systems are presented.

Besides being structurally unstable, the Lotka–Volterra predator-prey model has another shortcoming due to the invalidity of the principle of mass action when the populations are very small. This leads to extremely large populations recovering from unrealistically small ones. The effects of linear modifications to structurally unstable continuous-time predator-prey models in a (small) neighbourhood of the origin are investigated here. In particular, it is shown that typically either a global attractor or repeller arises depending on the choice of coefficients.

The analysis is based on Poincaré mappings, which allow an explicit representation for the classical Lotka–Volterra equations.