Shilnikov homoclinic orbit bifurcations in the Chua's circuit

We analytically describe the complex scenario of homoclinic bifurcations in the Chua's circuit. We obtain a general scaling law that gives the ratio between bifurcation parameters of different nearby homoclinic orbits. As an application of this theoretical approach, we estimate the number of higher order subsidiary homoclinic orbits that appear between two consecutive lower order subsidiary orbits. Our analytical finds might be valid for a large class of dynamical systems and are numerically confirmed in the parameter space of the Chua's circuit.     

Slow passage through a transcritical bifurcation for Hamiltonian systems and the change in action due to a nonhyperbolic homoclinic orbit

One-degree of freedom conservative slowly varying Hamiltonian systems are analyzed in the case in which a saddle-center pair undergo a transcritical bifurcation. We analyze the case in which the method of averaging predicts the solution crosses the unperturbed homoclinic orbit at the precise time at which the transcritical bifurcation occurs. For the slow passage through the nonhyperbolic homoclinic orbit associated with a transcritical bifurcation, the solution consists of a large sequence of nonhyperbolic homoclinic orbits surrounded by autonomous nonlinear saddle approaches. The change in action is computed by matching these solutions to those obtained by averaging, valid before and after crossing the nonhyperbolic homoclinic orbit. For initial conditions near the stable manifold of the nonhyperbolic saddle point, one saddle approach has particularly small energy and instead satisfies a nonautonomous nonlinear equation, which provides a transition between nonhyperbolic homoclinic orbits, centers, and saddles. (c) 2000 American Institute of Physics.     

Subharmonic and homoclinic bifurcations in a parametrically forced pendulum

Depending on the parameters of a parametrically forced pendulum system the boundaries of subharmonic and homoclinic bifurcations are calculated on the basis of the Melnikov method and of averaging methods. It is shown that, as a parameter is varied, repeated resonances of successively higher periods occur culminating in homoclinic orbits. According to the theorem of Smale homoclinic bifurcation is the source of the unstable chaotic motions observed. For some selected parameter sets the theoretical predictions are tested by numerical calculations. Very good agreement is found between analytical and numerical results.     

Renormalization invariance of a Hamiltonian system near the saddle point

The rescaling property of a one degree of freedom Hamiltonian system near the saddle point is analytically studied as regards the transformations of time-periodic perturbations. Two kinds of perturbation functions are considered: (a) linear functions and (b) homogeneous polynomial functions of canonical variables with time-periodic coefficients. The simple rescaling law of the phase-space of the Hamiltonian system near the hyperbolic fixed point with respect to a transformation of the amplitude and phase of the time-periodic perturbation is derived.     

Experimental bifurcations and homoclinic chaos in a laser with a saturable absorber

The shape and the peak values of the pulses from a passive Q-switching CO2 laser with SF6 as saturable absorber were detected while the laser was tuned in frequency across a longitudinal mode. A succession of stability windows, typical for bifurcation diagrams in the homoclinic scenario, was observed and the widths of those windows were measured. The expansion rate of the undulations in individual pulses was also obtained and compared to Floquet's multipliers given by the ratio of widths in consecutive windows. The dynamics is consistent with a homoclinic tangency to a periodic orbit.     

非线性传送带系统的复杂分岔

讨论了一类单自由度非线性传送带系统. 首先通过分段光滑动力系统理论得出系统滑动区域的解析分析和平衡点存在性条件; 其次利用数值方法, 对系统几种类型的周期轨道进行单参数和双参数延拓, 得到系统的余维一滑动分岔曲线和若干余维二滑动分岔点, 以及系统在参数空间中的全局分岔图. 通过对系统分岔行为的研究, 反映出传送带速度和摩擦力振幅对系统动力学行为有较大影响, 揭示了非线性传送带系统的复杂动力学现象. 关键词： 传送带系统 滑动分岔 周期运动     

Pulse dynamics in a three-component system: Stability and bifurcations

In this article, we analyze the stability and the associated bifurcations of several types of pulse solutions in a singularly perturbed three-component reaction-diffusion equation that has its origin as a model for gas discharge dynamics. Due to the richness and complexity of the dynamics generated by this model, it has in recent years become a paradigm model for the study of pulse interactions. A mathematical analysis of pulse interactions is based on detailed information on the existence and stability of isolated pulse solutions. The existence of these isolated pulse solutions is established in previous work. Here, the pulse solutions are studied by an Evans function associated to the linearized stability problem. Evans functions for stability problems in singularly perturbed reaction-diffusion models can be decomposed into a fast and a slow component, and their zeroes can be determined explicitly by the NLEP method. In the context of the present model, we have extended the NLEP method so that it can be applied to multi-pulse and multi-front solutions of singularly perturbed reaction-diffusion equations with more than one slow component. The brunt of this article is devoted to the analysis of the stability characteristics and the bifurcations of the pulse solutions. Our methods enable us to obtain explicit, analytical information on the various types of bifurcations, such as saddle-node bifurcations, Hopf bifurcations in which breathing pulse solutions are created, and bifurcations into travelling pulse solutions, which can be both subcritical and supercritical.     

Relativistic fluid dynamics as a Hamiltonian system

The equations of ideal relativistic fluid dynamics in the laboratory frame form a noncanonical hamiltonian system with the same Poisson bracket as for nonrelativistic fluids, but with dynamical variables and hamiltonian obtained via a regular deformation of their nonrelativistic counterparts.     

Autonomous bursting in a homoclinic system

The output of a dynamical system in a regime of homoclinic chaos transforms from a continuous train of irregularly spaced spikes to clusters of regularly spaced spikes with quiescent periods in between (bursting), provided a low frequency portion of the output is fed back. We provide experimental evidence of such an autonomous bursting by a CO2 laser with feedback. The phenomena here presented are extremely robust against noise and display qualitative analogies with bursting phenomena in neurons.     

Shell structure and orbit bifurcations in finite fermion systems

We first give an overview of the shell-correction method which was developed by V.M. Strutinsky as a practicable and efficient approximation to the general self-consistent theory of finite fermion systems suggested by A.B. Migdal and collaborators. Then we present in more detail a semiclassical theory of shell effects, also developed by Strutinsky following original ideas of M.C. Gutzwiller. We emphasize, in particular, the influence of orbit bifurcations on shell structure. We first give a short overview of semiclassical trace formulae, which connect the shell oscillations of a quantum system with a sum over periodic orbits of the corresponding classical system, in what is usually called the “periodic orbit theory”. We then present a case study in which the gross features of a typical double-humped nuclear fission barrier, including the effects of mass asymmetry, can be obtained in terms of the shortest periodic orbits of a cavity model with realistic deformations relevant for nuclear fission. Next we investigate shell structures in a spheroidal cavity model which is integrable and allows for far-going analytical computation. We show, in particular, how period-doubling bifurcations are closely connected to the existence of the so-called “superdeformed” energy minimum which corresponds to the fission isomer of actinide nuclei. Finally, we present a general class of radial power-law potentials which approximate well the shape of a Woods-Saxon potential in the bound region, give analytical trace formulae for it and discuss various limits (including the harmonic oscillator and the spherical box potentials).     

Complex bifurcations in a periodically forced normal form

The effect of a periodic forcing on the normal form of a two-dimensional dynamical system, in which both roots of the characteristic equation can vanish simultaneously, is analyzed. In the space spanned by the system's parameters, the onset of nonperiodic behavior and subharmonic behavior are determined analytically using standard perturbation theory. Moreover it is shown that complex behavior can already appear in the immediate vicinity of singular points. An example of physico-chemical system amenable to the normal form is also constructed.     

Hamiltonian approach to the dynamics of a chemostat

Based on the Hamiltonian approach to an analysis of the system of Monod equations describing the chemostat dynamics, their partial analytical solution is found for a certain class of initial conditions. It is shown that this class of initial conditions can be easily realized in microbiological practice, and the solution obtained is generally described by the attractor of the system trajectories. A methodical approach, which allows the given Hamiltonian formalism to be used to analyze the kinetics of growth of microorganisms in the chemostat, is developed and experimentally checked. Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 46–53, July, 2000.     

Complex dynamical invariant for a PT- symmetric Hamiltonian system in higher dimensions

With a view to get further insight into the integrability of a dynamical system, we investigate the complex invariant for a three-dimensional Hamiltonian system using the extended complex phase space approach (ECPSA) characterized by $x=x_1+i{p}_4,y=x_2+i{p}_5,z=x_3+i{p}_6,p_x=p_1+i{x}_4,p_y=p_2+i{x}_5$ and $p_z=p_3+i{x}_6$. For this purpose the rationalization method is utilized and the invariant obtained is expected to play an important role in the study of the complex trajectories for the system of concern.     

Bifurcations of the trajectories at the saddle level in a Hamiltonian system generated by two coupled Schrodinger equations

Bifurcations of the complex homoclinic loops of an equilibrium saddle point in a Hamiltonian dynamical system with two degrees of freedom are studied. It arises to pick out the stationary solutions in a system of two coupled nonlinear Schrodinger equations. Their relation to bifurcations of hyperbolic and elliptic periodic orbits at the saddle level is studied for varying structural parameters of the system. Series of complex loops are described whose existence is related to periodic orbits.     

Primary resonant optimal control for nested homoclinic and heteroclinic bifurcations in single-dof nonlinear oscillators

A unified control theorem is presented in this paper, whose aim is to suppress the transversal intersections of stable and unstable manifolds of homoclinic and heteroclinic orbits in the Poincarè map embedding in system dynamics. Based on the control theorem, a primary resonant optimal control technique (PROCT for short) is applied to a general single-dof nonlinear oscillator. The novelty of this technique is able to obtain the unified analytical expressions of the control gain and the control parameters for suppressing the homoclinic and heteroclinic bifurcations, where the control gain can guarantee that the control region where the homoclinic and heteroclinic bifurcations do not occur can be enlarged as much as possible at least cost. The technique is applied to a nonlinear oscillator with a pair of nested homoclinic and heteroclinic orbits. By the PROCT, the transversal intersections of homoclinic and heteroclinic orbits can be suppressed, respectively. The hopping phenomenon that there coexist two kinds of chaotic attractors of Duffing-type and pendulum-type can be suppressed. On the contrary, if the first amplitude coefficient is greater than the critical heteroclinic bifurcation value, then another degenerate hopping behavior of chaos will take place again. Therefore, the phenomenon of hopping is the dominant type of chaos in this oscillator, whose suppressing or inducing is admissible from the points of practical and theoretical view.     

Stability, bifurcations, and dynamics of global variables of a system of bursting neurons

An approximate mean field model of an ensemble of delayed coupled stochastic Hindmarsh-Rose bursting neurons is constructed and analyzed. Bifurcation analysis of the approximate system is performed using numerical continuation. It is demonstrated that the stability domains in the parameter space of the large exact systems are correctly estimated using the much simpler approximate model.     

A global qualitative view of bifurcations and dynamics in the Rössler system

The aim of the Letter is a global study of the well-known Rössler system to point out the main complex dynamics that it can exhibit. The structural analysis is based on the periodic solutions of the system investigated by a harmonic balance technique. Simplified expressions of such limit cycles are first derived and characterized, then their local bifurcations are denoted, also giving indications to predict possible homoclinic orbits with the same unifying approach. These analytical results give a general picture of the system behaviours in the parameter space and numerical analysis and simulations confirm the qualitative accuracy of the whole. Such predictions have also an important role in applying efficiently the above numerical procedures.     

Complex chaos and bifurcations of semiconductor lasers subjected to optical injection

This paper presents the nonlinear dynamics and bifurcations of optically injection semiconductor lasers in the frame of relative weak injection strength. We consider the new modified rate equations model established recently and the behavior of the system is explored by means of bifurcation diagrams. However, the exact nature of the involved dynamics is well described by a detailed study of the changes of dynamics as a function of the effective gain coefficient. As results, we notice symmetry spectra of intensity, the sudden transition between chaos and stable limit cycle, double scroll attractors together with the phenomenon of a sequence of period-doubling route of chaos, strict crisis between the two basins attraction and the boundary crisis as well as the effects of frequency detuning and linewidth enhancement factor on the nonlinear behaviors.     

The symmetry of the fundamental equations of the dynamics of a Hamiltonian system

This paper constructs covariant defining equations for infinitesimal operators of the Lie symmetry groups of the Hamilton-Jacobi equations, and Hamilton's and Lagrange's systems whose Hamiltonians are a quadratic function of the generalized momenta; a study is made of the relation between the groups, and in particular the relation is considered between the differential laws of conservation and the symmetry of the systems.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 12–16, February, 1977.