Heteroclinic cycles arising in generic unfoldings of nilpotent singularities

In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in \mathbbR⁴{\mathbb{R}^4} unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in \mathbbR³{\mathbb{R}^3} unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.     

Homoclinic-Doubling Cascades

Cascades of period-doubling bifurcations have attracted much interest from researchers of dynamical systems in the past two decades as they are one of the routes to onset of chaos. In this paper we consider routes to onset of chaos involving homoclinic-doubling bifurcations. We show the existence of cascades of homoclinic-doubling bifurcations which occur persistently in two-parameter families of vector fields on ?³. The cascades are found in an unfolding of a codimension-three homoclinic bifurcation which occur an orbit-flip at resonant eigenvalues. We develop a continuation theory for homoclinic orbits in order to follow homoclinic orbits through infinitely many homoclinic-doubling bifurcations.     

Perturbation analysis in parametrically excited two-degree-of-freedom systems with quadratic and cubic nonlinearities

Based on temporal rescaling and harmonic balance, an extended asymptotic perturbation method for parametrically excited two-degree-of-freedom systems with square and cubic nonlinearities is proposed to study the nonlinear dynamics under 1:2 internal resonance. This asymptotic perturbation method is employed to transform the two-degree-of-freedom nonlinear systems into a four-dimensional nonlinear averaged equation governing the amplitudes and phases of the approximation solutions. Linear stable analysis at equilibrium solutions of the averaged equation is done to show bifurcations of periodic motion and homoclinic motions. Furthermore, analytical expressions of homoclinic orbits and heteroclinic cycles for the averaged equation without dampings are obtained. Considering the action of the damping, the bifurcations of limit cycles are also investigated. A concrete example is further provided to discuss the correctness and accuracy of the extended asymptotic perturbation method in the case of small-amplitude motion for the two-degree-of-freedom nonlinear system.     

Weak Solutions of a Generalized Boussinesq System

We study the convergence of homoclinic orbits and heteroclinic orbits in the dynamical system governing traveling wave solutions of a perturbed Boussinesq systems modeling two-directional propagation of water waves. Nonanalytic weak solutions are found to be limits of these orbits, including compactons, peakons, and rampons, as well as infinitely many mesaons occurring at the same fixed point in the dynamical system. Singularities of solitary wave solutions in the system are also studied to understand the important impact of both linear and nonlinear dispersion terms on the regularity of these solutions.     

The Sliding Bifurcations in Planar Piecewise Smooth Differential Systems

In this paper we are mainly interested in the bifurcation phenomena for a class of planar piecewise smooth differential systems, where a new phenomenon, i.e. sliding heteroclinic bifurcation, is found. Furthermore we will show that the involved systems can present many interesting bifurcation phenomena, such as the (sliding) heteroclinic bifurcation, sliding (homoclinic) cycle bifurcation and semistable limit cycle bifurcation and so on. The system can have two hyperbolic limit cycles, which are bifurcated in one way from a semistable limit cycle, and in another way from a heteroclinic cycle and a sliding cycle. In the proof of our main results, we will use the geometric singular perturbation theory to analyze the dynamics near the sliding region.     

Orbits homoclinic to resonances in a harmonically excited and undamped circular plate

Orbits homoclinic to resonances in mode interactions of an imperfect circular plate with 1:1 internal resonance are investigated. The case of primary resonance is considered. The damping force is not included in the analysis. The energy-phase criterion is used to give a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonances. A saddle-node bifurcation of homoclinic orbits occurs. The existence of homoclinic orbits in the unperturbed system may lead to chaos in the sense of Smale horseshoes under perturbation.     

Homoclinic Twist Bifurcation in a System of Two Coupled Oscillators

We analyse the dynamics of two identical Josephson junctions coupled through a purely capacitive load in the neighborhood of a degenerate symmetric homoclinic orbit. A bifurcation function is obtained applying Lin's version of the Lyapunov–Schmidt reduction. We locate in parameter space the region of existence of n-periodic orbits, and we prove the existence of n-homoclinic orbits and bounded nonperiodic orbits. A singular limit of the bifurcation function yields a one-dimensional mapping which is analyzed. Numerical computations of nonsymmetric homoclinic orbits have been performed, and we show the relevance of these computations by comparing the results with the analysis.     

Limit cycles and homoclinic orbits and their bifurcation of Bogdanov-Takens system

A quantitative analysis of limit cycles and homoclinic orbits, and the bifurcation curve for the Bogdanov-Takens system are discussed. The parameter incremental method for approximate analytical-expressions of these problems is given. These analytical-expressions of the limit cycle and homoclinic orbit are shown as the generalized harmonic functions by employing a time transformation. Curves of the parameters and the stability characteristic exponent of the limit cycle versus amplitude are drawn. Some of the limit cycles and homoclinic orbits phase portraits are plotted. The relationship curves of parameters μ and A with amplitude a and the bifurcation diagrams about the parameter are also given. The numerical accuracy of the calculation results is good.     

Non-existence of Shilnikov chaos in continuous-time systems

In this paper,a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional,continuous-time,and smooth systems is obtained.Based on this result and an elementary example,it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation(ODE) systems characterized by the nonexistence of homoclinic and heteroclinic orbits.     

Dynamics of the general Lorenz family

The present work is devoted to investigating the dynamical entities of the general Lorenz family, which contains four independent parameters. The classical Lorenz system, the Chen system, and the Lü system are all contained by the system considered in this paper as special cases. First, the properties of the equilibria, in particular, the stability of the non-hyperbolic equilibrium obtained by using the center manifold theorem and the technique of the polar transformation, the pitchfork bifurcation and the degenerate pitchfork bifurcation, Hopf bifurcations, and the local stable and unstable manifold character, are all analyzed when the parameters are varied in the space of parameters. Based on the theoretic analysis and numerical simulations, the dynamics of the system are discussed subtly under all kind of the critical state. Second, the properties of the existence of homoclinic and heteroclinic orbits for the system are rigorously studied. Finally, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated.     

Homoclinic Tubes in Discrete Nonlinear Schrödinger Equation under Hamiltonian Perturbations

In this paper, we study the discrete cubic nonlinear Schrödinger lattice under Hamiltonian perturbations. First we develop a complete isospectral theory relevant to the hyperbolic structures of the lattice without perturbations. In particular, Bäcklund–Darboux transformations are utilized to generate heteroclinic orbits and Melnikov vectors. Then we give coordinate-expressions for persistent invariant manifolds and Fenichel fibers for the perturbed lattice. Finally based upon the above machinery, existence of codimension 2 transversal homoclinic tubes is established through a Melnikov type calculation and an implicit function argument. We also discuss symbolic dynamics of invariant tubes each of which consists of a doubly infinite sequence of curve segments when the lattice is four dimensional. Structures inside the asymptotic manifolds of the transversal homoclinic tubes are studied, special orbits, in particular homoclinic orbits and heteroclinic orbits when the lattice is four dimensional, are studied.     

Homoclinic orbits for some （2＋1）-dimensional nonlinear Schrodinger-like equations

Chaos is closely associated with homoclinic orbits in deterministic nonlinear dynamics. In this paper, analytic expressions of homoclinic orbits for some （2＋1）- dimensional nonlinear Schrodinger-like equations are constructed based on Hirota＇s bilinear method, including long wave-short wave resonance interaction equation, generalization of the Zakharov equation, Mel＇nikov equation, and g-Schrodinger equation are constructed based on Hirota＇s bilinear method.     

Homoclinic flip bifurcation with a nonhyperbolic equilibrium

In this paper, the problem of homoclinic bifurcation accompanied by a transcritical bifurcation is investigated for high-dimensional systems. With the aid of a suitable local coordinate system, the Poincaré map is constructed. Under certain nongeneric conditions (orbit flip and inclination flip homoclinic orbits), the existence, nonexistence, coexistence and uniqueness of homoclinic and periodic orbits are studied. Some known results are extended.     

Rich dynamics caused by diffusion

It is an awfully difficult task to design an efficient numerical method for bifurcation diagrams, the graphs of Lyapunov exponents, or the topological entropy about discrete dynamical systems by linear/nonlinear diffusion with the Direchlet/Neumann- boundary conditions. Until now there are less works concerned with such a problem. In this paper, we propose a scheme about bifurcating analysis in a series of discrete-time dynamical systems with linear/nonlinear diffusion terms under the periodic boundary conditions. The complexity of dynamical behaviors caused by the diffusion term are to be determined. Bifurcation diagrams are shown by numerical simulation and chaotic behavior (chaotic Turing patterns) is demonstrated by computing the largest Lyapunov exponent. Our theoretical model can give an interesting case study about the phenomenon: the individuals exhibit a very simple dynamics but the groups with linear/nonlinear coupling can own a complex dynamics including fluctuation, periodicity and even chaotic behavior. We find that diffusion can trigger chaotic behavior in the present system and there exist multiple Turing patterns. It is interesting as regular or chaotic patterns can be reported in this study. Chaotic orbits emerge when exploring further in the diffusion coefficient space, and such a behavior is entirely absent in the corresponding continuous time-space system. The method proposed in the present paper is innovative and the conclusion is novel.

     

Analysis of Hopf and Takens–Bogdanov Bifurcations in a Modified van der Pol–Duffing Oscillator

We analyze a modified van der Pol–Duffing electronic circuit, modeled by a tridimensional autonomous system of differential equations with Z₂-symmetry. Linear codimension-one and two bifurcations of equilibria give rise to several dynamical behaviours, including periodic, homoclinic and heteroclinic orbits. The local analysis provides, in first approximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four Hopf bifurcation is also pointed out. In the case of the Takens–Bogdanov bifurcation, several degenerate situations of codimension-three are analyzed in both homoclinic and heteroclinic cases. The existence of a Hopf–Shil'nikov singularity is also shown.     

STUDIES OF MELNIKOV METHOD AND TRANSVERSAL HOMOCLINIC ORBITS IN THE CIRCULAR PLANAR RESTRICTED THREE－BODY PROBLEM

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Horseshoes in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

We consider a class of two-degree-of-freedom Hamiltonian systems which have saddle-centers with homoclinic orbits and do not take the form of small perturbations of integrable systems. Using a Melnikov-type global perturbation technique, we present a condition under which orbits transversely homoclinic to periodic orbits exist and horseshoes are created on surfaces of energy level near the saddle-centers. We apply the theory to systems with potentials and give concrete examples. In particular, we can detect the presence of horseshoes in a class of systems to which previous results did not apply. Accepted January 4, 2000?Published online August 15, 2000     

On the existence of heteroclinic cycles in some class of 3-dimensional piecewise affine systems with two switching planes

Since complicated dynamical behavior can occur easily near homoclinic trajectory or heteroclinic cycle in dynamical systems with dimension not less than three, this paper investigates the existence of heteroclinic cycles in some class of 3-dimensional three-zone piecewise affine systems with two switching planes. Based on the exact determination of the stable manifold, unstable manifold and analytic solution, a rigorous analytic methodology of designing chaos generators is proposed, which may be of potential applications to chaos secure communication. Furthermore, we obtain three sufficient conditions for the existence of a single or two heteroclinic cycles in three different cases. Finally, some examples are given to illustrate our theoretical results.