The purpose of this paper is to prove a new, more general version of the Morse index theorem for heteroclinic, homoclinic, and half-clinic solutions in general Lagrangian systems. In the final section, we compute the Morse index for specific heteroclinic and half-clinic solutions in classical mechanical models such as the mathematical pendulum, the Nagumo equation, and a four-dimensional competition-diffusion system.
Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few results are known in the case of homoclinic orbits of Hamiltonian systems. Moreover, to the authors' knowledge, no results have been yet proved in the case of heteroclinic and halfclinic (i.e. parametrised by a half-line) orbits. Motivated by the importance played by these motions in understanding several challenging problems in Classical Mechanics, we develop a …
In this paper, we present a method to generate homoclinic and heteroclinic motions in impulsive systems. We rigorously prove the presence of such motions in the case that the systems are under the influence of a discrete map that possesses homoclinic and heteroclinic orbits. Simulations that support the theoretical results are represented by means of a Duffing equation with impacts.
In this paper we present sufficient conditions for the existence of heteroclinic or homoclinic solutions for second order coupled systems of differential equations on the real line. We point out that it is required only conditions on the homeomorphisms and no growth or asymptotic conditions are assumed on the nonlinearities. The arguments make use of the fixed point theory, L^1-Carathéodory functions and Schauder's fixed point theorem. An application to a family of second order nonlinear coupled systems of two degrees of freedom, shows the applicability of the main theorem.
The Nielsen-Thurston theory of surface diffeomorphisms shows that useful dynamical information can be obtained about a surface diffeomorphism from a finite collection of periodic orbits.In this paper, we extend these results to homoclinic and heteroclinic orbits of saddle points. These orbits are most readily computed and studied as intersections of unstable and stable manifolds comprising homoclinic or heteroclinic tangles in the surface. We show how to compute a map of a one-dimensional space similar to a train-track which represents the isotopy-stable dynamics of the surface diffeomorphism …
Systems of $N$ identical globally coupled phase oscillators can demonstrate a multitude of complex behaviours. Such systems can have chaotic dynamics for $N>4$ when a coupling function is biharmonic. The case $N = 4$ does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies showed that some of chaotic attractors in this system are organized by heteroclinic networks. In present paper we discuss which heteroclinic cycles are forbidden and which are supported by this particular system. We also …
Consider a family of planar systems depending on two parameters $(n,b)$ and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when $Φ(n,b)=0.$ We present a method that allows to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set ${Φ(n,b)=0}.$ The method is applied to two quadratic families, one of them is the well-known Bogdanov-Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of $n$, given by $b=\frac5 7 n^{1/2}+{72/2401}n- {30024/4…
In this paper, we consider the scalar reaction-diffusion equations $\partial_t u=Δu + f(x,u,\nabla u)$ on a bounded domain $Ω\subset\mathbb{R}^d$ of class $C^2$. We show that the heteroclinic and homoclinic orbits connecting hyperbolic equilibria and hyperbolic periodic orbits are transverse, generically with respect to f. One of the main ingredients of the proof is an accurate study of the singular nodal set of solutions of linear parabolic equations. Our main result is a first step for proving the genericity of Kupka-Smale property, the generic hyperbolicity of periodic orbits remaining unpr…
In the theory of surface diffeomorphisms relative to homoclinic and heteroclinic orbits, it is possible to compute a one-dimensional representative map for any irreducible isotopy class. The topological entropy of this graph representative is equal to the growth rate of the number of essential Nielsen classes of a given period, and hence is a lower bound for the topological entropy of the diffeomorphism. In this paper, we show that this entropy bound is the infemum of the topological entropies of diffeomorphisms in the isotopy class, and give necessary and sufficient conditions for the infemal…
In this paper we consider finite dimensional dynamical systems generated by a Lipschitz function. We prove a version of the Whitney's Extension Theorem on compact manifolds to obtain a version of the well-known Lambda Lemma for Lipschitz functions. The notions of Lipschitz transversality and hyperbolicity are investigated in the context of finite dimension with a norm weaker than $C^1$-norm and stronger than $C^0$-norm.
An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same level of a Hamiltonian and two non-symmetric heteroclinic orbits permuted by the involution. This is a codimension one structure and therefore it can be met generally in one-parameter families of reversible Hamiltonian systems. There exist two possible types of such connections in dependence on how the involution acts near the equilibrium. We prove a series…
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 $\mathbb{R}^4$ 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 $\mathbb{R}^3$ unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic a…
Shunting inhibitory cellular neural networks (SICNNs) with continuous as well as discontinuous external inputs are investigated. The descriptions of homoclinic and heteroclinic motions are provided in the functional sense for the multidimensional dynamics of SICNNs, and it is demonstrated that the networks under investigation exhibit such motions. Homoclinic and heteroclinic outputs that are asymptotic to quasi-periodic outputs are illustrated.
We present a method for proving the existence of symmetric periodic, heteroclinic or homoclinic orbits in dynamical systems with the reversing symmetry. As an application we show that the Planar Restricted Circular Three Body Problem (PCR3BP) corresponding to the Sun-Jupiter-Oterma system possesses an infinite number of symmetric periodic orbits and homoclinic orbits to the Lyapunov orbits. Moreover, we show the existence of symbolic dynamics on six symbols for PCR3BP and the possibility of resonance transitions of the comet. This extends earlier results by Wilczak and Zgliczynski - Heteroclin…
This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are allowed. The main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time, and their parameter values. A leading order expansion is given in terms of the time spent near vertices and, if applicable, the location on the non-trivial tangent directions. The only difference between a pe…
Homoclinic and heteroclinic motions in dynamics equations on time scales is investigated. The utilized time scale is a specific one such that it is a union of disjoint compact intervals. A numerical example that supports the theoretical results is presented.
Lorenz attractors are important objects in the modern theory of chaos. The reason from one side is that they are met in various natural applications (fluid dynamics, mechanics, laser dynamics, etc.). At the same time, Lorenz attractors are robust, in the sense that they are generally not destroyed by small perturbations (autonomous, non-autonomous, stochastic). This allows us to be sure that the observed in the experiment object is exactly the chaotic attractor, rather than a long-time periodic orbit. Discrete-time analogs of the Lorenz attractor possess even more complicated structure -- they…
We present an extension of the theory known as Lin's method to heteroclinic chains that connect hyperbolic equilibria and hyperbolic periodic orbits. Based on the construction of a so-called Lin orbit, that is, a sequence of continuous partial orbits that only have jumps in a certain prescribed linear subspace, estimates for these jumps are derived. We use the jump estimates to discuss bifurcation equations for homoclinic orbits near heteroclinic cycles between an equilibrium and a periodic orbit (EtoP cycles).
We show that heterodimensional cycles can be born at the bifurcations of a pair of homoclinic loops to a saddle-focus equilibrium for flows in dimension 4 and higher. In addition to the classical heterodimensional connection between two periodic orbits, we found two new types of heterodimensional connections: one is a heteroclinic between a periodic orbit of index 2 and a homoclinic loop, and the other connects a periodic orbit of index 3 to the saddle-focus equilibrium.
In this paper, we study the discrete cubic nonlinear Schroedinger lattice under Hamiltonian perturbations. First we develop a complete isospectral theory relevant to the hyperbolic structures of the lattice without perturbations. In particular, Backlund-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 ca…
Heteroclinic orbits for one-parameter families of nonautonomous vectorfields appear in a very natural way in many physical applications. Inspired by some recent bifurcation results for homoclinic trajectories of nonautonomous vectorfield, we define a new $\mathbf Z_2$-index and we construct a index theory for heteroclinic orbits of nonautonomous vectorfield. We prove an index theorem, by showing that, under some standard transversality assumptions, the $\mathbf Z_2$-index is equal to the parity, a homotopy invariant for paths of Fredholm operators of index 0. As a direct consequence of the ind…
Homoclinic snaking refers to the sinusoidal snaking continuation curve of homoclinic orbits near a heteroclinic cycle connecting an equilibrium E and a periodic orbit P. Along this curve the homoclinic orbit performs more and more windings about the periodic orbit. Typically this behaviour appears in reversible Hamiltonian systems. Here we discuss this phenomenon in systems without any particular structure. We give a rigorous analytical verification of homoclinic snaking under certain assumptions on the behaviour of the stable and unstable manifolds of E and P. We show how the snaking behaviou…
We consider the planar circular restricted three-body problem (PCRTBP), as a model for the motion of a spacecraft relative to the Earth-Moon system. We focus on the Lagrange equilibrium points $L_1$ and $L_2$. There are families of Lyapunov periodic orbits around either $L_1$ or $L_2$, forming Lyapunov manifolds. There also exist homoclinic orbits to the Lyapunov manifolds around either $L_1$ or $L_2$, as well as heteroclinic orbits between the Lyapunov manifold around $L_1$ and the one around $L_2$. The motion along the homoclinic/heteroclinic orbits can be described via the scattering map, w…
When a real saddle equilibrium in a three-dimensional vector field undergoes a homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or non-orientable way. We are interested in the interaction between global invariant manifolds of saddle equilibria and saddle periodic orbits for a vector field close to a codimension-two homoclinic flip bifurcation, that is, the point of transition between having an orientable or non-orientable two-dimensional surface. Here, we focus on homoclinic flip bifurcations of case $\textbf{B}$, wh…
Localized patterns in singularly perturbed reaction-diffusion equations typically consist of slow parts -- in which the associated solution follows an orbit on a slow manifold in a reduced spatial dynamical system -- alternated by fast excursions -- in which the solution jumps from one slow manifold to another, or back to the original slow manifold. In this paper we consider the existence and stability of stationary and traveling slow localized patterns that do not exhibit such jumps, i.e. that are completely embedded in a slow manifold of the singularly perturbed spatial dynamical system. The…
We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in $\mathbb{R}^3$ that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary $n$-homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity. We present an adaptation of the technique known as Lin's method that enables us to compute such connect…
We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolic equilibrium point having a loop or homoclinic orbit (or, alternatively, two hyperbolic equilibrium points connected by a heteroclinic orbit), as a step towards understanding the behavior of nearly-integrable Hamiltonians near double resonances. We provide a constructive approach to study whether the unstable and stable invariant manifolds of the hyperbolic point intersect transversely along the loop, inside their common energy level. For the system considered, we establish a necessary and sufficient condition for the…
Rhythmic behaviors in neural systems often combine features of limit cycle dynamics (stability and periodicity) with features of near heteroclinic or near homoclinic cycle dynamics (extended dwell times in localized regions of phase space). Proximity of a limit cycle to one or more saddle equilibria can have a profound effect on the timing of trajectory components and response to both fast and slow perturbations, providing a possible mechanism for adaptive control of rhythmic motions. Reyn showed that for a planar dynamical system with a stable heteroclinic cycle (or separatrix polygon), small…
We study a homoclinic flip bifurcation of case~\textbf{C}, where a homoclinic orbit to a saddle equilibrium with real eigenvalues changes from being orientable to nonorientable. This bifurcation is of codimension two, and it is the lowest codimension for a homoclinic bifurcation of a real saddle to generate chaotic behavior in the form of (suspended) Smale horseshoes and strange attractors. We present a detailed numerical case study of how global stable and unstable manifolds of the saddle equilibrium and of bifurcating periodic orbits interact close to such bifurcation. This is a step forward…
Multi-shooting parameterization methods for invariant manifolds and heteroclinics of 2 DOF Hamiltonian Poincaré maps, with applications to celestial resonant dynamics
