Multiplicity Of Homoclinic Orbits Research Articles

Bifurcations and Chaos in Three-Coupled Ramp-Type Neurons

The bifurcations and chaos in autonomous systems of two- and three-coupled ramp-type neurons are considered. An asymmetric piecewise linear function is employed for the output function of neurons in order to examine changes in the bifurcations from a sigmoid output function to a ramp output function. Steady solutions in the systems are obtained exactly and they undergo discontinuous bifurcations because the systems are piecewise linear. Periodic solutions and homoclinic/heteroclinic orbits in the systems are obtained by connecting local solutions in linear domains at borders and solving transcendental equations. The bifurcations of the periodic solutions are calculated with the Poincaré maps and the Jacobian matrices, which are also derived rigorously. A stable periodic solution in a two-neuron oscillator of the Wilson–Cowan type with three couplings remains in the case of a ramp neuron. A chaotic attractor of Rössler type emerges in a network of three ramp neurons with six couplings, which is due to two saddle-focuses. The network consists of the two-neuron oscillator and one bypass neuron connected through three couplings. One-dimensional Poincaré maps show the generation of the chaotic attractor through a cascade of period-doubling bifurcations. Further, multiple homoclinic orbits based at a saddle are generated from the destabilization of two focuses when asymmetry in the output function is large. This homoclinicity causes qualitative change in the bifurcations of the periodic solutions as the output function of neurons changes from sigmoid to ramp.

Multiple homoclinic orbits for a class of the discrete p‐Laplacian with unbounded potentials

This paper is devoted to study the existence results of a sequence of infinitely many homoclinic orbits for the discrete p‐Laplacian with unbounded potentials without the Ambrosetti and Rabinowitz condition. The strategy of the proof for these results is to approach the problem using the mountain pass theorem, the fountain theorem, and dual fountain theorem.

Homoclinic Orbits for a Class of Fractional Hamiltonian Systems via Variational Methods

This paper is devoted to the existence and multiplicity of homoclinic orbits for a class of fractional-order Hamiltonian systems with left and right Liouville–Weyl fractional derivatives. Here, we present a new approach via variational methods and critical point theory to obtain sufficient conditions under which the Hamiltonian system has at least one homoclinic orbit or multiple homoclinic orbits. Some results are new even for second-order Hamiltonian systems.

Homoclinic orbits for first order Hamiltonian systems with some twist conditions

In this paper, we study the nonperiodic first-order Hamiltonian system \(\dot u = JL(t)u + JH'(t,u)\), where H ∈ C1(R × R2n). With some assumptions on L, the corresponding Hamiltonian operator has only discrete spectrum. By using the index theory for self-adjoint operator equation, we establish the existence of multiple homoclinic orbits for the asymptotically quadratic nonlinearty satisfying some twist conditions between infinity and origin.

Multiple homoclinic orbits for second order discrete Hamiltonian systems without symmetric condition

In this paper, we investigate the second order self-adjoint discrete Hamiltonian system $\Delta[p(n)\Delta u(n-1)]-L(n)u(n)+\lambda a(n)\nabla G(u(n))+\mu b(n)\nabla F(u(n))=0$ , where $p,L:\mathbb{Z}\rightarrow\mathbb{R}^{N\times N}$ are both positive definite for all $n\in\mathbb{Z}$ , and no symmetric condition on G and F is needed. We establish two new criteria to guarantee that the above system has at least two nontrivial homoclinic solutions or infinitely many homoclinic solutions via critical point theory.

Homoclinic Orbits for a Class of Subquadratic Second Order Hamiltonian Systems

The existence and multiplicity of homoclinic orbits are considered for a class of subquadratic second order Hamiltonian systems q¨t-Ltqt+∇Wt,qt=0. Recent results from the literature are generalized and significantly improved. Examples are also given in this paper to illustrate our main results.

Homoclinic orbits for second-order Hamiltonian systems with subquadratic potentials

In this paper we consider a class of subquadratic second-order Hamiltonian systems and new results about the existence and multiplicity of homoclinic orbits are obtained by using the Minimizing Theorem and the Clark’s Theorem respectively and a new compact imbedding theorem is also proved.

Existence and Multiplicity of Homoclinic Orbits for Second-Order Hamiltonian Systems with Superquadratic Potential

We investigate the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems with local superquadratic potential by using the Mountain Pass Theorem and the Fountain Theorem, respectively.

Homoclinic Orbits for a Class of Nonperiodic Hamiltonian Systems with Some Twisted Conditions

By the Maslov index theory, we will study the existence and multiplicity of homoclinic orbits for a class of asymptotically linear nonperiodic Hamiltonian systems with some twisted conditions on the Hamiltonian functions.

Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first order periodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we establish some new criteria to guarantee that Hamiltonian systems, with asymptotically quadratic terms and spectrum point zero, have at least one and infinitely many homoclinic orbits under certain conditions.

Homoclinic Orbits for Second‐Order Hamiltonian Systems with Some Twist Condition

We study the existence and multiplicity of homoclinic orbits for second‐order Hamiltonian systems , where L(t) is unnecessarily positive definite for all t ∈ ℝ, and ∇qW(t, q) is of at most linear growth and satisfies some twist condition between the origin and the infinity.

Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first-order nonperiodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we give some new criteria to guarantee that Hamiltonian systems with asymptotically quadratic terms and spectrum point zero have at least one and a finite number of pairs of homoclinic orbits under some adequate conditions, respectively.

Multiplicity of homoclinic orbits in quasi-linear autonomous Lagrangian systems

The existence of at least two homoclinic orbits is proved by A. Ambrosetti and V. Coti Zelati (Multiple homoclinic orbits for a class of conservative systems, Rend. Sem. Mat. Univ. Padova, 89 (1993), 177-194) for autonomous Lagrangian systems

MULTIPLE HOMOCLINICS IN A HAMILTONIAN SYSTEM WITH ASYMPTOTICALLY OR SUPER LINEAR TERMS

This paper is concerned with homoclinic orbits in the Hamiltonian system [Formula: see text] where H is periodic in t with Hz(t, z) = L(t)z + Rz(t, z), Rz(t, z) = o(|z|) as z → 0. We find a condition on the matrix valued function L to describe the spectrum of operator [Formula: see text] so that a proper variational formulation is presented. Supposing Rz is asymptotically linear as |z| → ∞ and symmetric in z, we obtain infinitely many homoclinic orbits. We also treat the case where Rz is super linear as |z| → ∞ with assumptions different from those studied previously in relative work, and prove existence and multiplicity of homoclinic orbits. Our arguments are based on some recent information on strongly indefinite functionals in critical point theory.

Multiple homoclinic orbits in conservative and reversible systems

We study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows. We consider the existence of two homoclinic orbits in the bellows configuration, where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times. In conservative systems one finds one parameter families of suspended horseshoes, parameterized by the level of the first integral. A somewhat similar picture occurs in reversible systems, with two homoclinic orbits that are both symmetric. The lack of a first integral implies that complete horseshoes do not exist. We provide a description of orbits that necessarily do exist. A second possible configuration in reversible systems occurs if a non-symmetric homoclinic orbit exists and forms a bellows together with its symmetric image. We describe the nonwandering set in an unfolding. The nonwandering set is shown to simultaneously contain one parameter families of periodic orbits, hyperbolic periodic orbits of different index, and heteroclinic cycles between these periodic orbits.

Multiple homoclinic orbits for a class of Hamiltonian systems

In this paper, we obtain the existence of at least two nontrivial homoclinic orbits for a class of second order autonomous Hamiltonian systems. This multiplicity result is obtained by a new variational method based on the relative category: to overcome the lack of compactness of the problem, we first solve perturbed nonautonomous problems and study the limit of the solutions as the nonautonomous perturbation goes to 0. This method allows to get rid of some assumptions on the potential used in the work of Ambrosetti and Coti-Zelati.

Semiclassical States of Nonlinear Schrödinger Equations

The existence of semiclassical states to some nonlinear Schrodinger equations that concentrate near the critical points of the potential V is studied by means of a local approach, variational in nature. We also discuss stability and necessary conditions for concentration. The same method is used to find multiple homoclinic orbits to a class of second‐order Hamiltonian systems.

Darboux transformations for the nonlinear Schrödinger equations

Darboux transformations for the AKNS/ZS system are constructed in terms of Grammian-type determinants of vector solutions of the associated Lax pairs with an operator spectral parameter. A study of the reduction of the Darboux transformation for the nonlinear Schr¨ odinger equations with standard and anomalous dispersion is presented. Two different families of new solutions for a given seed solution of the nonlinear Schr ¨ odinger equation are given, being one family related to a new vector Lax pair for it. In the first family and associated to diagonal matrices we present topological solutions, with different asymptotic argument for the amplitude and nonzero background. For the anomalous dispersion case they represent continuous deformations of the bright n-soliton solution, which is recovered for zero background. In particular these solutions contain the combination of multiple homoclinic orbits of the focusing nonlinear Schrequation. Associated with Jordan blocks we find rational deformations of the just described solutions as well as pure rational solutions. The second family contains not only the solutions mentioned above but also broader classes of solutions. For example, in the standard dispersion case, we are able to obtain the dark soliton solutions.

MULTIPLE HOMOCLINIC BIFURCATIONS FROM ORBIT-FLIP I: SUCCESSIVE HOMOCLINIC DOUBLINGS

The purpose of this and forthcoming papers is to obtain a better understanding of complicated bifurcations for multiple homoclinic orbits. We shall take one particular type of codimension two homoclinic orbits called orbit-flip and study bifurcations to multiple homoclinic orbits appearing in a tubular neighborhood of the original orbit-flip. The main interest of the present paper lies in the occurrence of successive homoclinic doubling bifurcations under an appropriate condition, which is a part of the entire bifurcation for multiple homoclinic orbits. Since this is a totally global bifurcation, we need the aid of numerical experiments for which we must choose a concrete set of ordinary differential equations that exhibits the desired bifurcation. In this paper we employ a family of continuous piecewise-linear vector fields for such a model equation. In order to explain the cascade of homoclinic doubling bifurcations theoretically, we also derive a two-parameter family of unimodal maps as a singular limit of the Poincaré maps along homoclinic orbits. We locate bifurcation curves for this family of unimodal maps in the two-dimensional parameter space, which basically agree with those for the piecewise-linear vector fields. In particular, we show, using a standard technique from the theory of unimodal maps, that there exists an infinite sequence of doubling bifurcations which corresponds to the sequence of homoclinic doubling bifurcations for the piecewise-linear vector fields described above. Since our unimodal map has a singularity at a boundary point of its domain of definition, the doubling bifurcation is slightly different from that for standard quadratic unimodal maps, for instance the Feigenbaum constant associated with the accumulation of the doubling bifurcations is different from the standard value 4.6692.…

A note on the existence of multiple homoclinic orbits for a perturbed radial potential

A note on the existence of multiple homoclinic orbits for a perturbed radial potential