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This study provides a comprehensive analysis of the dynamics of a three-level vertical food chain model, specifically focusing on the interactions between vegetation, herbivores, and predators in a Snowshoe hare-Canadian lynx system. By simplifying the model through dimensional analysis, we determine conditions for equilibrium existence and identify various types of bifurcations, including Saddle-Node and Hopf bifurcations. Additionally, the study explores codimension-two bifurcations such as Bogdanov–Takens (BT) and zero-Hopf bifurcations. Coefficient formulas of normal forms are derived through the use of center manifold reduction and normal form theory. The study also presents an approximation of homoclinic orbits near a BT bifurcation of the system by computing explicit asymptotics based on regular perturbation methods. Utilizing the MATLAB package MATCONT, a family of limit cycles and their associated bifurcations are computed, including limit point cycles, period-doubling bifurcations, cusp points of cycles, fold-flip bifurcations, and various resonance bifurcations (R1, R2, R3, and R4). The biological implications of the findings are discussed in detail, highlighting how the identified bifurcations and dynamics can impact the population dynamics of vegetation, herbivores, and predators in real-world ecosystems. Numerical experiments validate the theoretical results and provide further support for the conclusions. [ABSTRACT FROM AUTHOR]

In this paper, a 5D Lorenz system is discussed. The discrete mappings are developed to solve the periodic motions in the 5D Lorenz system. Then the stability and bifurcations are determined by eigenvalue analysis. A bifurcation tree is presented to demonstrate that the discrete mapping method can provide not only stable orbits but also unstable motions. Finally, trajectory illustrations are given to show bifurcation influences on periodic orbits and homoclinic orbits in the 5D Lorenz system. [ABSTRACT FROM AUTHOR]

The neuron model with conductance-resistance symmetry was recently derived by Deng, which is similar to the Hodgkin-Huxley equation, referred to as CRS neuron model. In this paper, we will consider a 2-dimensional reduction model qualitatively similar to the FitzHugh-Nagumo equation. We first give the derivation of the CRS neuron model in propagating action potential. And then we prove the existence of solitary wave solution for the 2-dimensional reduced CRS neuron model by using phase diagram analysis.

We obtain the existence of nontrivial homoclinic orbits for nonautonomous second order Hamiltonian systems by using critical point theory under some different superlinear conditions from those previously used in Hamiltonian systems. In particular, an example is given to illustrate our result.

Conditions for the long term coexistence of the prey and predator populations of a diffusive Rosenzweig-MacArthur model are studied. The coexistence is represented by traveling pulses, which approach a coexistence equilibrium state as the moving coordinate approaches to infinities. Three different pulses, according to their speeds, are analyzed by regular perturbation as well as geometric singular perturbation methods. We further show that the pulses are connected by a bifurcation curve (surface) in parametric space. The paper concludes with several numerical simulations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the nonperiodic Hamiltonian system u ′ = J L (t) u + J H u ′ (t , u) , t ∈ R. Inspired by Dolbeault, Esteban and Séré [On the eigenvalues of operators with gaps. Application to Dirac operators. J Funct Anal. 2000;174:208–226], we will reduce the associated linear system to a new linear system with finite Morse index. We will develop an index theory and define the relative Morse index by investigating the Morse index of the reduced linear Hamiltonian system. Combining the index theory with a generalized linking theorem developed by Bartsch and Ding [Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math Nachr. 2006;279:1267–1288.], the existence and multiplicity of solutions are obtained for asymptotically quadratic nonlinearity. [ABSTRACT FROM AUTHOR]

Consider the geodesic flow on a real-analytic closed hypersurface M of ℝn, equipped with the induced metric. How commonly can we expect such flows to have a transverse homoclinic orbit? In this paper, we give the following two partial answers to this question: • If M is a real-analytic closed hypersurface in ℝn (with n ≥ 3) on which the geodesic flow with respect to the induced metric has a nonhyperbolic periodic orbit, then Cω-generically the geodesic flow on M with respect to the induced metric has a hyperbolic periodic orbit with a transverse homoclinic orbit; and • There is a Cω-open and dense set of real-analytic, closed, and strictly convex surfaces M in ℝ³ on which the geodesic flow with respect to the induced metric has a hyperbolic periodic orbit with a transverse homoclinic orbit. These are among the first perturbation-theoretic results for real-analytic geodesic flows. [ABSTRACT FROM AUTHOR]

We develop methods giving sufficient conditions for the existence of bounded solutions near homoclinic orbits of planar R.D.E.s. We provide families of bounded solutions near the homoclinic by decomposing the corresponding variational equation of the system on the space of solutions. We apply the results for controlling a power-law oscillatory system with a retarded feedback control. Our results are numerically verifiable and do not meet the restrictions of classic methods (such as Melnikov's integrals and Poincare's map). [ABSTRACT FROM AUTHOR]

The complexity of a dynamical system exhibiting a homoclinic orbit is given by its dynamical core which, due to Cantwell, Conlon and Fenley, is a set uniquely determined in the isotopy class, up to a topological conjugacy, of the end-periodic map relative to that orbit. In this work we prove that a sufficient condition to determine the dynamical core of a homoclinic orbit of a Smale diffeomorphism on the -disk is the non-existence of bigons relative to this orbit. Moreover, we propose a pruning method for eliminating bigons that can be used to find a Smale map without bigons and hence for finding the dynamical core. [ABSTRACT FROM AUTHOR]

Abstract By using variational methods, we obtain the existence of homoclinic orbits for perturbed Hamiltonian systems with sub-linear terms. To the best of our knowledge, there is no published result focusing on the perturbed and sub-linear Hamiltonian systems.

We propose a verification method for specification of homoclinic orbits as application of our previous work for constructing local Lyapunov functions by verified numerics. Our goal is to specify parameters appeared in the given systems of ordinary differential equations (ODEs) which admit homoclinic orbits to equilibria. Here we restrict ourselves to cases that each equilibrium is independent of parameters. The feature of our methods consists of Lyapunov functions, integration of ODEs by verified numerics, and Brouwer's coincidence theorem on continuous mappings. Several techniques for constructing continuous mappings from a domain of parameter vectors to a region of the phase space are shown. We present numerical examples for problems in 3 and 4-dimensional cases. [ABSTRACT FROM AUTHOR]

In this paper we study the first-order Hamiltonian system z ˙ = J H z (t , z). Moreover precisely, assuming that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition, we obtain new existence results of ground state homoclinic orbits and infinitely many geometrically distinct homoclinic orbits by using a variational method. An interesting problem is that the nonlinearity may be super-quadratic on some domains and asymptotically quadratic on other domains under local super-quadratic condition. Since we are without more global information on the nonlinearity, we apply a perturbation approach and some special techniques in the proofs. [ABSTRACT FROM AUTHOR]

We analyze a singularly Kuramoto-Sivashinsky perturbed Camassa-Holm equation with methods of the geometric singular perturbation theory. Especially, we study the persistence of smooth and peaked solitons. Whether a solitary wave of the original Camassa-Holm equation is smooth or peaked depends on whether there is linear dispersion, i.e. whether 2 k = 0. If 2 k > 0 , then a unique smooth solitary wave persists with selected wave speed under singular Kuramoto-Sivashinsky perturbation just as what happens in the KS-KdV equation. On the other hand, we show that if there is no linear dispersion, i.e. 2 k = 0 , then any observable peaked soliton fails to persist. This case is non-typical since the related slow manifold blows up and the classical geometric singular perturbation theory is not available. [ABSTRACT FROM AUTHOR]

In this paper, the bifurcation dynamics of a family of -periodic motions to chaos with infinite homoclinic orbits , in the Lorenz system is studied through the discrete mapping method. The bifurcation trees of -periodic motion to chaos are presented through discrete nodes and harmonic amplitudes. The stability and bifurcations of periodic motions are determined through eigenvalue analysis. The bifurcation scenarios of -period-1 motions to chaos are similar each other. The critical values for existence of -related periodic motions are determined for saddle-node and period-doubling bifurcations. The homoclinic orbits are associated with unstable periodic motions on the bifurcation trees of the -periodic motions to chaos. The homoclinic obits and periodic motions are illustrated from the bifurcation trees of the -periodic motions to chaos. The numerical and analytical trajectories of unstable periodic motions were presented for comparison. If the numerical simulations did not have any computational errors, the numerical and analytical solutions of unstable periodic motions in the Lorenz system should be identical. Thus, one observed so-called strange attractors in the Lorenz system through numerical simulations, which are not real strange attractors. This paper is specially dedicated to the good friend and colleague in memory of Gennady A. Leonov for his contributions on nonlinear dynamics. [ABSTRACT FROM AUTHOR]

Abstract In this paper, bifurcations and chaotic behaviours of Kopel oligopoly model with different adjustment speed are discussed. The results imply that the Kopel oligopoly model undergoes flip bifurcation, Neimark–Sacker bifurcation, 1:3 and 1:4 resonances, which could induce complex dynamics, especially global behaviours between different orbits. The conditions for the occurrence of three different kinds of bifurcation are derived. Furthermore, the numerical simulations provide us the case study of theoretical analysis and the corresponding dynamical behaviours, especially the occurrence of global orbits.

In 1958 started the study of the families of algebraic limit cycles in the class of planar quadratic polynomial differential systems. In the present we known one family of algebraic limit cycles of degree 2 and four families of algebraic limit cycles of degree 4, and that there are no limit cycles of degree 3. All the families of algebraic limit cycles of degree 2 and 4 are known, this is not the case for the families of degree higher than 4. We also know that there exist two families of algebraic limit cycles of degree 5 and one family of degree 6, but we do not know if these families are all the families of degree 5 and 6. Until today it is an open problem to know if there are algebraic limit cycles of degree higher than 6 inside the class of quadratic polynomial differential systems. Here we investigate the birth and death of all the known families of algebraic limit cycles of quadratic polynomial differential systems. [ABSTRACT FROM AUTHOR]

By using variational methods, we obtain the existence of homoclinic orbits for perturbed Hamiltonian systems with sub-linear terms. To the best of our knowledge, there is no published result focusing on the perturbed and sub-linear Hamiltonian systems. [ABSTRACT FROM AUTHOR]

The so-called Dixon system is often cited as an example of a two-dimensional (continuous) dynamical system that exhibits chaotic behavior, if its two parameters take their values in a certain domain. We provide first a rigorous proof that there is no chaos in Dixon's system. Then we perform a complete bifurcation analysis of the system showing that the parameter space can be decomposed into 16 different regions in each of which the system exhibits qualitatively the same behavior. In particular, we prove that in some regions two elliptic sectors with infinitely many homoclinic orbits exist. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study the effect of time-dependent, non-conservative perturbations on the dynamics along homoclinic orbits to a normally hyperbolic invariant manifold. We assume that the unperturbed system is Hamiltonian, and the normally hyperbolic invariant manifold is parametrized via action-angle coordinates. The homoclinic excursions can be described via the scattering map, which gives the future asymptotic of an orbit as a function of its past asymptotic. We provide explicit formulas, in terms of convergent integrals, for the perturbed scattering map expressed in action-angle coordinates. We illustrate these formulas for perturbations of both uncoupled and coupled rotator-pendulum systems. [ABSTRACT FROM AUTHOR]

For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree d, it is proved that any bifurcation which preserves the multiplicity of equilibrium points admits a decomposition into a finite number of elementary bifurcations, and the elementary bifurcations are characterized. [ABSTRACT FROM AUTHOR]

Recent studies using the classical Lorenz model and generalized Lorenz models present abundant features of both chaotic and oscillatory solutions that may change our view on the nature of the weather as well as climate. In this study, the mathematical universality of solutions in different physical systems is presented. Specifically, the main goal is to reveal mathematical similarities for solutions of homoclinic orbits and solitary waves within a three-dimensional nondissipative Lorenz model (3D-NLM), the Korteweg–de Vries (KdV) equation, and the Nonlinear Schrodinger (NLS) equation. A homoclinic orbit for the X , Y , and Z state variables of the 3D-NLM connects the unstable and stable manifolds of a saddle point. The X and Z solutions for the homoclinic orbit can be expressed in terms of a hyperbolic secant function (sech) and a hyperbolic secant squared function ( sech 2 ), respectively. Interestingly, these two solutions have the same mathematical form as solitary solutions for the NLS and KdV equations, respectively. After introducing new independent variables, the same second-order ordinary differential equation (ODE) and solutions for the Z component and the KdV equation were obtained. Additionally, the ODE for the X component has the same form as the NLS for the solitary wave envelope. Finally, how a logistic equation, also known as the Lorenz error growth model, and an improved error growth model can be derived by simplifying the 3D-NLM is also discussed. Future work will compare the solutions of the 3D-NLM and KdV equation in order to understand the different physical role of nonlinearity in their solutions and the solutions of the error growth model and the 3D-NLM, as well as other Lorenz models, to propose an improved error growth model for better representing error growth at linear and nonlinear stages for both oscillatory and nonoscillatory solutions. [ABSTRACT FROM AUTHOR]

Magal et al. [13] studied both spatial and non-spatial host-parasitoid models motivated by biological control of horse-chestnut leaf miners that have spread through Europe. In the non-spatial model, they considered pest control by predation of leaf miners by a generalist parasitoid with a Holling type II functional (Monod) response. They showed that there can be at most six equilibrium points and discussed local stability. We revisit their model in the non-spatial case, identify cases missed in their investigation and discuss consequences for possible pest control strategies. Both the local stability of equilibria and global properties are considered. We use a bifurcation theoretical approach and provide analytical expressions for fold and Hopf bifurcations and for the criticality of the Hopf bifurcations. Our numerical results show very interesting dynamics resulting from codimension one bifurcations including: Hopf, fold, transcritical, cyclic-fold, and homoclinic bifurcations as well as codimension two bifurcations including: Bautin and Bogdanov-Takens bifurcations, and a codimension three Bogdanov-Takens bifurcation. [ABSTRACT FROM AUTHOR]

We characterize the families of periodic orbits of two discontinuous piecewise differential systems in ℝ 3 separated by a plane using their first integrals. One of these discontinuous piecewise differential systems is formed by linear differential systems, and the other by nonlinear differential systems. [ABSTRACT FROM AUTHOR]

The ionization of hydrogen in a circularly polarized electric field consists of the electron escaping over a rotating Stark saddle. The escape is mediated by the stable and unstable manifolds of a normally hyperbolic invariant manifold associated with the saddle. General principles guarantee the existence of this normally hyperbolic invariant manifold which is referred to as a transition state for energies close to the saddle. We use a numerical procedure to continue the transition state to high energies and study the dynamics restricted to the transition state. We find that for the strength of the electric field under consideration, the transition state persists to very high energies and the restricted dynamics remains amazingly regular. Our analysis moreover includes a study of homoclinic orbits connecting the transition state to itself. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear fifth-order KdV equation which has wide physical applications. We first establish the existence of solitary wave solutions for this equation without delay by using the Hamilton function method. Then we obtain the desired homoclinic orbits by constructing a locally invariant manifold. Finally we obtain the existence of solitary wave solutions for this equation with local and nonlocal delay convolution kernels by using the geometric singular perturbation analysis, implicit function theorem and Fredholm theory. [ABSTRACT FROM AUTHOR]

This paper investigates the homoclinic orbits and chaos in the generalized Lorenz system. Using center manifold theory and Lyapunov functions, we get non-existence conditions of homoclinic orbits associated with the origin. The existence conditions of the homoclinic orbits are obtained by Fishing Principle. Therefore, sufficient and necessary conditions of existence of homoclinic orbits associated with the origin are given. Furthermore, with the broken of the homoclinic orbits, we show that the chaos is in the sense generalized Shil'nikov homoclinic criterion. [ABSTRACT FROM AUTHOR]

In this paper, bifurcations and chaotic behaviours of Kopel oligopoly model with different adjustment speed are discussed. The results imply that the Kopel oligopoly model undergoes flip bifurcation, Neimark–Sacker bifurcation, 1:3 and 1:4 resonances, which could induce complex dynamics, especially global behaviours between different orbits. The conditions for the occurrence of three different kinds of bifurcation are derived. Furthermore, the numerical simulations provide us the case study of theoretical analysis and the corresponding dynamical behaviours, especially the occurrence of global orbits. [ABSTRACT FROM AUTHOR]

We consider the stability of nonlinear travelling waves in a class of activator-inhibitor systems. The eigenvalue equation arising from linearizing about the wave is seen to preserve the manifold of Lagrangian planes for a nonstandard symplectic form. This allows us to define a Maslov index for the wave corresponding to the spatial evolution of the unstable bundle. We formulate the Evans function for the eigenvalue problem and show that the parity of the Maslov index determines the sign of the derivative of the Evans function at the origin. The connection between the Evans function and the Maslov index is established by a 'detection form,' which identifies conjugate points for the curve of Lagrangian planes. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we are concerned about homoclinic orbits of the Lorenz system { x ˙ = − σ x + σ y , y ˙ = r x − y − x z , z ˙ = − b z + x y , where σ , b , r > 0 are three parameters. Firstly, by Fredholm alternative theorem and Taylor expansions, one can figure out the local invariant manifolds of the trivial equilibrium E 0 = (0 , 0 , 0) , including the stable and unstable ones. Secondly, the undetermined coefficient method is used to find out the negative part of the homoclinic orbit, which is supposed to have the form of exponential type series. Such negative semiorbit together with the positive semiorbit that starts from the locally stable manifold consists of a homoclinic loop. As these two semiorbits should be continuous at the intersection point, and the corresponding series should be convergent, one can establish the conditions for the existence of homoclinic orbits. Finally, as an application, we will exhibit the existence theorem of homoclinic orbits for the Lorenz system with the parameters σ = 10 , b = 8 / 3 , r = 13.926. [ABSTRACT FROM AUTHOR]

We are concerned with conservative systems q = ∇V(q), q ∈ RN for a general class of potentials V ∈ C1(RN). Assuming that a given sublevel set {V ≤ c} splits in the disjoint union of two closed subsets Vc− and Vc+, for some c ∈ R, we establish the existence of bounded solutions qc to the above system with energy equal to −c whose trajectories connect Vc− and Vc+. The solutions are obtained through an energy constrained variational method, whenever mild coerciveness properties are present in the problem. The connecting orbits are classified into brake, heteroclinic or homoclinic type, depending on the behavior of ∇V on ∂Vc±. Next, we illustrate applications of the existence result to double-well potentials V, and for potentials associated to systems of duffing type and of multiple-pendulum type. In each of the above cases we prove some convergence results of the family of solutions (qc). [ABSTRACT FROM AUTHOR]

Abstract Using mainly variational methods and an approximation technique we obtain one homoclinic orbit for a Hamiltonian system type (P) u ̈ + V u (t , u) = 0. Our approach allows us to consider new examples of nonlinearities which do not satisfy the classical Ambrosetti–Rabinowitz condition and neither strict growth hypotheses that allow to use Nehari Manifold. For instance, in our system we can consider a periodic potential V given by V (t , u) = − 〈 L (t) u , u 〉 + f (t) | u | 2 ln (1 + | u |) , which clearly does not verify those conditions when L (t) is positive definite and symmetric matrix and f is a nontrivial nonnegative continuous function. We notice that f may vanish in some part of its domain. [ABSTRACT FROM AUTHOR]

Homoclinic orbits of a buckled beam subjected to transverse uniform harmonic excitation are investigated in the case of 1:1 internal resonance. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect the equilibria in a resonance band of the slow manifold. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The results obtained imply the existence of the amplitude modulated chaos for the Smale horseshoe sense in the class of buckled beam systems. [ABSTRACT FROM AUTHOR]

The mechanism of a rogue water wave is still unknown. One popular conjecture is that the Peregrine wave solution of the nonlinear Schrodinger equation (NLS) provides a mechanism. A Peregrine wave solution can be obtained by taking the infinite spatial period limit to the homoclinic solutions. In this article, from the perspective of the phase space structure of these homoclinic orbits in the infinite dimensional phase space where the NLS defines a dynamical system, we examine the observability of these homoclinic orbits (and their approximations). Our conclusion is that these approximate homoclinic orbits are the most observable solutions, and they should correspond to the most common deep ocean waves rather than the rare rogue waves. We also discuss other possibilities for the mechanism of a rogue wave: rough dependence on initial data or finite time blow up.

We obtain the existence of homoclinic orbits at infinity for a class of second-order Hamiltonian systems with fixed energy. We use the limit for a sequence of approximate solutions which are obtained by variational methods.

We establish existence criteria for homoclinic orbits of second-order nonlinear difference equations by using the critical point theory in combination with periodic approximations.

Stable and unstable fibre bundles with respect to a fixed point or a bounded trajectory are of great dynamical relevance in (non)autonomous dynamical systems. These sets are defined via an infinite limit process. However, the dynamics of several real world models are of interest on a short time interval only. This task requires finite time concepts of attraction and repulsion that have been recently developed in the literature. The main idea consists in replacing the infinite limit process by a monotonicity criterion and in demanding the end points to lie in a small neighbourhood of the reference trajectory. Finite time areas of attraction and repulsion defined in this way are fat sets and their dimension equals the dimension of the state space. We propose an algorithm for the numerical approximation of these sets and illustrate its application to several two- and three-dimensional dynamical systems in discrete and continuous time. Intersections of areas of attraction and repulsion are also calculated, resulting in finite time homoclinic orbits. [ABSTRACT FROM AUTHOR]

In the 1960s, L.P. Shilnikov showed that certain homoclinic orbits for smooth families of differential equations imply the existence of chaos, and there are complicated sequences of bifurcations near the parameter value at which the homoclinic orbit exists. We describe how this analysis is modified if the differential equations are piecewise smooth and the homoclinic orbit has a sliding segment. Moreover, we show that the Shilnikov mechanism appears naturally in the unfolding of boundary equilibrium bifurcations in R3. [ABSTRACT FROM AUTHOR]

We give a complete description of the shapes and the behavior of all homoclinic orbits in the Euler problem of two fixed centers. [ABSTRACT FROM AUTHOR]

We consider a mechanical system consisting of n-penduli and a d-degree-of-freedom rotator. The phase space of the rotator defines a normally hyperbolic invariant manifold Λ0. We apply a time-dependent perturbation, which is not assumed to be either Hamiltonian, or periodic, or quasi-periodic, as we allow for rather general time dependence. The strength of the perturbation is given by a parameter ε∈R. For all |ε| sufficiently small, the augmented flow—obtained by making the time into a new variable—has a normally hyperbolic locally invariant manifold Λ~ε. For ε=0, Λ~0=Λ0×R. We define a Melnikov-type vector, which gives the first-order expansion of the displacement of the stable and unstable manifolds of Λ~0 under the perturbation. We provide an explicit formula for the Melnikov vector in terms of convergent improper integrals of the perturbation along homoclinic orbits of the unperturbed system. We show that if the perturbation satisfies some explicit non-degeneracy conditions, then the stable and unstable manifolds of Λ~ε, Ws(Λ~ε) and Wu(Λ~ε), respectively, intersect along a transverse homoclinic manifold, and, moreover, the splitting of Ws(Λ~ε) and Wu(Λ~ε) can be explicitly computed, up to the first order, in terms of the Melnikov-type vector. This implies that the excursions along some homoclinic trajectories yield a non-trivial increase of order O(ε) in the action variables of the rotator, for all sufficiently small perturbations. The formulas that we obtain are independent of the unperturbed motions in Λ0, and give, at the same time, the effects on periodic, quasi-periodic, or general-type orbits. When the perturbation is Hamiltonian, we express the effects of the perturbation, up to the first order, in terms of a Melnikov potential. In addition, if the perturbation is periodic, we obtain that the non-degeneracy conditions on the Melnikov potential are generic. [ABSTRACT FROM AUTHOR]

We study the following second-order differential system -u¨(t)-Mu˙(t)+L(t)u(t)=Hu(t,u(t)),t∈R,which can be regarded as a second-order Hamiltonian system with a damped term. Here, the nonlinearity H(t, u) is superquadratic as |u|→∞. We do not need any periodic conditions, and we obtain infinitely many nontrivial homoclinic orbits of this system by variational methods. Our result improves and extends the corresponding results existed. [ABSTRACT FROM AUTHOR]

Based on the mathematical analysis of the stable manifolds, unstable manifolds and solutions of subsystems, this paper provides methodologies to ensure the existence of homoclinic orbits (degenerate and non-degenerate) which cross the switching manifold transversally at two points in n -dimensional two-zone piecewise affine systems. In addition, two examples are provided to illustrate the effectiveness of the methods, moreover, the simulation results of homoclinic orbits and chaotic invariant sets are provided for the two examples. [ABSTRACT FROM AUTHOR]