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In this paper we are concerned with the study of the existence and multiplicity of connecting orbits for a singular planar Newtonian system $${\ddot{q} + V_q(t, q) = 0}$$ with a periodic strong force V( t, q), an infinitely deep well of Gordon's type at one point and two stationary points at which a potential V ( t, q) achieves a strict global maximum. To this end we minimize the corresponding actiön functional over the classes of functions in the Sobolev space $${W^{1, 2}_{\rm loc}(\mathbb{R}, \mathbb{R}^2)}$$ that turn a given number of times around the singularity. [ABSTRACT FROM AUTHOR]

Some dynamical and geometrical properties of controls dynamic for a drift-free left invariant control system from the Poisson geometry point of view are described. The integrability of such system are also studied. [ABSTRACT FROM AUTHOR]

The existence of homoclinic solutions is obtained for a class of nonautonomous second order Hamiltonian systems ü(t) + ...V(t,u (t)) = f (t) as he limit of the 2kT-periodic solutions which are obtained by the Mountain ass theorem, where V (t, x) = -K(t, x) + W (t, x) is T-periodic with respect to t, T > 0, and W (t, x) satisfies the superquadratic condition: W(t, x)/|x|2 → + ∞ as |x| →∞ uniformly in t, which needs not to satisfy the global Ambrosetti-Rabinowitz condition. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the dynamics and ergodic theory of certain economic models which are implicitly defined. We consider 1-dimensional and 2-dimensional overlapping generations models, a cash-in-advance model, heterogeneous markets and a cobweb model with adaptive adjustment. We consider the inverse limit spaces of certain chaotic invariant fractal sets and their metric, ergodic and stability properties. The inverse limits give the set of intertemporal perfect foresight equilibria for the economic problem considered. First we show that the inverse limits of these models are stable under perturbations. We then employ utility functions on inverse limits in our case. We give two ways to rank such utility functions. First, when perturbing certain dynamical systems, we rank utility functions in terms of their average values with respect to invariant probability measures on inverse limits, especially with respect to measures of maximal entropy. For families of certain unimodal maps we can adjust both the discount factor and the system parameters in order to obtain maximal average value of the utility. The second way to rank utility functions (for more general maps on hyperbolic sets) will be to use equilibrium measures of these utility functions on inverse limits; they optimize average values of utility functions while at the same time keeping the disorder in the system as low as possible in the long run. [Copyright &y& Elsevier]

Abstract: In this paper, we study the following nonperiodic Hamiltonian system where is superquadratic in as . By applying a generalized linking theorem for strongly indefinite functionals, we obtain infinitely many homoclinic orbits for the above system. [Copyright &y& Elsevier]

The existence of horseshoes is proved in a class of 3-dim piecewise linear systems, in which a homoclinic orbit connecting the origin to itself is explicitly given. Based on these results, a mathematically rigorous methodology for design of chaos generators is proposed. Implementation of such chaos generators by circuit is easy and the chaotic attractor is robust under small perturbations. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents the existence of Ši’lnikov orbits in two different chaotic systems belong to the class of Lorenz systems, more exactly in the Lü system and in the Zhou’s system. Both systems have exactly two heteroclinic orbits which are symmetrical with respect to the z-axis by using the undetermined coefficient method. The existence of the homoclinic orbit for the Zhou’s system has been proven also by using the undetermined coefficient method. As a result, the Ši’lnikov criterion along with some technical conditions guarantees that Lü and Zhou’s systems have both Smale horseshoes and horseshoe type of chaos. Moreover, the geometric structures of attractors are determined by these heteroclinic orbits. [Copyright &y& Elsevier]

In this paper the stability of homoclinic loops of saddle equilibrium states in high dimensional systems is analyzed. By constructing local moving frame along the unperturbed homoclinic orbit, the refined Poincaré map is well established, and simple criteria are given for the stability of the saddle homoclinic loop. Some known results are extended. [ABSTRACT FROM AUTHOR]

Abstract: We study some aspects of the dynamics of unstable breathers in a three-site discrete cubic NLS chain with Dirichlet boundary conditions. We view breathers as fixed points of the energy in the reduced phase space obtained by eliminating directions related to the global phase symmetry of the system. We use a combination of numerical calculations and Morse-theoretical arguments to see that there are two breathers that correspond to critical energies where the energy hypersurface changes its connectivity. These breathers are elliptic–hyperbolic fixed points of the reduced four-dimensional system. We compute the periodic orbits in their center manifolds (Lyapunov orbits) and see evidence for homoclinic intersections of their stable and unstable manifolds. We also examine the possibility of heteroclinic connections between Lyapunov orbits, these however appear not to exist for the energies near the energy where the energy hypersurface becomes connected. [Copyright &y& Elsevier]

Abstract: In this paper, we prove the existence of homoclinic solutions for a class of noncoercive first order Hamiltonian systems , by the minimax methods in critical point theory, specially, a Generalized Mountain Pass Theorem, when is a linear operator with adjoint and satisfies the superquadratic condition as , uniformly in , and need not satisfy the global Ambrosetti–Rabinowitz condition. [Copyright &y& Elsevier]

In this paper we consider the first order discrete Hamiltonian systems , where $$x\left( n \right) = \left( {_{x_2 \left( n \right)}^{x_1 \left( n \right)} } \right) \in \mathbb{R}^{2N} , H\left( {n,z} \right) = \frac{1} {2}S\left( n \right)z \cdot z + R\left( {n,z} \right)$$ is periodic in n and superlinear as | z| → ∞. We prove the existence and infinitely many (geometrically distinct) homoclonic orbits of the system by critical point theorems for strongly indefinite functionals. [ABSTRACT FROM AUTHOR]

Abstract: The existence of homoclinic orbits is obtained for a class of the second order Hamiltonian systems , by the local linking lemma, where is superquadratic and need not be nonnegative globally. [Copyright &y& Elsevier]

We consider an autoparametric system which consists of an oscillator coupled with a parametrically excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in principal parametric resonance. The system contains the most general type of quadratic and cubic non-linearities. The method of second-order averaging is used to yield a set of autonomous equations of the second-order approximations to the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Shilnikov-type multi-pulse homoclinic orbits in the averaged equations. The results obtained above mean the existence of amplitude-modulated chaos in the Smale horseshoe sense in the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse chaotic motions of the parametric excited system with autoparametric resonance are also found by using numerical simulation. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

The travelling wave solutions of a generalized Camassa-Holm-Degasperis-Procesi equation u − u + (1+ b) u u = bu + uu are considered where b > 1 and m are positive integers. The qualitative analysis methods of planar autonomous systems yield its phase portraits. Its soliton wave solutions, kink or antikink wave solutions, peakon wave solutions, compacton wave solutions, periodic wave solutions and periodic cusp wave solutions are obtained. Some numerical simulations of these solutions are also given. [ABSTRACT FROM AUTHOR]

The paper concerns three-dimensional (3D) traveling gravity-capillary waves in water of finite depth. It was well known that the waves are determined by two constants: nondimensional wave speed F (called Froude number) and nondimensional surface tension b (called Bond number). For two-dimensional (2D) waves, it was known that F = 1 is a critical value, and there were many existence results for 2D waves with large or small b. However, there was still no existence of 3D waves when F is near 1 and b is small. In this paper, the existence of 3D waves in this case is discussed. It is shown that the exact Euler equations have a generalized solitary-wave solution (solitary waves with small oscillations at infinity) which is uniformly translating in the propagation direction and periodic in the transverse direction. The first-order approximation for this 3D wave is the solution of a system of coupled Schrödinger-KdV equations. [ABSTRACT FROM AUTHOR]

We consider a four-dimensional Hamiltonian system representing the reduced-order (two-mode) dynamics of a buckled beam, where the nonlinearity comes from the axial deformation in moderate displacements, according to classical theories. The system has a saddle-center equilibrium point, and we pay attention to the existence and detection of the stable-unstable nonlinear manifold and of homoclinic solutions, which are the sources of complex and chaotic dynamics observed in the system response. The system has also a coupling nonlinear parameter, which depends on the boundary conditions, and is zero, e.g., for the beam hinged-hinged ends and different from zero, e.g., for the beam fixed-fixed ends. The invariant manifold in the latter case is detected assuming that it can be represented as a graph over the plane spanned by the unstable (principal) variable and its velocity. We show by a series solution that the manifold exists but has a limited extension, not sufficient for the deployment of the homoclinic orbit. Thus, the homoclinic orbit is addressed directly, irrespective of its belonging to the invariant manifold. By means of the perturbation method, it is shown that it exists only on some curves of the governing parameters space, which branch from a fundamental path. This shows that the homoclinic orbit is not generic. These results have been confirmed by numerical simulations and by a different analytical technique. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of homoclinic orbits for the second-order Hamiltonian system with obstacle item, ü( t) − A $$ \dot u $$ ( t) = ∇ F( t, u), where F( t, u) is T-periodic in t with ∇ F( t, u) = L( t) u + ∇ R( t, u). By using a generalized linking theorem for strongly indefinite functionals, we prove the existence of homoclinic orbits for both the super-quadratic case and the asymptotically linear one. [ABSTRACT FROM AUTHOR]

We construct a theory-based numerical method for starting the continuation of homoclinic tangencies near 1 : 1 resonances, for systems with arbitrary dimension ≥ 2. The core of the method is numerical center manifold reduction and flow approximation. The reduction is implemented by means of the homological equation. The starting procedure is applied in numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the following nonperiodic second order Hamiltonian systems where may not be uniformly positive definite for all . Under more general conditions on , we prove that the above system has infinitely many homoclinic orbits provided R is even in u. [Copyright &y& Elsevier]

Abstract: In this paper we study the following nonperiodic second order Hamiltonian system where may not be uniformly positive definite for all , and is asymptotically quadratic in u as . By applying a generalized linking theorem in Bartsch and Ding (2006) , we obtain the existence and multiplicity of homoclinic orbits for the above system. [Copyright &y& Elsevier]

Abstract: This paper gives a direct, short and unified proof of Rabinowitz''s Theorem, Grossinho–Minhós–Tersian''s Theorem and Tersian–Chaparova''s Theorems on existence of homoclinic orbits for second order periodic Hamiltonian systems, fourth and sixth order periodic ordinary differential equations, respectively, by Brezis–Nirenberg type Mountain Pass Lemma. [Copyright &y& Elsevier]

Abstract: Two-dimensional travelling waves on an ideal fluid with gravity and surface tension over a periodically moving bottom with a small amplitude are studied. The bottom and the wave travel with a same speed. The exact Euler equations are formulated as a spatial dynamic system by using the stream function. A manifold reduction technique is applied to reduce the system into one of ordinary differential equations with finite dimensions. A homoclinic solution to the normal form of this reduced system persists when higher-order terms are added, which gives a generalized solitary wave—the homoclinic solution connecting a periodic solution. [Copyright &y& Elsevier]

This manuscript reports numerical investigations about the relative abundance and structure of chaotic phases in autonomous dissipative flows, i.e. in continuous-time dynamical systems described by sets of ordinary differential equations. In the first half, we consider flows containing "periodicity hubs", which are remarkable points responsible for organizing the dynamics regularly over wide parameter regions around them. We describe isolated hubs found in two forms of Rössler's equations and in Chua's circuit, as well as surprising infinite hub cascades that we found in a polynomial chemical flow with a cubic nonlinearity. Hub cascades converge orderly to accumulation points lying on specific parameter paths. In sharp contrast with familiar phenomena associated with unstable orbits, hubs and infinite hub cascades always involve stable periodic and chaotic orbits which are, therefore, directly measurable in experiments. In the last part, we consider flows having no hubs but unusual phase diagrams: a cubic polynomial model containing T-points and wide regions of dense chaos, a nonpolynomial model of the Belousov–Zhabotinsky reaction and the Hindmarsh–Rose model of neuronal bursting, both having chaotic phases with "fountains of chaos". The chaotic regions for the flows discussed here are different from what is known for discrete-time maps. This forcefully shows that knowledge about phase diagrams is quite fragmentary and that much work is still needed to classify and to understand them. [ABSTRACT FROM AUTHOR]

Abstract: Homoclinic and heteroclinic solutions are two important concepts that are used to investigate the complex properties of nonlinear evolutionary equations. In this Letter, we perform hyperbolic and linear stability analysis, and prove the existence of homoclinic and heteroclinic solutions for two-dimensional cubic Ginzburg–Landau equation with periodic boundary condition and even constraint. Then, using the Hirota''s bilinear transformation, we find the closed-form homoclinic and heteroclinic solutions. Moreover, we find that the homoclinic tubes and two families of heteroclinic solutions are asymptotic to a periodic cycle in one dimension. [Copyright &y& Elsevier]

Abstract: Diffusionless Lorenz equations (DLE) are a simple one-parameter version of the well-known Lorenz model, which was obtained in the limit of high Rayleigh and Prandtl numbers, physically corresponding to diffusionless convection. A simple control method is presented to control chaos by using periodic parameter perturbation in DLE. By using the generalized Melnikov method, the parameter conditions could be obtained to guide the controlled DLE to a low-periodic motion. Moreover, the existence conditions of periodic orbits and homoclinic orbits in the system are given. Some results of the numerical simulation are also explained clearly by a rigorous analysis. [Copyright &y& Elsevier]

In this paper we study the asymptotic solutions of the ( N+1)-body ring planar problem, N of which are finite and ν= N−1 are moving in circular orbits around their center of masses, while the Nth+1 body is infinitesimal. ν of the primaries have equal masses m and the Nth most-massive primary, with m0= β m, is located at the origin of the system. We found the invariant unstable and stable manifolds around hyperbolic Lyapunov periodic orbits, which emanate from the collinear equilibrium points L1 and L2. We construct numerically, from the intersection points of the appropriate Poincaré cuts, homoclinic symmetric asymptotic orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal points asymptotic orbits which intersect the x-axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points. All these families, for a fixed value of the mass parameter β=2, are found and presented. The eighteen (more geometrically simple) families and the corresponding eighteen terminating homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is computed and also presented. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a class of second order discrete Hamiltonian systems with periodicity assumptions is considered. Based on the critical point theory, some sufficient conditions for the existence of subharmonic solutions and homoclinic orbits are obtained. The results obtained extend the results in 2007 by relaxing the assumptions on the sign of the potential. [Copyright &y& Elsevier]

Abstract: This paper considers three-dimensional gravity–capillary waves on water of finite-depth, which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction. The exact Euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is a time-like variable. The existence of the solutions of the system is determined by two non-dimensional constants, the Bond number and the Froude number , which in turn give the number of eigenvalues on the imaginary axis of the complex plane for the corresponding linearized operator around a uniform flow. Assume that , is the curve in the -plane on which the first two eigenvalues for three-dimensional waves collide at the imaginary axis, and the intersection point of with is . In this paper, the case for and near is considered. A center-manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations. Using the existence of a homoclinic orbit connecting to a two-dimensional periodic (called generalized solitary-wave, thereafter) solution for the reduced system, it is shown that such a generalized solitary-wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants. [Copyright &y& Elsevier]

Abstract: We obtain a sufficient condition for the existence of nontrivial homoclinic orbits for fourth-order difference equations by using Mountain Pass Theorem, a weak convergence argument and a discrete version of Lieb’s lemma. [Copyright &y& Elsevier]

Abstract: We study the existence of infinitely many homoclinic orbits for some second-order Hamiltonian systems: , by the variant fountain Theorem, where satisfies the super-quadratic condition as uniformly in t, and need not satisfy the global Ambrosetti–Rabinowitz condition. [Copyright &y& Elsevier]

Abstract: In this paper we discuss how to use the critical point theory to study the existence of a nontrivial homoclinic orbit for nonlinear difference equations containing both advance and retardation without any periodic assumptions. Moreover, if the nonlinearity is an odd function, the existence of an unbounded sequence of homoclinic orbits is obtained. [Copyright &y& Elsevier]

In this paper, by using the critical point theory, we obtain the existence of a nontrivial homoclinic orbit which decays exponentially at infinity for nonlinear difference equations containing both advance and retardation without any periodic assumptions. Moreover, if the nonlinearity is an odd function, the existence of an unbounded sequence of nontrivial homoclinic orbits which decay exponentially at infinity is obtained. [ABSTRACT FROM AUTHOR]

A new vibro-impact bistable nonlinear energy sink (VI-BNES) with relatively light weight attached to a damped linear oscillator (LO) is proposed to form a two-degree-of-freedom nonlinear mechanical system with bilateral rigid constraints. The state of the system after instantaneous impact is described by the simplified shock theory and the momentum conservation. When the LO subjected to impulse excitations with different levels is considered as the driver of the VI-BNES, a two-dimensional non-smooth reduced impact system with a harmonic perturbation and an irrational nonlinear restoring force is obtained to study the threshold for the homoclinic bifurcations and chaotic responses of the VI-BNES. Bistability of this mechanical model makes the reduced unperturbed system have a pair of symmetric homoclinic orbits as the pseudo-separatrix for in-well and cross-well oscillations. The Melnikov method developed by us for non-smooth planar systems with bilateral rigid constraints is firstly employed to detect the threshold of the amplitude of excitations of the LO for the onset of chaotic oscillations. Furthermore, cubic polynomial approximation of the restoring force on the premise of maintaining the equilibria of the system is carried out such that the analytical Melnikov analysis for the global dynamics becomes possible. The performance of the VI-BNES is also studied and compared with other NESs in detail through numerical analysis and experiments to show the high efficiency of vibration reduction of the VI-BNES due to the dual beneficial role of the bistability and the impacts with the bilateral rigid constraints. • The structure of a vibro-impact bistable nonlinear energy sink, abbreviated as VI-BNES, is novel to combine both the characters of bistability and vibro-impacts. • The Melnikov method for non-smooth systems developed by us is firstly introduced to analyze the global dynamics and detect the chaotic threshold and homoclinic bifurcations of the non-smooth two-degree-of-freedom coupled systems. • Numerical method for vibro-impacts coupled systems is improved to show the good performance in reducing vibration of the VI-BNES, and experiments are designed to verify the high efficiency of vibration reduction for the VI-BNES. [ABSTRACT FROM AUTHOR]

In this work, a bidimensional continuous-time differential equations system is analyzed which is derived of Leslie-type predator-prey schemes by considering a nonmonotonic functional response and Allee effect on population prey. For the system obtained we describe the bifurcation diagram of limit cycles that appears in the first quadrant, the only quadrant of interest for the sake of realism. We show that, under certain conditions over the parameters, the system allows the existence of three limit cycles: The first two cycles are infinitesimal ones generated by Hopf bifurcation; the third one arises from a homoclinic bifurcation. Furthermore, we give conditions over the parameters such that the model allows long-term extinction or survival of both populations. In particular, the presence of a weak Allee effect does not imply extinction of populations necessarily for our model. [ABSTRACT FROM AUTHOR]

The authors study the bifurcation of homoclinic orbits from a degenerate homoclinic orbit in reversible system. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in two-dimensional manifolds. A perturbation technique for the detection of symmetric and nonsymmetric homoclinic orbits near the primary homoclinic orbits is developed. Some known results are extended. [ABSTRACT FROM AUTHOR]

In this paper, a class of second order discrete Hamiltonian systems without any periodicity assumptions are considered. Base on the critical point theory, some sufficient conditions for the existence of homoclinic orbits are obtained. The results obtained extend the results in [] by relaxing the assumptions on the sign of the potential. [ABSTRACT FROM AUTHOR]

Abstract: We show the existence of a nontrivial homoclinic orbit and subharmonic solutions for a class of second order difference equations by applying the “Mountain Pass” theorem relying on Ekeland’s variational principle and the diagonal method, and the homoclinic orbit as the limit of the subharmonics. A completely new way is provided for dealing with the existence of solutions for difference equations. [Copyright &y& Elsevier]

Abstract: We present a spectral method for the computation of homoclinic orbits in ordinary differential equations. The method is based on Hermite–Fourier expansions of the complete homoclinic solution and exhibits exponential convergence. In addition, our method can be used to approximate nonlinear functionals which depend on the complete homoclinic solution. This is demonstrated using examples from phase separation dynamics and metastability. [Copyright &y& Elsevier]

Abstract: We study -semi-static homoclinic orbits to Aubry sets in positive definite Lagrangian systems. [Copyright &y& Elsevier]

For a three-parametric family of continuous piecewise linear differential systems introduced by Arneodo et al. [1981] and considering a situation which is reminiscent of the Hopf-Zero bifurcation, an analytical proof on the existence of a two-parametric family of homoclinic orbits is provided. These homoclinic orbits exist both under Shil'nikov (0 < δ < 1) and non-Shil'nikov assumptions (δ ≥ 1). As it is well known for the case of differentiable systems, under Shil'nikov assumptions there exist infinitely many periodic orbits accumulating to the homoclinic loop. We also prove that this behavior persists at δ = 1. Moreover, for δ > 1 and sufficiently close to 1 we show that these periodic orbits persist but then they do not accumulate to the homoclinic orbit. [ABSTRACT FROM AUTHOR]

The stable and unstable invariant manifolds associated with Lyapunov orbits about the libration point L1 between the primaries in the planar circular restricted three-body problem with equal masses are considered. The behavior of the intersections of these invariant manifolds for values of the energy between that of L1 and the other collinear libration points L2, L3 is studied using symbolic dynamics. Homoclinic orbits are classified according to the number of turns about the primaries. [ABSTRACT FROM AUTHOR]

For a three-parametric family of continuous piecewise linear differential systems introduced by Arneodo et al. [1981] and considering a situation which is reminiscent of the Hopf-Zero bifurcation, an analytical proof on the existence of a two-parametric family of homoclinic orbits is provided. These homoclinic orbits exist both under Shil'nikov (0 < δ < 1) and non-Shil'nikov assumptions (δ ≥ 1). As it is well known for the case of differentiable systems, under Shil'nikov assumptions there exist infinitely many periodic orbits accumulating to the homoclinic loop. We also prove that this behavior persists at δ = 1. Moreover, for δ > 1 and sufficiently close to 1 we show that these periodic orbits persist but then they do not accumulate to the homoclinic orbit. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equations without any periodicity assumptions on , and f, providing that grows superlinearly both at origin and at infinity or is an odd function with respect to , and satisfies some additional assumptions. [Copyright &y& Elsevier]

A host-parasite model is proposed that incorporates a nonlinear incidence rate. Under the influence of multiple infectious attacks, the model admits bistable regions such that the infection dies out if initial states lie in one region, and the population and parasites coexist if initial states lie in the other region. It is also found that parasites can drive the population to extinction for suitable parameters. It is verified that the model has a saddle-node bifurcation, Hopf bifurcations and a cusp of codimension 2 or higher codimension. Stable limit cycles and unstable limit cycles are examined as the infection-reduced reproduction rate varies. It is shown that the model goes through the change of stages of infection extinction, infection persistence, infection extinction, and the extinction of both parasites and the population as the contact coefficient increases. [ABSTRACT FROM AUTHOR]

We consider a one-parameter family of discrete dynamical systems which has, for a critical value of the parameter, a homoclinic point with a quadratic tangency. We use Lin's method to prove the existence of shift dynamics near this homoclinic orbit. [ABSTRACT FROM AUTHOR]

To give rigorous mathematical proofs of chaotic behaviors in a given system, it is necessary to identify the homoclinic structures in the system. In this tutorial review, methods for constructing explicit solutions for nonlinear partial differential equations are presented, with more emphasis placed on those utilizing complete integrability associated with soliton equations. As an extended application, homoclinic orbits to spatial uniform plane waves of coupled modified nonlinear Schrödinger equations are obtained via the dressing method. During the procedure, it is necessary to introduce the Lax pair for these coupled equations, as well as its Floquet spectral analysis and corresponding Bloch functions. [ABSTRACT FROM AUTHOR]

Abstract: Rogue waves in deep water are investigated in the framework of the nonlinear Schrödinger (NLS) and the modified Dysthe (MD) equations. We observe that a chaotic regime increases the likelihood of rogue wave formation and that enhanced focusing occurs due to chaotic evolution of the phases. A Melnikov analysis indicates persistence of a homoclinic solution in the MD system which is -close to an optimally phase modulated solution of the NLS. The correlation of the Melnikov analysis and the numerical experiments indicates that one approach to predicting rogue waves in realistic oceanic states is to determine the proximity of a sea state to homoclinic data of the NLS. Using the inverse spectral theory of the NLS equation, we show that the development of extreme waves in random oceanic sea states characterized by JONSWAP power spectra is well predicted by the proximity to homoclinic data of the NLS. [Copyright &y& Elsevier]

Abstract: Systems for which each constant input gives rise to a unique globally attracting equilibrium are considered. A counterexample is provided to show that inputs which are only asymptotically constant may not result in states converging to equilibria (failure of the converging-input converging state, or “CICS” property). [Copyright &y& Elsevier]

This paper deals with Gause-type predator-prey models with a non-smooth prey growth rate. Our models have a unique positive equilibrium and are under the influence of an Allee effect. A necessary and sufficient condition is given for the existence of homoclinic orbits whose α- and ω-limit sets are the positive equilibrium. The argument used here is based on some results of a system of Liénard type. The relation between homoclinic orbits and the Anee effect is clarified. A simple example is included to illustrate the main result. Some global phase portraits are also attached. [ABSTRACT FROM AUTHOR]