Heteroclinic Solutions Research Articles

On the solvability of a boundary value problem on the real line

Abstract We investigate the existence of heteroclinic solutions to a class of nonlinear differential equations ( a ( x ) Φ ( x ′ ( t ) ) ) ′ = f ( t , x ( t ) , x ′ ( t ) ) , a . e . t ∈ ℝ governed by a nonlinear differential operator Φ extending the classical p- Laplacian, with right-hand side f having the critical rate of decay -1 as |t| → +∞, that is f ( t , ⋅ , ⋅ ) ≈ 1 t . We prove general existence and non-existence results, as well as some simple criteria useful for right-hand side having the product structure f(t, x, x') = b(t, x)c(x, x'). Mathematical subject classification: Primary: 34B40; 34C37; Secondary: 34B15; 34L30.

Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays

For a Nicholson’s blowflies model with patch structure and multiple discrete delays, we study some aspects of its global dynamics. Conditions for the absolute global asymptotic stability of both the trivial equilibrium and a positive equilibrium (when it exists) are given. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. We further consider a diffusive Nicholson-type model with patch structure, and establish a criterion for the existence of positive travelling wave solutions, for large wave speeds. Several applications illustrate the results, improving some criteria in the recent literature.

Heteroclinic dynamics and attitude motion chaotization of coaxial bodies and dual-spin spacecraft

Heteroclinic dynamics of a free coaxial bodies system and dual-spin spacecraft is examined. New analytical solutions for heteroclinic orbits, corresponded to the polhodes-separatrices in the space of the angular moment components, are obtained. On the base of these analytical heteroclinic solutions analysis of possibility of the system motion chaotization with the help of Melnikov method is conducted. The analysis shows the polhode-separatrix-orbit splitting at presence of small harmonical perturbation torques between the coaxial bodies. The separatrix splitting generates the chaotic layer near the unperturbed separatrix region. This fact proves possibility of realization of non-regular dynamics and chaotic tilting motion of the dual-spin spacecraft.

Heteroclinic Solutions to Asymptotically Autonomous Equations via Continuation Methods

Abstract By means of a continuation argument, we prove the existence of at least one increasing heteroclinic solution to a scalar equation of the kind ẍ = a(t)V′(x), where V is a non-negative double well potential, and a(t) is a positive, measurable coefficient, which is definitively monotone with respect to |t|, converges to l as |t| diverges and fulfils one of the two following assumptions: a(t) ≥ l everywhere, or a(t) − l converges to 0, as |t| → +∞, more slowly than a suitable exponential term.

Heteroclinic solutions of boundary-value problems on the real line involving general nonlinear differential operators

We discuss the solvability of the following strongly nonlinear non-autonomous boundary-value problem: \[ (P) \quad \left \{ \begin{array}{ll} (a(x(t))\Phi(x'(t)))' = f(t,x(t),x'(t)) \ \ \mbox{a.e. } t\in {\mathbb R} & \\ x(-\infty)=\nu^- ,\ \ x(+\infty)= \nu^+ & \end{array} \right . \] with $\nu^-< \nu^+$, where $\Phi:{\mathbb R} \to {\mathbb R}$ is a general increasing homeomorphism, with $\Phi(0)=0$, $a$ is a positive, continuous function and $f$ is a Caratheódory nonlinear function. We provide some sufficient conditions for the solvability of $(P)$ which turn out to be optimal for a large class of problems. In particular, we highlight the role played by the behavior of $f(t,x,\cdot)$ and $\Phi(\cdot)$ as $y\to 0$ related to that of $f(\cdot,x,y)$ as $|t|\to +\infty$. We also show that the dependence on $x$, both of the differential operator and of the right-hand side, does not influence in any way the existence or non-existence of solutions.

A note on hybrid heteroclinic solutions for a class of semilinear elliptic PDEs

Variational methods will be used to find solutions of a semilinear elliptic PDE by ‘‘glueing’’ mountain pass critical points of the corresponding functional to minima or to other mountain pass critical points.

Existence of heteroclinic solutions for discrete [formula omitted]-Laplacian problems with a parameter

In this paper, we study the existence of a heteroclinic solution for a discrete p -Laplacian problem depending on a parameter. The proof is deduced from a previous multiplicity result given by Candito and Giovannelli for a Dirichlet discrete boundary value problems with a parameter.

Existence of subsonic heteroclinic waves for the Frenkel–Kontorova model with piecewise quadratic on-site potential

We give an existence proof for heteroclinic wave solutions to the Frenkel–Kontorova model of dislocation dynamics. The on-site potential is assumed to be piecewise quadratic. The framework employed here allows us to determine explicitly a regime of subsonic velocities for which travelling waves exist.

Heteroclinic solutions of boundary value problems on the real line involving singular Φ-Laplacian operators

We discuss the solvability of the following strongly nonlinear BVP: { ( a ( x ( t ) ) Φ ( x ′ ( t ) ) ) ′ = f ( t , x ( t ) , x ′ ( t ) ) , t ∈ R , x ( − ∞ ) = α , x ( + ∞ ) = β where α < β , Φ : ( − r , r ) → R is a general increasing homeomorphism with bounded domain ( singular Φ-Laplacian), a is a positive, continuous function and f is a Carathéodory nonlinear function. We give conditions for the existence and non-existence of heteroclinic solutions in terms of the behavior of y ↦ f ( t , x , y ) and y ↦ Φ ( y ) as y → 0 , and of t ↦ f ( t , x , y ) as | t | → + ∞ . Our approach is based on fixed point techniques suitably combined to the method of upper and lower solutions.

Standing Waves Joining with Turing Patterns in FitzHugh–Nagumo Type Systems

An article by Kondo and Asai demonstrated that the pattern formation and change on the skin of tropical fishes can be predicted well by reaction-diffusion models of Turing type. As being observed, a common pattern structure is the rearrangement of stripe pattern, and defect like heteroclinic solution appeared between the patterns with different number of stripes. We consider FitzHugh–Nagumo type reaction-diffusion systems with anisotropic diffusion. Under a sufficient condition in diffusivity, we apply variational arguments to show the existence of standing waves joining with Turing patterns.

Erratum to: Fast and heteroclinic solutions for a second order ODE related to Fisher-Kolmogorov’s equation

The proof of Proposition 11 in [1] has a gap. The use of differential inequalities in line 7 of the statement in the beginning of page 333, is not correct. The aim of this note is to present a correct proof of this result. The question is to prove that if the equation (1) has a non fast heteroclinic solution for some c1 > 2 √ f ′(0) , then c1 > c∗, where c∗ := inf{c ∈ R : (1) has an heteroclinic solution}. So, we only have to prove there exists an heteroclinic solution for values below c1. By the claim in page 332 in [1] it is sufficient to find a solution of (19) for some c 0, u ∈]0, 1]. Let y1 be the solution of (19) with c = c1 corresponding to the non fast heteroclinic solution of (1). Then y1 satisfies y1(0) = y1(1) = 0. Consider 2 √ f ′(0) y1(u) for small values

Bounded traveling waves of the Burgers–Huxley equation

In order to investigate bounded traveling waves of the Burgers–Huxley equation, bifurcations of codimension 1 and 2 are discussed for its traveling wave system. By reduction to center manifolds and normal forms we give conditions for the appearance of homoclinic solutions, heteroclinic solutions and periodic solutions, which correspondingly give conditions of existence for solitary waves, kink waves and periodic waves, three basic types of bounded traveling waves. Furthermore, their evolutions are discussed to investigate the existence of other types of bounded traveling waves, such as the oscillatory traveling waves corresponding to connections between an equilibrium and a periodic orbit and the oscillatory kink waves corresponding to connections of saddle–focus.

Positive travelling fronts for reaction–diffusion systems with distributed delay

We give sufficient conditions for the existence of positive travelling wave solutions for multi-dimensional autonomous reaction–diffusion systems with distributed delay. To prove the existence of travelling waves, we give an abstract formulation of the equation for the wave profiles in some suitable Banach spaces and apply known results about the index of some associated Fredholm operators. After a Lyapunov–Schmidt reduction, these waves are obtained via the Banach contraction principle, as perturbations of a positive heteroclinic solution for the associated system without diffusion, whose existence is proven under some requirements. By a careful analysis of the exponential decay of the travelling wave profiles at −∞, their positiveness is deduced. The existence of positive travelling waves is important in terms of applications to biological models. Our method applies to systems of delayed reaction–diffusion equations whose nonlinearities are not required to satisfy a quasi-monotonicity condition. Applications are given, and include the delayed Fisher–KPP equation.

Monotone heteroclinic solutions to non-autonomous equations via phase plane analysis

We study the existence of at least one increasing heteroclinic solution to a scalar equation of the kind ẍ = a(t)V′(x), where V is a non-negative double well potential, and a(t) is a positive, measurable coefficient. We first provide with a complete answer in the definitively autonomous case, when a(t) takes a constant value l outside a bounded interval. Then we consider the case in which a(t) is definitively monotone, converges from above, as t → ±∞, to two positive limits l* and l*, and never goes below min(l*, l*). Furthermore, the convergence to max(l*, l*) is supposed to be not too fast (slower than a suitable exponential term).

Robust heteroclinic cycles in delay differential equations

We consider the existence of heteroclinic cycles in Γ-equivariant delay-differential equations which emerge from symmetry-breaking bifurcations from an equilibrium solution with maximal isotropy subgroup. We begin by describing the existence of robust heteroclinic cycles on finite-dimensional centre manifolds and show that these are also robust to Γ-equivariant perturbations of the delay-differential equation. We then present the first example of a delay-differential equation which supports a heteroclinic cycle not contained within a finite-dimensional submanifold. This system is a delayed version of the Guckenheimer and Holmes equivariant three-dimensional example realized as a coupled cell system. We prove the existence of the heteroclinic cycle and show that it is structurally stable to Γ-equivariant perturbations which preserve certain codimension one subspaces of phase space associated with fixed point subspaces. By letting the cell dynamics be delay-dependent, we show that for a large enough delay, we obtain a heteroclinic cycle joining periodic solutions. Numerical simulations are presented and discussed.

Heteroclinic Orbits in Slow–Fast Hamiltonian Systems with Slow Manifold Bifurcations

Motivated by a problem in which a heteroclinic orbit represents a moving inter- face between ordered and disordered crystalline states, we consider a class of slow-fast Hamiltonian systems in which the slow manifold loses normal hyperbolicity due to a trans- critical or pitchfork bifurcation as a slow variable changes. We show that under assumptions appropriate to the motivating problem, a singular heteroclinic solution gives rise to a true heteroclinic solution. In contrast to previous approaches to such problems, our approach uses blow-up of the bifurcation manifold, which allows geometric matching of inner and outer solutions.

Heteroclinic Breather-Wave Solutions for Davey–Stewartson Equation

Exact heteroclinic breather-wave solutions for Davey–Stewartson (DSI, DSII) system with periodic boundary condition are constructed using Hirota's bilinear form method and generalized ansatz method. The heteroclinic structure of wave is investigated.

Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by hyperbolic Lindstedt-Poincaré method

The hyperbolic Lindstedt-Poincare method is applied to determine the homoclinic and heteroclinic solutions of cubic strongly nonlinear oscillators of the form $$ \ddot x + c_1 x + c_3 x^3 = \varepsilon f(\mu ,x,\dot x) $$ . In the method, the hyperbolic functions are employed instead of the periodic functions in the Lindstedt-Poincare procedure. Critical value of parameter µ under which there exists homoclinic or heteroclinic orbit can be determined by the perturbation procedure. Typical applications are studied in detail. To illustrate the accuracy of the present method, its predictions are compared with those of Runge-Kutta method.

Mechanical effects coupled with calcium waves

In the paper we find explicit formulae for heteroclinic travelling wave solutions in the system of equations describing the dynamics of cytosolic calcium concentration and the accompanying mechanical phenomena.

On the existence of heteroclinic trajectories for asymptotically autonomous equations

By means of a minimax argument, we prove the existence of at least one heteroclinic solution to a scalar equation of the kind $\ddot x=a(t)V'(x)$, where $V$ is a double well potential, $0< l\le a(t)\le L$, $a(t)\to l$ as $|t|\to\infty$ and the ratio $L/l$ is suitably bounded from above.