Homoclinic Trajectories Research Articles

Dust trapping in vortex pairs

The motion of tiny heavy particles transported in a co-rotating point vortex pair, with or without particle inertia and sedimentation, is investigated. The dynamics of non-inertial sedimenting particles is shown to be chaotic, under the combined effects of gravity and of the circular displacement of the vortices. This phenomenon is very sensitive to the particles’ inertia, if any. By using a nearly hamiltonian dynamical system theory for the particles’ motion equation written in the rotating reference frame, one can show that small inertia terms of the particles’ motion equation strongly modify the Melnikov function of the homoclinic trajectories and heteroclinic cycles of the unperturbed system, as soon as the particles’ response time is of the order of the settling time (Froude number of order unity). The critical Froude number above which chaotic motion vanishes and a regular centrifugation takes place is obtained from this Melnikov analysis and compared to numerical simulations. Particles with a finite inertia, and in the absence of gravity, are not necessarily centrifuged away from the vortex system. Indeed, these particles can have various equilibrium positions in the rotating reference frame, like the Lagrange points of celestial mechanics, according to whether their Stokes number is smaller or larger than some critical value. An analytical stability analysis reveals that two of these points are stable attracting points, so that permanent trapping can occur for inertial particles injected in an isolated co-rotating vortex pair. Particle trapping is observed to persist when viscosity, and therefore vortex coalescence, is taken into account. Numerical experiments at large but finite Reynolds number show that particles can indeed be trapped temporarily during vortex roll-up, and are eventually centrifuged away once vortex coalescence occurs.

The qualitative analysis of a class of planar Filippov systems

We use the theory of differential inclusions, Filippov transformations and some appropriate Poincaré maps to discuss the special case of two-dimensional discontinuous piecewise linear differential systems with two zones. This analysis applies to uniqueness and non-uniqueness for the initial value problem, stability of stationary points, sliding motion solutions, number of closed trajectories, existence of heteroclinic trajectories connecting two saddle points forming a heteroclinic cycle and existence of the homoclinic trajectory

Solitary Waves in Massive Nonlinear SN-Sigma Models

The solitary waves of massive (1+1)-dimensional nonlinear S^N-sigma models are unveiled. It is shown that the solitary waves in these systems are in one-to-one correspondence with the separatrix trajectories in the repulsive N-dimensional Neumann mechanical problem. There are topological (heteroclinic trajectories) and non-topological (homoclinic trajectories) kinks. The stability of some embedded sine-Gordon kinks is discussed by means of the direct estimation of the spectra of the second-order fluctuation operators around them, whereas the instability of other topological and non-topological kinks is established applying the Morse index theorem.

Entropy, horseshoes and homoclinic trajectories on trees, graphs and dendrites

AbstractIt is known that the positiveness of topological entropy, the existence of a horseshoe and the existence of a homoclinic trajectory are mutually equivalent, for interval maps. The aim of the paper is to investigate the relations between the properties for continuous maps of trees, graphs and dendrites. We consider three different definitions of a horseshoe and two different definitions of a homoclinic trajectory. All the properties are mutually equivalent for tree maps, whereas not for maps on graphs and dendrites. For example, positive topological entropy and the existence of a homoclinic trajectory are independent and neither of them implies the existence of any horseshoe in the case of dendrites. Unfortunately, there is still an open problem, and we formulate it at the end of the paper.

Vanishing beyond all orders: Stokes lines in a water-wave model equation

We study the existence of solutions homoclinic to a saddle centre in a family of singularly perturbed fourth order differential equations, originating from a water-wave model. Due to a reversibility symmetry, the occurrence of such embedded solitons is a codimension-1 phenomenon. By varying a parameter a countable family of solitary waves is found. We examine the asymptotic frequency at which this phenomenon of persistence in the singular limit occurs, by performing a refined Stokes line analysis. In the limit where the parameter tends to infinity, each Stokes line splits into a pair, and the contributions of these two Stokes lines cancel each other for a countable set of parameter values. More generally, we derive the full leading order asymptotics for the Stokes constant, which governs the (exponentially small) amplitude of the (minimal) oscillations in the tails of nearly homoclinic solutions. True homoclinic trajectories are characterized by the Stokes constant vanishing. This formal asymptotic analysis is supplemented with numerical calculations.

Homoclinic points and topological entropy of a continuous mapping of a dendrite

The authors proves sufficient conditions for homoclinic trajectories of a continuous dendrite mapping under which the topological entropy of the mapping is positive. It is shown that the obtained conditions are not necessary.

Dynamics of black hole pairs. II. Spherical orbits and the homoclinic limit of zoom-whirliness

Spinning black hole pairs exhibit a range of complicated dynamical behaviors. An interest in eccentric and zoom-whirl orbits has ironically inspired the focus of this paper: the constant radius orbits. When black hole spins are misaligned, the constant radius orbits are not circles but, rather, they lie on the surface of a sphere and have acquired the name ``spherical orbits.'' The spherical orbits are significant as they energetically frame the distribution of all orbits. In addition, each unstable spherical orbit is asymptotically approached by an orbit that whirls an infinite number of times, known as a homoclinic orbit. A homoclinic trajectory is an infinite whirl limit of the zoom-whirl spectrum and has a further significance as the separatrix between inspiral and plunge for eccentric orbits. We work in the context of two spinning black holes of comparable mass as described in the third-order post-Newtonian Hamiltonian with spin-orbit coupling included. As such, the results could provide a testing ground for the accuracy of the post-Newtonian expansion. Further, the spherical orbits could provide useful initial data for numerical relativity. Finally, we comment that the spinning black hole pairs should give way to chaos around the homoclinic orbit when spin-spin coupling is incorporated.

Chaotic particle sedimentation in a rotating flow with time-periodic strength

Particle sedimentation in the vicinity of a fixed horizontal vortex with time-dependent intensity can be chaotic, provided gravity is sufficient to displace the particle cloud while the vortex is off or weak. This "stretch, sediment, and fold" mechanism is close to the so-called blinking vortex effect, which is responsible for chaotic transport of perfect tracers, except that in the present case the vortex motion is replaced by gravitational settling. In the present work this phenomenon is analyzed for heavy Stokes particles moving under the sole effect of gravity and of a linear drag. The vortex is taken to be a fixed isolated point vortex, the intensity of which varies under the effect of either boundary conditions or volume force. When the unsteadiness of the vortex is weak and the free-fall velocity is of the order of the fluid velocity, and the particle response time is small, the particle motion equation can be written asymptotically as a perturbed Hamiltonian system, the phase portrait of which displays a homoclinic trajectory. A homoclinic bifurcation is therefore likely to occur, and the contribution of particle inertia to the occurrence of this bifurcation is analyzed asymptotically by using Melnikov's method.

Solutions Classification to the Extended Reduced Ostrovsky Equation

An alternative to the Parkes' approach (SIGMA 4 (2008), 053, 17 pages) is suggested for the solutions categorization to the extended reduced Ostrovsky equation (the exROE in Parkes' terminology). The approach is based on the application of the qualitative theory of differential equations which includes a mechanical analogy with the point particle motion in a potential field, the phase plane method, analysis of homoclinic trajectories and the like. Such an approach is seemed more vivid and free of some restrictions contained in (SIGMA 4 (2008), 053, 17 pages).

Generalized MSTB models: Structure and kink varieties

In this paper, we describe the structure of a class of two-component scalar field models in a (1+1) Minkowskian space–time which generalize the well-known Montonen–Sarker–Trullinger–Bishop — hence MSTB-model. This class includes all the field models whose static field equations are equivalent to the Newton equations of two-dimensional type I Liouville mechanical systems, with a discrete set of instability points. We offer a systematic procedure to characterize these models and to identify the solitary wave or kink solutions as homoclinic or heteroclinic trajectories in the analogous mechanical system. This procedure is applied to a one-parametric family of generalized MSTB models with a degree-eight polynomial as potential energy density.

Bifurcation of homoclinics of Hamiltonian systems

We obtain sufficient conditions for bifurcation of homoclinic trajectories of nonautonomous Hamiltonian vector fields parametrized by a circle, together with estimates for the number of bifurcation points in terms of the Maslov index of the asymptotic stable and unstable bundles of the linearization at the stationary branch.

Bifurcation of homoclinics

We show that homoclinic trajectories of nonautonomous vector fields parametrized by a circle bifurcate from the stationary solution when the asymptotic stable bundles of the linearization at plus and minus infinity are “twisted” in different ways.

Homoclinic Trajectories in Discontinuous Systems

We study bifurcations of bounded solutions from homoclinic orbits for time-perturbed discontinuous systems. Functional analytic method is used. An illustrative example of a periodically perturbed piecewise linear differential equation in $$\mathbb{R}^3$$ is presented.

Scenarios of the onset of unsteady regimes in the problem of plane convective flow through a porous medium

The problem of plane convective flow through a porous medium in a rectangular vessel with a linear temperature profile steadily maintained on the boundary is considered. The onset of unsteady regimes is investigated numerically. It is shown that their onset scenarios depend on the vessel dimensions and the seepage Rayleigh number and may be as follows: the generation of stable and unstable periodic regimes as a result of a one-sided bifurcation, the generation of a stable periodic regime as a result of an Andronov-Hopf cosymmetric bifurcation, the formation of a chaotic attractor, the branching-out of a stable quasi-periodic regime from a point of a single-parameter family of steady-state regimes, and the generation of unstable periodic regimes as a result of disintegration of homoclinic trajectories. The specifics of most of the bifurcations mentioned above are attributable to the cosymmetry of the problem considered.

Strange distributionally chaotic triangular maps III

In the class T of triangular maps of the square we consider the strongest notion of distributional chaos, DC1, originally introduced by Schweizer and Smítal [Trans Amer Math Soc 1994;344:737–854] for continuous maps of the interval. We show that a map F ∈ T is DC1 if F has a periodic orbit with period ≠ 2 n , for any n ⩾ 0. Consequently, a map in T is DC1 if it has a homoclinic trajectory. This result is important since in general systems like T , positive topological entropy itself does not imply DC1. It contributes to the solution of a long-standing open problem of A. N. Sharkovsky concerning classification of triangular maps of the square.

Determination of the Chaos Onset in Mechanical Systems with Several Equilibrium Positions

Determination of the chaos onset in some mechanical systems with several equilibrium positions are analyzed. Namely, the snap-through truss and the oscillator with a nonlinear dissipation force, under the external periodical excitation, are considered. Two approaches are used for the chaos onset determination. First, Pade and quasi-Pade approximants are used to construct closed homoclinic trajectories for a case of small dissipation. Convergence condition used earlier in the theory of nonlinear normal vibration modes as well conditions at infinity make possible to evaluate initial amplitude values for the trajectories with admissible precision. Mutual instability of phase trajectories is used as criterion of chaotic behavior in nonlinear systems for a case of not small dissipation. The numerical realization of the Lyapunov stability definition gives us a possibility to observe a process of appearance and fast enlargement of the chaotic behavior regions if some selected parameters of the dynamical systems under consideration are changing.

Non-linear analysis and modelling of the cellular mechanisms that regulate arterial vasomotion

Non-linearity intrinsic to the ion transport systems that regulate intracellular [Ca2+] and smooth muscle tone allows the emergence of fluctuations in vascular diameter and resistance in isolated arteries (vasomotion) that can be classified as chaotic. Correlation analysis suggests an underlying low dimensional system and a four-dimensional model of vasomotion has been formulated to simulate the effects of pharmacological manipulation of smooth muscle tone or nitric oxide (NO) synthesis by the vascular endothelium. The oscillatory patterns observed experimentally and in modelling studies may be considered ‘universal’ in the sense that they also occur in many physico-chemical systems and include: (a) period-doubling, a feature of the Feigenbaum route to chaos; (b) mode-locking and quasiperiodicity, which reflect the interaction of two nonlinear oscillatory subsystems; and (c) intermittency, in which segments of nearly periodic oscillations of variable duration are interrupted by short chaotic bursts. Low dimensionality allows the construction of iterative maps that confirm the existence of types I and III Pomeau-Manneville intermittency in vascular dynamics, with attractor reconstructions indicating that the reinjection mechanism underlying the type III scenario involves a Shil'nikov-type homoclinic trajectory. Dimensional analysis of experimental data and corresponding surrogate time series generated by randomization of Fourier phase also provide evidence for underlying nonlinear structure in fluctuations in red cell velocity and arteriolar calibre in vivo.

Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits in the sun-earth and earth-moon systems

In this paper a method for finding homoclinic and heteroclinic connections between Lyapunov orbits using invariant manifolds in a given energy surface of the planar restricted circular three body problem is developed. Moreover, the systematic application of this method to a range of Jacobi constants provides a classification of the connections in bifurcation families. The models used correspond to the Sun-Earth+Moon and the Earth-Moon cases.

Exponentially small separatrix splittings and almost invisible homoclinic bifurcations in some billiard tables

We present a numerical study of some billiard tables depending on a perturbative parameter ɛ ≥ 0 and a hyperbolicity parameter h > 0 . These tables are ellipses for ɛ = 0 and circumferences in the limit h → 0 + . Elliptic billiard tables are integrable and have four separatrices, which break up when ɛ > 0 . We conjecture, based on numerical experiments, that as h → 0 + the area of the main lobes of the resulting turnstile (which can be interpreted as the difference of the lengths of the symmetric primary homoclinic billiard trajectories) behaves like an exponential term ɛ e − π 2 / h times an asymptotic series ∑ j ≥ 0 α j ɛ h 2 j such that α 0 ɛ ≠ 0 . This series is Gevrey-1 of type ρ = 1 / 2 π 2 , so that its Borel transform is convergent on a disk of radius 2 π 2 . In the limit ɛ → 0 , the series ∑ j ≥ 0 α j 0 h 2 j is an analytic function which can be explicitly computed with a discrete Melnikov method. The asymptotic series ∑ j ≥ 0 ω j ɛ h 2 j associated to the second exponential term ɛ e − 2 π 2 / h has the same properties. Finally, we have detected some almost invisible homoclinic bifurcations that take place in an exponentially small region of the parameter space. Our computations have been performed in multiple-precision arithmetic (namely, with several thousands decimal digits) and rely strongly on the expansion of the local invariant curves up to very high orders (namely, with several hundreds Taylor coefficients). Our programs have been written using the PARI system.

Triangular Maps Non-Decreasing on the Fibers

There is a list of about 50 properties which characterize continuous maps of the interval with zero topological entropy. Most of them were proved by A.N. Sharkovsky [cf., e.g., Sharkovsky et al., Dynamics of One-Dimensional Mappings, Kluwer 1997]. It is also well known that only a few of these properties remain equivalent for continuous maps of the square. Recall, e.g., the famous Kolyada's example of a triangular map of type $2^\infty$ with positive topological entropy. In 1989 Sharkovsky formulated the problem to classify these conditions in a special case of triangular maps of the square. The present paper is a step toward the solution. In particular, we give a classification of 23 conditions in the case of triangular maps of the square which are non-decreasing on the fibers. We show that the weakest is"no homoclinic trajectory", the two strongest, mutually incomparable, are "map restricted to the set of chain recurrent points is not Li and Yorke chaotic" and "every $\omega$-limit set contains a unique minimal set"