Homoclinic Trajectories Research Articles

Chaotic Orbits for Systems of Nonlocal Equations

We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinc, homoclinic and chaotic trajectories are constructed. This is the first attempt to consider a nonlocal version of this type of dynamical systems in a variational setting and the first result regarding symbolic dynamics in a fractional framework.

On Saturn’s rotation relative to a center of mass under the action of the gravitational moments of the Sun and Jupiter

Saturn’s rotation relative to a center of mass is considered within an elliptic restricted three-body problem. It is assumed that Saturn is a solid under the action of gravity of the Sun and Jupiter. The motions of Saturn and Jupiter are considered elliptic with small eccentricities eS and eJ, respectively; the mean motion of Jupiter nJ is also small. We obtain the averaged Hamiltonian function for a small parameter of e = nJ and integrals of evolution equations. The main effects of the influence of Jupiter on Saturn’s rotation are described: (α) the evolution of the constant parameters of regular precession for the angular momentum vector I2; (β) the occurrence of new libration zones of oscillations I2 near the plane of the celestial equator parallel to the plane of the Jupiter’s orbit; (γ) the occurrence of additional unstable equilibria of vector I2 at the points of the north and south poles of the celestial sphere and, as a result, the existence of homoclinic trajectories; and (δ) the existence of periodic trajectories with arbitrarily large periods near the homoclinic trajectory. It is shown that the effects of (β), (γ), and (δ) are caused by the eccentricity e of the Jupiter’s orbit and are practically independent of Jupiter’s mass (within satellite approximation).

Necessary and sufficient conditions of the existence of homoclinic trajectories and cascade of bifurcations in Lorenz-like systems: birth of strange attractor and 9 homoclinic bifurcations

Necessary and sufficient conditions of the existence of homoclinic trajectories in Lorenz-like systems are obtained. The Shimizu–Morioka, Lu, and Chen systems are considered. For special Lorenz-like system with zero saddle value, cascade of bifurcations is obtained. Birth of strange attractor, blue sky catastrophe and 9 homoclinic bifurcations are discovered.

Existence Conditions of Homoclinic Trajectories in Tigan System

For Tigan system existence conditions of homoclinic trajectories are obtained. Algorithms for computing bifurcation values of parameters corresponding to homoclinic trajectories are formulated.

Counterexamples of continuous maps on dendrites

We take a list of properties of continuous maps on compact metric spaces, e.g. the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, the existence of an omega-limit set containing two minimal sets and others. Our aim is to find the relations between the properties in the case of dendrite maps. To achieve the aim at least partially, we consider several examples of dendrite maps and check which of the properties particular examples have. It also helps us to understand the dynamics of dendrite maps.

Spectral flow, crossing forms and homoclinics of Hamiltonian systems

We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions for bifurcation of homoclinic trajectories of one-parameter families of nonautonomous Hamiltonian vector fields.

Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity

In this paper a Lorenz-like system, describing convective fluid motion in rotating cavity, is considered. It is shown numerically that this system, like the classical Lorenz system, possesses a homoclinic trajectory and a chaotic self-excited attractor. However, for the considered system, unlike the classical Lorenz system, along with self-excited attractor a hidden attractor can be localized. Analytical-numerical localization of hidden attractor is demonstrated.

Existence criterion of homoclinic trajectories in the Glukhovsky–Dolzhansky system

Abstract Existence criterion of homoclinic trajectories in the Glukhovsky–Dolzhansky system, describing three-mode model of rotating fluid convection, is obtained. New applications of the Fishing principle are developed.

Bifurcation and chaos analysis of spur gear pair in two-parameter plane

A developed algorithm is designed based on the simple cell mapping method and escape time algorithm to examine every state cell and the dynamic characteristics of the multi-parameter coupling in torsion-vibration gear system. Two different types of bifurcation caused by the intersection of the period-doubling bifurcation curves are researched by analyzing the distribution map and the bifurcation diagram of system’s dynamic characteristic in the parameter plane, $$\omega -F$$ . The occurrence processes of periodic bubbles and saltatory periodic bifurcation are studied. The stationary solution and its phase trajectory of the fractal of the periodic motion attractor boundary are researched too. The sufficient condition of the fractal structure of the periodic attractor domain is achieved. Homoclinic or heteroclinic trajectory in phase space is found caused by the intersection of the different periodic motion trajectories in non-smooth system.

Localized Pulsating Solutions of the Generalized Complex Cubic-Quintic Ginzburg-Landau Equation

We study the dynamics of the localized pulsating solutions of generalized complex cubic-quintic Ginzburg-Landau equation (CCQGLE) in the presence of intrapulse Raman scattering (IRS). We present an approach for identification of periodic attractors of the generalized CCQGLE. Using ansatz of the travelling wave and fixing some relations between the material parameters, we derive the strongly nonlinear Lienard-Van der Pol equation for the amplitude of the nonlinear wave. Next, we apply the Melnikov method to this equation to analyze the possibility of existence of limit cycles. For a set of fixed parameters we show the existence of limit cycle that arises around a closed phase trajectory of the unperturbed system and prove its stability. We apply the Melnikov method also to the equation of Duffing-Van der Pol oscillator used for the investigation of the influence of the IRS on the bandwidth limited amplification. We prove the existence and stability of a limit cycle that arises in a neighborhood of a homoclinic trajectory of the corresponding unperturbed system. The condition of existence of the limit cycle derived here coincides with the relation between the critical value of velocity and the amplitude of the solitary wave solution (Uzunov, 2011).

Fishing principle for homoclinic and heteroclinic trajectories

In the present paper, a Fishing principle for the existence of homoclinic and heteroclinic trajectories is formulated and proved. This principle is applied to Lorenz-like systems and Phase locked loops.

О существовании гомоклинической траектории с симметрией в трехмерных системах

О существовании гомоклинической траектории с симметрией в трехмерных системах

Dynamics and chaos control of asymmetric gyrostat satellites

The motion of a free gyrostat consisting of a platform with a triaxial ellipsoid of inertia and a rotor with a slight asymmetry with respect to the axis of rotation is considered. Dimensionless equations of motion for a system with perturbations caused by the small asymmetries of the rotor are written in Andoyer-Deprit variables. These perturbations result in a chaotic layer in the separatrix vicinity. Heteroclinic and homoclinic trajectories are written in analytical form for gyrostats with different ratios of their moments of inertia. These trajectories are used to construct a modified Melnikov function, and to produce control that eliminates separatrix chaos. The Poincare sections and Melnikov function are constructed via numerical modeling that demonstrates the effectiveness of control.

Dynamics of Bimodality in Vehicular Traffic Flows

A model equation has been proposed to describe bimodal features in vehicular traffic flows. The dynamics of the bimodal distribution reveals the existence of a fixed point that is connected to itself by a homoclinic trajectory. The mathematical conditions associated with bimodality have been established. The critical factors necessary for both a breaking of symmetry and a transition from bimodal to uni-modal behaviour, in the manner of a bifurcation, have been analysed.

Horseshoes, Entropy, Homoclinic Trajectories, and Lyapunov Stability

We consider six properties of continuous maps, such as the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, or Lyapunov instability on the set of periodic points. The relations between the considered properties are provided in the case of graph maps, dendrite maps and maps on compact metric spaces. For example, by [Llibre & Misiurewicz, 1993] in the case of graph maps, the existence of an arc horseshoe implies the positivity of topological entropy, but we construct a continuous map on a Peano continuum with an arc horseshoe and zero topological entropy. We also formulate one open problem.

Robust Pulse Generators in an Excitable Medium with Jump-Type Heterogeneity

Heterogeneity is one of the most important and ubiquitous types of external perturbations. We study a spontaneous pulse-generating mechanism in an excitable medium with jump-type heterogeneity. Such a pulse generator (PG) has attracted considerable interest due to the computational potential of pulse waves in physiological signal processing. We first investigate the conditions for the onset of robust-type PGs, and then we show the global bifurcation structure of heterogeneity-induced patterns, including the complex ordered sequence of pulse-generating manners. We devise numerical frameworks to trace the long-term behavior of PGs as periodic solutions, and we detect the associated terminal homoclinic orbits that are homoclinic to a special type of heterogeneity-induced ordered pattern with a hyperbolic saddle. These numerical approaches assist us in identifying a candidate for the organizing center, and producing a variety of PGs as a codimension-two gluing bifurcation, in which two homoclinic trajectories...

Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator

Systems of operator-differential equations with hysteresis operators can have unstable equilibrium points with an open basin of attraction. Such equilibria can have homoclinic orbits attached to them, and these orbits are robust. In this paper a population dynamics model with hysteretic response of the prey to variations of the predator is introduced. In this model the prey moves between two patches, and the derivative of the Preisach operator is used to describe the hysteretic flow between the patches. A numerical example of a robust homoclinic loop is presented, and a mechanism creating this homoclinic trajectory is discussed.

Construction of codimension one homoclinic cycles

We give an explicit construction of families of Dm-equivariant polynomial vector fields in possessing a codimension one homoclinic cycle. The homoclinic cycle consists of m homoclinic trajectories all connected to the equilibrium at the origin. The constructed vector fields can provide a setting for a (numerical) bifurcation study of these homoclinic cycles, in particular for m equal to a multiple of 4, where the bifurcations form an open problem.

Homoclinic orbits in three-dimensional Shilnikov-type chaotic systems

In this paper, the Padé approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic trajectories in three-dimensional nonlinear dynamical systems. The PID controller system with quadratic and cubic nonlinearities, the simplified solar-wind-driven-magnetosphere-ionosphere system, and the human DNA sequence system are considered. With the aid of presenting a new condition, the solutions of solving the boundary-value problems which are formulated for the trajectory and evaluating the initial amplitude values become available. At the same time, the value of the bifurcation parameter is obtained directly, which is almost consistent with the numerical result.

Large deviations of Lyapunov exponents

Generic dynamical systems have ‘typical’ Lyapunov exponents, measuring the sensitivity to small perturbations of almost all trajectories. A generic system also has trajectories with exceptional values of the exponents, corresponding to unusually stable or chaotic situations. From a more mathematical point of view, large deviations of Lyapunov exponents characterize phase-space topological structures such as separatrices, homoclinic trajectories and degenerate tori. Numerically sampling such large deviations using the Lyapunov Weighted Dynamics allows one to pinpoint, for example, stable configurations in celestial mechanics or collections of interacting chaotic ‘breathers’ in nonlinear media. Furthermore, we show that this numerical method allows one to compute the topological pressure of extended dynamical systems, a central quantity in the thermodynamic of trajectories of Ruelle.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.