Attractors for Weak Solutions of a Regularized Model of Viscoelastic Mediums Motion With Memory in Non-Autonomous Case

We study existence of attractors for weak solutions of the regularized model for viscoelastic medium motion with memory in non-autonomous case. We apply the theory of trajectory attractors for non-invariant trajectory spaces and prove the existence of trajectory attractor, global attractor, uniform trajectory attractor, and uniform global attractor for this system.     

Richness of chaotic dynamics in nonholonomic models of a celtic stone

We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.     

The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration

This paper is concerned with the Chaplygin sleigh with time-varying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found.     

Regular and Chaotic Dynamics of a Chaplygin Sleigh due to Periodic Switch of the Nonholonomic Constraint

The main goal of the article is to suggest a two-dimensional map that could play the role of a generalized model similar to the standard Chirikov–Taylor mapping, but appropriate for energy-conserving nonholonomic dynamics. In this connection, we consider a Chaplygin sleigh on a plane, supposing that the nonholonomic constraint switches periodically in such a way that it is located alternately at each of three legs supporting the sleigh. We assume that at the initiation of the constraint the respective element (“knife edge”) is directed along the local velocity vector and becomes instantly fixed relative to the sleigh till the next switch. Differential equations of the mathematical model are formulated and an analytical derivation of mapping for the state evolution on the switching period is provided. The dynamics take place with conservation of the mechanical energy, which plays the role of one of the parameters responsible for the type of the dynamic behavior. At the same time, the Liouville theorem does not hold, and the phase volume can undergo compression or expansion in certain state space domains. Numerical simulations reveal phenomena characteristic of nonholonomic systems with complex dynamics (like the rattleback or the Chaplygin top). In particular, on the energy surface attractors associated with regular sustained motions can occur, settling in domains of prevalent phase volume compression together with repellers in domains of the phase volume expansion. In addition, chaotic and quasi-periodic regimes take place similar to those observed in conservative nonlinear dynamics.     

Regular and chaotic motions of a Chaplygin sleigh under periodic pulsed torque impacts

For a Chaplygin sleigh on a plane, which is a paradigmatic system of nonholonomic mechanics, we consider dynamics driven by periodic pulses of supplied torque depending on the instant spatial orientation of the sleigh. Additionally, we assume that a weak viscous force and moment affect the sleigh in time intervals between the pulses to provide sustained modes of the motion associated with attractors in the reduced three-dimensional phase space (velocity, angular velocity, rotation angle). The developed discrete version of the problem of the Chaplygin sleigh is an analog of the classical Chirikov map appropriate for the nonholonomic situation. We demonstrate numerically, discuss and classify dynamical regimes depending on the parameters, including regular motions and diffusive-like random walks associated, respectively, with regular and chaotic attractors in the reduced momentum dynamical equations.     

Dynamic complexities in predator–prey ecosystem models with age-structure for predator

Natural populations, whose generations are non-overlapping, can be modelled by difference equations that describe how the populations evolve in discrete time-steps. In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, in former studies most of the investigations of complex population dynamics were mainly concentrated on single populations instead of higher dimensional ecological systems. This paper reports a recent study on the complicated dynamics occurring in a class of discrete-time models of predator–prey interaction based on age-structure of predator. The complexities include (a) non-unique dynamics, meaning that several attractors coexist; (b) antimonotonicity; (c) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties, consisting of pattern of self-similarity and fractal basin boundaries; (d) intermittency; (e) supertransients; and (f) chaotic attractors.     

Measure attractors and random attractors for stochastic partial differential equations

In the theory of stochastic differential equations we can distinguish between two kinds of attractors. The first one is the attractor (measure attractor) with respect to the Markov semigroup generated by a stochastic differential equation. The second meaning of attractors (random attractors) is to be understood with respect to each trajectory of the random equation. The aim of this paper is to bring together the two meanings of attractors. In particular, we show the existence of measure attractors if random attractors exist. We can also show the uniqueness of the stationary distributions of the stochastic Navier-Stokes equation if the viscosity is large     

Strange attractors and mixed dynamics in the problem of an unbalanced rubber ball rolling on a plane

We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincaré map is enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos — the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given.     

Attractors for discrete periodic dynamical systems

A mathematical framework is introduced to study attractors of discrete, nonautonomous dynamical systems which depend periodically on time. A structure theorem for such attractors is established which says that the attractor of a time-periodic dynamical system is the union of attractors of appropriate autonomous maps. If the nonautonomous system is a perturbation of an autonomous map, properties that the nonautonomous attractor inherits from the autonomous attractor are discussed. Examples from population biology are presented.     

Bifurcation analysis of some forced Lu systems

Bifurcation behaviour of a forced Lu system is analyzed as the system parameter c and a forcing parameter F are varied. The Lu system belongs to a family of generalized Lorenz system. Members of this family are known to exhibit different types of chaotic attractors. Some of these attractors have been named Lorenz type L, Lu or Transition type T, Chen type T and Transverse 8 Type S. These different types of chaotic attractors are visually distinct when the parameters are widely separated. However, there is a need for identifying the precise point where transition from one type of chaotic attractor to another takes place. We identified signatures in the return map, which could be used for determining the point of transition and classifying the different types of chaotic attractors. These signatures helped to identify the point in coordinate space associated with such transitions. We find that such transitions take place when a chaotic attractor comes very close to a one-dimensional manifold on which the time derivatives of two of the variables is zero. We also find that just before coming to this point in coordinate space associated with the transition, the trajectory had approached, very closely, the equilibrium point at the origin.     

Noise in a Small Genetic Circuit that Undergoes Bifurcation

Based on the consideration of Boolean dynamics, it has been hypothesized that cell types may correspond to alternative attractors of a gene regulatory network. Recent stochastic Boolean network analysis, however, raised the important question concerning the stability of such attractors. In this paper a detailed numerical analysis is performed within the framework of Langevin dynamics. While the present results confirm that the noise is indeed an important dynamical element, the cell type as represented by attractors can still be a viable hypothesis. It is found that the stability of an attractor depends on the strength of noise related to the distance of the system to the bifurcation point and it can be exponentially stable depending on biological parameters.     

ON THE ATTRACTOR FOR A SEMILINEAR WAVE EQUATION WITH CRITICAL EXPONENT AND NONLINEAR BOUNDARY DISSIPATION

ABSTRACT

Long time behavior of a semilinear wave equation with nonlinear boundary dissipation and critical exponent is considered. It is shown that weak solutions generated by the wave dynamics converge asymptotically to a global and compact attractor. In addition, regularity and structure of the attractor are discussed in the paper. While this type of results are known for wave dynamics with interior dissipation this is, to our best knowledge, first result pertaining to boundary and nonlinear dissipation in the context of global attractors and their properties.     

Approximating topological approach to the existence of attractors in fluid mechanics

The aim of this paper is to demonstrate how the approximating topological method can be effectively combined with the theory of attractors of trajectory spaces in problems of fluid mechanics. First we give an exposition of the theory. Then we consider the model of motion of weak aqueous polymer solutions and prove that it has the minimal trajectory attractor and the global one. Finally we prove that the attractors of approximating problem converge to the attractors of the unperturbed one.     

Structure of attractors for skew product semiflows

In this work we study the continuity and structural stability of the uniform attractor associated with non-autonomous perturbations of differential equations. By a careful study of the different definitions of attractor in the non-autonomous framework, we introduce the notion of lifted-invariance on the uniform attractor, which becomes compatible with the dynamics in the global attractor of the associated skew product semiflow, and allows us to describe the internal dynamics and the characterization of the uniform attractors. The associated pullback attractors and their structural stability under perturbations will play a crucial role.     

Trajectory and global attractors for evolution equations with memory

Our aim in this note is to analyze the relation between two notions of attractors for the study of the long time behavior of equations with memory, namely, the global attractor in the so-called past history approach, and the more recently proposed notion of trajectory attractor.     

Dynamics of a generalized Ricker–Beverton–Holt competition model subject to Allee effects

In this article, we propose and study a generalized Ricker–Beverton–Holt competition model subject to Allee effects to obtain insights on how the interplay of Allee effects and contest competition affects the persistence and the extinction of two competing species. By using the theory of monotone dynamics and the properties of critical curves for non-invertible maps, our analysis show that our model has relatively simple dynamics, i.e. almost every trajectory converges to a locally asymptotically stable equilibrium if the intensity of intra-specific competition intensity exceeds that of inter-specific competition. This equilibrium dynamics is also possible when the intensity of intra-specific competition intensity is less than that of inter-specific competition but under conditions that the maximum intrinsic growth rate of one species is not too large. The coexistence of two competing species occurs only if the system has four interior equilibria. We provide an approximation to the basins of the boundary attractors (i.e. the extinction of one or both species) where our results suggests that contest species are more prone to extinction than scramble ones are at low densities. In addition, in comparison to the dynamics of two species scramble competition models subject to Allee effects, our study suggests that (i) Both contest and scramble competition models can have only three boundary attractors without the coexistence equilibria, or four attractors among which only one is the persistent attractor, whereas scramble competition models may have the extinction of both species as its only attractor under certain conditions, i.e. the essential extinction of two species due to strong Allee effects; (ii) Scramble competition models like Ricker type models can have much more complicated dynamical structure of interior attractors than contest ones like Beverton–Holt type models have; and (iii) Scramble competition models like Ricker type competition models may be more likely to promote the coexistence of two species at low and high densities under certain conditions: At low densities, weak Allee effects decrease the fitness of resident species so that the other species is able to invade at its low densities; While at high densities, scramble competition can bring the current high population density to a lower population density but is above the Allee threshold in the next season, which may rescue a species that has essential extinction caused by strong Allee effects. Our results may have potential to be useful for conservation biology: For example, if one endangered species is facing essential extinction due to strong Allee effects, then we may rescue this species by bringing another competing species subject to scramble competition and Allee effects under certain conditions.     

A three-dimensional nonlinear system with a single heteroclinic trajectory

The study for singular trajectories of three-dimensional (3D) nonlinear systems is one of recent main interests. To the best of our knowledge, among the study for most of Lorenz or Lorenz-like systems, a pair of symmetric heteroclinic trajectories is always found due to the symmetry of those systems. Whether or not does there exist a 3D system that possesses a single heteroclinic trajectory? In the present note, based on a known Lorenz-type system, we introduce such a 3D nonlinear system with two cubic terms and one quadratic term to possess a single heteroclinic trajectory. To show its characters, we respectively use the center manifold theory, bifurcation theory, Lyapunov function and so on, to systematically analyse its complex dynamics, mainly for the distribution of its equilibrium points, the local stability, the expression of locally unstable manifold, the Hopf bifurcation, the invariant algebraic surface, and its homoclinic and heteroclinic trajectories, etc. One of the major results of this work is to rigorously prove that the proposed system has a single heteroclinic trajectory under some certain parameters. This kind of interesting phenomenon has not been previously reported in the Lorenz system family (because the huge amount of related research work always presents a pair of heteroclinic trajectories due to the symmetry of studied systems). What"s more key, not like most of Lorenz-type or Lorenz-like systems with singularly degenerate heteroclinic cycles and chaotic attractors, the new proposed system has neither singularly degenerate heteroclinic cycles nor chaotic attractors observed. Thus, this work represents an enriching contribution to the understanding of the dynamics of Lorenz attractor.     

On three types of dynamics and the notion of attractor

We propose a theoretical framework for explaining the numerically discovered phenomenon of the attractor–repeller merger. We identify regimes observed in dynamical systems with attractors as defined in a paper by Ruelle and show that these attractors can be of three different types. The first two types correspond to the well-known types of chaotic behavior, conservative and dissipative, while the attractors of the third type, reversible cores, provide a new type of chaos, the so-called mixed dynamics, characterized by the inseparability of dissipative and conservative regimes. We prove that every elliptic orbit of a generic non-conservative time-reversible system is a reversible core. We also prove that a generic reversible system with an elliptic orbit is universal; i.e., it displays dynamics of maximum possible richness and complexity.