Nontrivial Attractors Research Articles

Templex-based dynamical units for a taxonomy of chaos.

Discriminating different types of chaos is still a very challenging topic, even for dissipative three-dimensional systems for which the most advanced tool is the template. Nevertheless, getting a template is, by definition, limited to three-dimensional objects based on knot theory. To deal with higher-dimensional chaos, we recently introduced the templex combining a flow-oriented BraMAH cell complex and a directed graph (a digraph). There is no dimensional limitation in the concept of templex. Here, we show that a templex can be automatically reduced into a "minimal" form to provide a comprehensive and synthetic view of the main properties of chaotic attractors. This reduction allows for the development of a taxonomy of chaos in terms of two elementary units: the oscillating unit (O-unit) and the switching unit (S-unit). We apply this approach to various well-known attractors (Rössler, Lorenz, and Burke-Shaw) as well as a non-trivial four-dimensional attractor. A case of toroidal chaos (Deng) is also treated.

Near-critical behavior of the Zhong-Zhang model.

The Zhong-Zhang (ZZ) model is a one-degree-of-freedom dynamical system describing the motion of an insulating plate of length d floating on the upper surface of a convecting fluid, with locking at the boundaries. In the absence of noise, the system away from the boundaries is described by linear differential equationswith a delay time τ. The d,τ plane consists of two domains separated by a critical curve. For asymptotically long times, subcritical orbits approach a nontrivial periodic attractor, while the supercritical ones tend to a stationary state at the origin. We investigate near-critical behavior using a modified fourth-order Runge-Kutta integration scheme. We then construct a piecewise analytic decomposition of the periodic attractor, which makes possible a far higher level of accuracy. Our results provide solid evidence for an asymptotic power-law approach to criticality of several observables. The power laws are fed back to determine the piecewise-analytic structure deep into the near-critical regime. In an Appendix, we explore the effect of introducing noise using modified order-3/2 Kloeden-Platen-Schurz stochastic integration, following several observable quantities through the near-critical parameter domain.

Robustness of global attractors: Abstract framework and application to dissipative wave equations

<p style='text-indent:20px;'>We establish local input-to-state stability and the asymptotic gain property for a class of infinite-dimensional systems with respect to the global attractor of the respective undisturbed system. We apply our results to a large class of dissipative wave equations with nontrivial global attractors.</p>

Bifurcations and Synchronization in Networks of Unstable Reaction–Diffusion Systems

This article is devoted to the analysis of the dynamics of a complex network of unstable reaction–diffusion systems. We demonstrate the existence of a non-empty parameter regime for which synchronization occurs in non-trivial attractors. We establish a lower bound of the dimension of the global attractor in an innovative manner, by proving a novel theorem of continuity of the unstable manifold, for which we invoke a principle of spectrum perturbation of non-bounded operators. Finally, we exhibit a co-dimension 2 bifurcation of the unstable manifold which shows that synchronization is compatible with instabilities.

Study of the Existence of Global Attractors for the Wezewska, Czyewska and Lasota Models

In this research we present a study of global attractors in mathematical models of differential equations, which are an important tool in mathematics; furthermore, taking advantage of the stability of the solutions, it was possible to determine the control of biomedical phenomena, among other aspects, present in various population groups. Likewise, differential equation models are used to simulate biological, epidemiological and medical phenomena, among others. The reference population groups used in this research are the family of population models given by the differential equations N'(t)=p(t, N (t)) - d(t, N (t)). A particular case of this family of differential equations is the mathematical model called the Wezewska, Czyewska and Lasota (WCL) model, whose form is given by: N'(t)=pe(-q) - μN (t). This model describes the survival of red blood cells (erythrocytes) in humans. The WCL model, in discrete variable, has a non-trivial global attractor. In this research we demonstrate, using the Schwarz derivative technique, the existence of at least one model global attractor. On the other hand, the results of the present investigation showed the existence of a single fixed point, as the only global attractor characterized by the equation N=pe(-qN) - μN.

Designing megastable systems with multidimensional lattice of chaotic attractors

Multistable systems and their dynamic properties are interesting topics in nonlinear dynamics. Small differences in the initial conditions (for example, due to rounding errors in numerical calculations) lead to completely different results for such dynamic systems, which makes a long-term prediction of their behavior practically impossible. This happens even if such systems are deterministic, that is, their future behavior is completely determined by the choice of initial conditions without the participation of random elements. In other words, the deterministic nature of these systems does not make them predictable. The behavior of the solutions of a dynamic system depends both on the choice of their initial conditions and on the values of the system parameters. The coexistence of several attractors, or multistability, corresponds to the simultaneous existence of more than one nontrivial attractor for the same set of system parameters. This phenomenon was discovered in almost all natural sciences, including electronics, optics, biology. In recent years, the efforts of many researchers have been aimed at creating so-called megastable systems, that is, systems that, at constant values of their parameters, have a countable number of coexisting attractors. Interest in such systems is due to a wide range of applications, for example, for hiding information in communication systems and audio encryption schemes, biomedical engineering and fuzzy control. The article proposes methods for the synthesis of megastable systems using systems in Lurie form. Megastable systems containing a 1-D lattice of chaotic attractors can be obtained by replacing the nonlinearity in the original system with a periodic function. By replacing some variables with periodic functions in the original system of order n, one can construct a megastable system containing an n-D lattice of chaotic attractors. As one example, a fourth-order system with a 4-D lattice of chaotic attractors is constructed for the first time. The Lyapunov exponents and Kaplan – Yorke dimension are calculated for attractors belonging to lattices

Input-to-state stability results w.r.t. global attractors of semi-linear reaction-diffusion equations

Abstract We establish local input-to-state stability and asymptotic gain results for a class of nonlinear infinite-dimensional systems with respect to the global attractor of the respective undisturbed system. We apply our results to a large class of reaction-diffusion equations with nontrivial global attractors.

Scenario of a Simple Transition from a Structurally Stable 3-Diffeomorphism with a Two-Dimensional Expanding Attractor to a DA Diffeomorphism

The Smale surgery on the three-dimensional torus allows one to obtain a so-called DA diffeomorphism from the Anosov automorphism. The nonwandering set of a DA diffeomor-phism consists of a single two-dimensional expanding attractor and a finite number of source periodic orbits. As shown by V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, the dynamics of an arbitrary structurally stable 3-diffeomorphism with a two-dimensional expanding attrac-tor generalizes the dynamics of a DA diffeomorphism: such a 3-diffeomorphism exists only on the three-dimensional torus, and the two-dimensional attractor is its unique nontrivial basic set, but its nonwandering set may contain isolated saddle periodic orbits together with source periodic orbits. In the present study, we describe a scenario of a simple transition (through elementary bifurcations) from a structurally stable diffeomorphism of the three-dimensional torus with a two-dimensional expanding attractor to a DA diffeomorphism. A key moment in the construction of the arc is the proof that the closure of the separatrices of boundary periodic points of a nontrivial attractor and of isolated saddle periodic points are tamely embedded. This result demonstrates the fundamental difference of the dynamics of such diffeomorphisms from the dynamics of three-dimensional Morse-Smale diffeomorphisms, in which the closure of the separatrices of saddle periodic points may be wildly embedded.

Remarks on nontrivial pullback attractors of the 2‐D Navier‐Stokes equations with delays

This paper is concerned with some further research on the pullback dynamics for 2‐D Navier‐Stokes equations with delays. By some new definition of generalized Grashof numbers, we presented some sufficient conditions when the pullback attractors of the 2‐D nonautonomous incompressible Navier‐Stokes equations with differential continuous delays become a single trajectory, which is a preparation for the fractal dimension of pullback attractors for our problem with constant or variable delays.

Nontrivial Attractors of the Perturbed Nonlinear Schrödinger Equation: Applications to Associative Memory and Pattern Recognition

AbstractComputers are dynamical systems that carry out information processing through their change of state with time. For instance, neural networks such as Hopfield's associative memory are dissipative dynamical systems in a finite dimensional configuration space with attractors that represent stored patterns. In particular, dissipative dynamical systems with an infinite dimensional configuration space are of broad interest and have the possibility to store and restore complex and strongly distorted data structures. In this work, the nonlinear Schrödinger equation (NLSE) with a dissipative perturbation which creates a frictional force acting on soliton is considered. It is shown that the control of the perturbative term allows one to decrease the velocity of soliton to zero and conserve a positive value of its amplitude. The existence of such a frictional force would suggest that the perturbation makes the zero‐velocity solitonic solution of non‐zero amplitude into an attractor for all evolution trajectories whose initial conditions are moving solitons. This allows a way to store information in the infinite dimensional dynamical system governed by NLSE using principles which are completely analogous to that of Hopfield's associative memory. This approach can be realized with solitons in Bose–Einstein condensates and nonlinear optical systems.

Dynamical transitions of the quasi-periodic plasma model

This study examines the stability and transitions of the quasi-periodic plasma perturbation model from the perspective of the dynamical transition. By analyzing the principle of exchange of stability, it is shown that this model undergoes three different types of dynamical transitions as the model control parameter increases. For the first transition, the model exhibits a continuous transition type and subsequently bifurcates to two stable steady states. As this model parameter further increases, the second and third transition can be either continuous or catastrophic. For the continuous transition, the model bifurcates from the two steady states that are resulted from the first transition to a stable periodic solution. If the second transition is catastrophic, there exists a singular separation of a periodic solution, and a nontrivial attractor emerging suddenly for a subcritical value of the control parameters. Numerical experiments confirm both continuous and catastrophic types of transitions, which depend also on the other model parameters. The continuous transition region and catastrophic transition region are also numerically demonstrated. The method used in this paper can be generalized to study other ODE systems.

Cherry flows with non-trivial attractors

We provide an example of a Cherry flow (i.e. a C-infinity flow on the 2-dimensional torus with a sink and a saddle) having a quasi-minimal set which is an attractor. The first return map for such a ...

A Complex Network Framework to Model Cognition: Unveiling Correlation Structures from Connectivity

Several approaches to cognition and intelligence research rely on statistics-based model testing, namely, factor analysis. In the present work, we exploit the emerging dynamical system perspective putting the focus on the role of the network topology underlying the relationships between cognitive processes. We go through a couple of models of distinct cognitive phenomena and yet find the conditions for them to be mathematically equivalent. We find a nontrivial attractor of the system that corresponds to the exact definition of a well-known network centrality and hence stresses the interplay between the dynamics and the underlying network connectivity, showing that both of the two are relevant. Correlation matrices evince there must be a meaningful structure underlying real data. Nevertheless, the true architecture regarding the connectivity between cognitive processes is still a burning issue of research. Regardless of the network considered, it is always possible to recover a positive manifold of correlations. Furthermore, we show that different network topologies lead to different plausible statistical models concerning the correlation structure, ranging from one to multiple factor models and richer correlation structures.

Open sets of Axiom A flows with exponentially mixing attractors

For any dimension d ≥ 3 d\geq 3 we construct C 1 \mathcal {C}^{1} -open subsets of the space of C 3 \mathcal {C}^{3} vector fields such that the flow associated to each vector field is Axiom A and exhibits a non-trivial attractor which mixes exponentially with respect to the unique SRB measure.

Controlling dissociation by trapping trajectories in highly energetic states

We consider the non-linear dynamics of a polar diatomic molecule under the action of laser–field interactions and in the presence of a dissipation mechanism, described by the classical damped and driven one-dimensional Morse oscillator. In the absence of laser fields and dissipation, the phase space consists of a negative-energy bound region and a positive-energy dissociative region. Laser–molecule interaction changes the phase space allowing transitions from the bound to the dissociative region through chaotic routes. We show that for a spatially dependent dipole force, resonances with positive energies allow the trapping of trajectories in pseudo-bound states. We also show that, upon the introduction of dissipation, there exist non-trivial point attractors as well as chaotic attractors, which capture the trajectories in pseudo-bound states. Consequently, in addition to the parameters associated with the laser–molecule interaction, the amplitude of the dissipation acts as a control parameter of the photo-dissociation dynamics.

DYNAMICAL BIFURCATION OF THE ONE DIMENSIONAL MODIFIED SWIFT-HOHENBERG EQUATION

Abstract. In this paper, we study the dynamical bifurcation of the mod-ified Swift-Hohenberg equation on a periodic interval as the system con-trol parameter crosses through a critical number. This critical numberdepends on the period. We show that there happens the pitchfork bifur-cation under the spatially even periodic condition. We also prove thatin the general periodic condition the equation bifurcates to an attractorwhich is homeomorphic to a circle and consists of steady states solutions. 1. IntroductionThe formation of patterns in non-equilibrium systems is closely related tothe instability ([9]) and the bifurcation analysis plays an important role inunderstanding the instability. Indeed, the instability arises when stable statesare driven into unstable states during phases transition. As a control parameterrelated to the instability passes through critical values, the trivial state losesits stability and bifurcates to nontrivial states which form patterns. Thenthe dynamics of the system after the threshold of bifurcation is completelydetermined by its behavior on the center manifold. In particular, Ma andWang showed in [13] that the system bifurcates to a nontrivial attractor on thecenter manifold which determines the final patterns of the system.The Swift-Hohenberg equation is a widely accepted model in the study of theformation of patterns [1, 12]. It was derived in [18] as an approximate model forthe Rayleigh-B´enardconvection describing the pattern formation in layer fluidsbetween horizontal plates. It has attracted a lot of interest in various areas ofapplication regardingpattern formations such as Taylor-Couetteflow and lasers[5]. In particular, there has been much efforts on the bifurcation analysis as away of understanding pattern formations. See [4, 8, 10, 11, 14, 15, 19, 20] forrecent development in this direction.

Maps with separable dynamics and the spectral properties of the operators generated by them

A map of a space into itself generates weighted shift operators in function spaces on . The spectral properties of such operators are intimately connected with the dynamics of . It was known previously that the spectrum of an operator depends only on the set of invariant ergodic measures for . Conditions for the right invertibility of the operators are obtained when is a spectral value. The main result states that right invertibility is only possible when a nontrivial attractor exists. Bibliography: 29 titles.

The evolution of pebble size and shape in space and time

We propose a mathematical model which suggests that the two main geological observations about shingle beaches, i.e. the emergence of predominant pebble size ratios and strong segregation by size, are interrelated. Our model is based on a system of ordinary differential equations (ODEs) called the box equations that describe the evolution of pebble ratios. We derive these ODEs as a heuristic approximation of Bloore's partial differential equation (PDE) describing collisional abrasion and verify them by simple experiments and by direct simulation of the PDE. Although representing a radical simplification of the latter, our system admits the inclusion of additional terms related to frictional abrasion. We show that non-trivial attractors (corresponding to predominant pebble size ratios) only exist in the presence of friction. By interpreting our equations as a Markov process, we illustrate by direct simulation that these attractors may only be stabilized by the ongoing segregation process.

Bifurcation theory of ac electric arcing

The performance of alternating current (ac) electric arcing devices is related to arc extinction or its re-ignition at zero crossings of the current (so-called ‘current zero’, CZ). Theoretical investigations thus usually focus on the transient behaviour of arcs near CZ, e.g. by solving the modelling differential equations in the vicinity of CZ. This paper proposes as an alternative approach to investigate global mathematical properties of the underlying periodically driven dynamic system describing the electric circuit containing the arcing device. For instance, the uniqueness of the trivial solution associated with the insulating state indicates the extinction of any arc. The existence of non-trivial attractors (typically a time-periodic state) points to a re-ignition of certain arcs. The performance regions of arcing devices, such as circuit breakers and arc torches, can thus be identified with the regions of absence and existence, respectively, of non-trivial attractors. Most important for applications, the boundary of a performance region in the model parameter space is then associated with the bifurcation of the non-trivial attractors. The concept is illustrated for simple black-box arc models, such as the Mayr and the Cassie model, by calculating for various cases the performance boundaries associated with the bifurcation of ac arcs.

Complex dynamics of survival and extinction in simple population models with harvesting

We study the effects of constant harvesting in a discrete population model that includes density-independent survivorship of adults in a population with overcompensating density dependence. The interaction between the survival parameter and other parameters of the model (harvesting rate, natural growth rate) reveal new phenomena of survival and extinction. The main differences with the dynamics of survival and extinction reported for semelparous populations with overcompensatory density dependence are that there can be multiple windows of extinction and conditional persistence as harvesting increases or the intrinsic growth rate is increased, and that, in case of bistability, the basin of attraction of the nontrivial attractor may consist of an arbitrary number of disjoint connected components.