Attractors Of Chaotic Systems Research Articles

Hidden Attractors in Chaotic Systems with Nonlinear Functions

In the present work, an interesting mini-review of hidden attractors in dynamical systems with associated nonlinear functions is carried out. Chaotic systems with nonlinear functions often possess hidden attractors due to their inherent complexity. These attractors can arise in various mathematical models, such as the Lorenz system, Rössler system, or Chua's circuit. The identification and comprehension of hidden attractors broaden our understanding of complex systems and provide new directions for future study and technological development. The discovery and characterization of hidden attractors in chaotic systems have profound implications for various scientific disciplines, including physics, biology, and engineering.

Heterogeneity of the attractor of the Lorenz ’96 model: Lyapunov analysis, unstable periodic orbits, and shadowing properties

It is well known that the predictability of weather and climate is strongly state-dependent. Special, easily recognisable, and extremely relevant atmospheric states like blockings are associated with anomalous instability. This reflects the general property that the attractors of chaotic systems can feature considerable heterogeneity in terms of dynamical properties, and specifically, of their instability. The attractor of a chaotic system is densely populated by unstable periodic orbits (UPOs) that can be used to approximate any forward trajectory through the so-called shadowing. Dynamical heterogeneity can lead to the presence of UPOs with different number of unstable dimensions. This phenomenon – unstable dimensions variability – is a serious breakdown of hyperbolicity and has considerable implications in terms of the structural stability of the system and of the possibility to model accurately its behaviour through numerical models. As a step in the direction of better understanding the properties of high-dimensional chaotic systems, here we provide an extensive numerical investigation of the variability of the dynamical properties across the attractor of the much studied Lorenz ’96 model. By combining the Lyapunov analysis of the tangent space with the study of the shadowing of the chaotic trajectory performed by a very large set of UPOs, we show that the observed variability in the number of unstable dimensions is associated with the presence of a substantial number of finite-time Lyapunov exponents that fluctuate about zero also when very long averaging times are considered. The transition between regions of the attractor with different degrees of instability is associated with a significant drop of the quality of the shadowing. By performing a coarse graining based on the shadowing UPOs, we are able to characterise the slow fluctuations of the system between regions featuring, on the average, anomalously high and anomalously low instability. In turn, such regions are associated, respectively, with states of anomalously high and low energy, thus providing a clear link between the microscopic and thermodynamical properties of the system.

Connecting Curves as a Tool to Localize Hidden Attractors in a New Chaotic Hyperjerk System with No Equilibria

Localizing hidden attractors of chaotic systems is practically and theoretically important. Differing from self-excited attractors, hidden ones do not have any equilibria on the boundaries of their basin of attraction. This characteristic makes hidden attractors hard to localize. Some theoretical and numerical methods have been developed to recognize these attractors, yet the problem remains highly uncertain. For this purpose, the theory of connecting curves is utilized in this work. These curves are one-dimensional set-points that describe the structure of chaotic attractors even in the absence of zero-dimensional fixed-points. In this study, a new four-dimensional chaotic system with hidden attractors is presented. Despite the controversial idea of connecting curves that pass through fixed-points, the connecting curves of a system with no equilibria are considered. This analysis confirms that connecting curves provide more critical information about attractors even if they are hidden.

СКРЫТЫЕ АТТРАКТОРЫ НЕКОТОРЫХ МУЛЬТИСТАБИЛЬНЫХ СИСТЕМ С БЕСКОНЕЧНЫМ ЧИСЛОМ СОСТОЯНИЙ РАВНОВЕСИЯ

It is well known that mathematically simple systems of nonlinear differential equations can exhibit chaotic behavior. Detection of attractors of chaotic systems is an important problem of nonlinear dynamics. Results of recent researches have made it possible to introduce the following classification of periodic and chaotic attractors depending on the presence of neighborhood of equilibrium into their basin of attraction – self-excited and hidden attractors. The presence of hidden attractors in dynamical systems has received considerable attention to both theoretical and applied research of this phenomenon. Revealing of hidden attractors in real engineering systems is extremely important, because it allows predicting the unexpected and potentially dangerous system response to perturbations in its structure. In the past three years after discovering by S. Jafari and J. C. Sprott chaotic system with a line and a plane of equilibrium with hidden attractors there has been much attention to systems with uncountable or infinite equilibria. In this paper it is offered new models of control systems with an infinite number of equilibrium possessing hidden chaotic attractors: a piecewise-linear system with a locally stable segment of equilibrium and a system with periodic nonlinearity and infinite number of equilibrium points. The original analytical-numerical method developed by the author is applied to search hidden attractors in investigated systems.

Multiple attractors and dynamic analysis of a no-equilibrium chaotic system

The hidden attractor holds attracting basin that does not intersect with any small neighborhoods of equilibrium. The attractors of chaotic systems with no-equilibrium, only stable equilibrium or a line of equilibrium belong to hidden attractors, and in which the Shilnikov method is not suitable to explain the chaos. This paper introduces a three-dimensional chaotic system with no equilibrium point. The fundamental dynamical properties as phase portrait, power spectral density, state trajectory, Lyapunov exponent, Kaplan-Yorke dimension and bifurcation are displayed and discussed, which show that the new system exhibits rich dynamics and can generate multiple attractors with different system parameters and initial values.

Simple Geometric Techniques to Delineate the Location, Extent, and Approximate Shapes of Attractors in Chaotic Systems

In this paper we examine the chaotic regimes of a variety of recently discovered hyperchaotic systems using a completely novel geometrical interpretation of the Competitive Modes analysis as simple criteria to map out the spatial location and extent, as well as the rough general shape, of the system attractor for any parameter sets corresponding to chaos. The accuracy of this mapping adds further evidence to the growing body of recent work on the correctness and usefulness of these Competitive Modes conjectures. Indeed, this may be taken as an ’a posteriori’ validation of the Competitive Modes conjectures.

Fractional standard map: Riemann–Liouville vs. Caputo

Properties of the phase space of the standard maps with memory obtained from the differential equations with the Riemann–Liouville and Caputo derivatives are considered. Properties of the attractors which these fractional dynamical systems demonstrate are different from properties of the regular and chaotic attractors of systems without memory: they exist in the asymptotic sense, different types of trajectories may lead to the same attracting points, trajectories may intersect, and chaotic attractors may overlap. Two maps have significant differences in the types of attractors they demonstrate and convergence of trajectories to the attracting points and trajectories. Still existence of the most remarkable new type of attractors, “cascade of bifurcation type trajectories”, is a common feature of both maps.

Use of fuzzy logic to calculate the statistical properties of strange attractors in chaotic systems

Nonlinear systems are important in many fields of science, mathematics, and engineering. The complexity of these systems, however, makes it difficult to ascertain their global behavior using classical methods of analysis. This is particularly true when the system is chaotic. In recent years, simple cell mapping (SCM) and generalized cell mapping (GCM) methods have been proposed and successfully used to analyze nonlinear systems. This paper combines fuzzy logic with cell to cell mapping techniques in order to analyze the global properties of nonlinear, dynamical systems. We refer to this new cell to cell mapping method as fuzzy cell mapping (FCM) to differentiate it from the other methods. Unlike SCM, FCM can handle chaotic systems and can do so without the added computational complexity required by GCM. In order to illustrate these different methods, we analyze a chaotic system using FCM and then compare our results with those obtained with SCM and GCM methods.

Multiscaling transformation in dynamical systems and turbulence

Abstract Multiscaling is a scaling law where the exponent is slowing varying with the length scale (pseudo-algebraic law). We discuss its origin as a consequences of multifractility and the existence of a lower cutoff in the calculation of correlation functions in different contexts. We derive some exact results in the case of two scale Cantor sets, which can be extended to other fractal structures such as strange attractors of chaotic systems. In fully developed turbulence, the cutoff is naturally introduced by the viscosity and our approach leads to the prediction of an intermediate dissipation range, which can be tested experimentally.