Chaotic Behaviour Of Dynamical Systems Research Articles

Traveling wave structures and analysis of bifurcation and chaos theory for Biswas–Milovic Model in conjunction with Kudryshov’s law of refractive index

The present study investigates the generalized Biswas–Milovic model using Kudryshov’s law of refractive index by employing the G′/(bG′+G+a)-Expansion method. This technique facilitates the extraction of singular exponential, trigonometric and kink soliton solutions for the model under examination. The outcomes are subsequently presented as 2D, 3D, and contour plots. Additionally, the model is transformed into a planner dynamical system, and the bifurcation theory is utilized to explore all potential parameter dependencies of the governing model. The chaotic behavior of dynamical systems is also studied by adding an external periodic force. The phase portraits of the obtained findings are also displayed. The results demonstrate the rich and sophisticated behavior of dynamical systems and emphasized the significance of exact solutions for determining their behavior.

Torsion Geometry 5-Fold Symmetry, Anholonomic Phases, Klein Bottle Logophysics, Chaos, Resonance: Applications Towards a Novel Paradigm for the Neurosciences and Consciousness

Torsion Geometry 5-Fold Symmetry, Anholonomic Phases, Klein Bottle Logophysics, Chaos, Resonance: Applications Towards a Novel Paradigm for the Neurosciences and Consciousness

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

After the discovery in early 1960s by E. Lorenz and Y. Ueda of the first example of a chaotic attractor in numerical simulation of a real physical process, a new scientific direction of analysis of chaotic behavior in dynamical systems arose. Despite the key role of this first discovery, later on a number of works have appeared supposing that chaotic attractors of the considered dynamical models are rather artificial, computer-induced objects, i.e., they are generated not due to the physical nature of the process, but only by errors arising from the application of approximate numerical methods and finite-precision computations. Further justification for the possibility of a real existence of chaos in the study of a physical system developed in two directions. Within the first direction, effective analytic-numerical methods were invented providing the so-called computer-assisted proof of the existence of a chaotic attractor. In the framework of the second direction, attempts were made to detect chaotic behavior directly in a physical experiment, by designing a proper experimental setup. The first remarkable result in this direction is the experiment of L. Chua, in which he designed a simple RLC circuit (Chua circuit) containing a nonlinear element (Chua diode), and managed to demonstrate the real evidence of chaotic behavior in this circuit on the screen of oscilloscope. The mathematical model of the Chua circuit (further, Chua system) is also known to be the first example of a system in which the existence of a chaotic hidden attractor was discovered and the bifurcation scenario of its birth was described. Despite the nontriviality of this discovery and cogency of the procedure for hidden attractor localization, the question of detecting this type of attractor in a physical experiment remained open. This article aims to give an exhaustive answer to this question, demonstrating both a detailed formulation of a radiophysical experiment on the localization of a hidden attractor in the Chua circuit, as well as a thorough description of the relationship between a physical experiment, mathematical modeling, and computer simulation.

Transitional regimes under route to chaos in vibroimpact system

In recent years in nonlinear dynamics particular attention was paid to studying the chaotic behaviour of dynamical systems and their routes to chaos. Sometimes this route may be intricate. We watched such intricate route to chaos when studying the quasi-periodic route to chaos in strongly nonlinear non-smooth discontinuous vibroimpact system that was two-body 2-DOF one. After Neimark-Sacker bifurcation many different regimes replace each other. There are transitional regimes with inconsistent characteristics among them. We analyze these regimes with continuous wavelet transform CWT applying. CWT plots confirm just their transition kind and give clear picture of different frequencies presence in time series and their distribution in time.

Wavelet transform using for analysis of vibroimpact system chaotic behavior

Chaotic behaviour of dynamical systems, their routes to chaos, and the intermittency are interesting and investigated subjects in nonlinear dynamics. The studying of these phenomena in non-smooth dynamical systems is of the special scientists’ interest. In this paper we apply relatively young mathematical tool – continuous wavelet transform CWT – for investigating the chaotic behavior and intermittency in particular in strongly nonlinear non-smooth discontinuous 2-DOF vibroimpact system. We show that CWT applying allows to detect and determine the chaotic motion and the intermittency with great confidence and reliability, gives the possibility to demonstrate route to chaos via intermittency, to distinguish and analyze the laminar and turbulent phases.

The study of Lorenz and Rössler strange attractors by means of quantum theory

We have developed a method for complementing an arbitrary classical dynamical system to a quantum system using the Lorenz and Rössler systems as examples. The Schrödinger equation for the corresponding quantum statistical ensemble is described in terms of the Hamilton–Jacobi formalism. We consider both the original dynamical system in the position space and the conjugate dynamical system corresponding to the momentum space. Such simultaneous consideration of mutually complementary position and momentum frameworks provides a deeper understanding of the nature of chaotic behavior in dynamical systems. We have shown that the new formalism provides a significant simplification of the Lyapunov exponents calculations. From the point of view of quantum optics, the Lorenz and Rössler systems correspond to three modes of a quantized electromagnetic field in a medium with cubic nonlinearity. From the computational point of view, the new formalism provides a basis for the analysis of complex dynamical systems using quantum computers.

Restricted feedback control in discrete-time dynamical systems with memory.

When an equilibrium state of a physical or biological system suffers a loss of stability (e.g., via a bifurcation), it may be both possible and desirable to stabilize the equilibrium via closed-loop feedback control. Significant effort has been devoted towards using such control to prevent oscillatory or chaotic behavior in dynamical systems, both continuous-time and discrete-time. Regarding control in discrete-time systems, most prior attempts to stabilize unstable equilibria require that the system be perturbed once during each time step. However, there are examples of systems for which this is neither feasible nor possible. In this paper, we analyze a restricted feedback control method for discrete-time systems (restricted in the sense that the controller's perturbations may be applied only in every other time step). We apply our theoretical analysis to a specific example from cardiac electrophysiology in which this sort of restricted feedback control is especially relevant. The example is a useful test case for the theory, and one for which an experimental setup is rather straightforward.

Characterization of Rössler and Duffing maps with Rényi entropy and generalized complexity measures

Rényi entropy and generalized complexity measures are used to describe the chaotic behaviour of dynamical systems. These measures are found to be sensitive to the fine details of the Rössler and the Duffing maps. They are good descriptors of chaotic behaviour. Periodic windows and the fractal character of the chaotic dynamics are nicely detected.

Examining the Chaotic Behavior in Dynamical Systems by Means of the 0-1 Test

We perform the stability analysis and we study the chaotic behavior of dynamical systems, which depict the 3-particle Toda lattice truncations through the lens of the 0-1 test, proposed by Gottwald and Melbourne. We prove that the new test applies successfully and with good accuracy in most of the cases we investigated. We perform some comparisons of the well-known maximum Lyapunov characteristic number method with the 0-1 method, and we claim that 0-1 test can be subsidiary to the LCN method. The 0-1 test is a very efficient method for studying highly chaotic Hamiltonian systems of the kind we study in our paper and is particularly useful in characterizing the transition from regularity to chaos.

Target-oriented chaos control

Designing intervention methods to control chaotic behavior in dynamical systems remains a challenging problem, in particular for systems that are difficult to access or to measure. We propose a simple, intuitive technique that modifies the values of the state variables directly toward a certain target. The intervention takes into account the difference to the target value, and is a combination of traditional proportional feedback and constant feedback methods. It proves particularly useful when the target corresponds to the equilibrium of the uncontrolled system, and is available or can be estimated from expert knowledge (e.g. in biology and economy).

OPTIMIZATION AND THE MIRANDA APPROACH IN DETECTING HORSESHOE-TYPE CHAOS BY COMPUTER

We report on experiences with an adaptive subdivision method supported by interval arithmetic that enables us to prove subset relations of the form [Formula: see text], and thus check certain sufficient conditions for chaotic behavior of dynamical systems in a rigorous way. Our proof of the underlying abstract theorem avoids referring to any results of applied algebraic topology and relies only on the Brouwer fixed point theorem. The second novelty is that the process of gaining the subset relations to be checked is, to a large extent, also automatized. The promising subset relations come from solving a constrained optimization problem via the penalty function approach. Abstract results and computational methods are demonstrated by finding embedded copies of the standard horseshoe dynamics in iterates of the classical Hénon mapping.

Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions

This paper deals with forced vibrations of two-DOF systems with more than one equilibrium positions. Such systems may be obtained by digitization of elastic post-buckling systems. A vibration mode, which is periodic at small force amplitudes and becomes chaotic as the force amplitudes are slowly increased, is selected. It is possible to formulate and solve the problem of stability of a periodic or chaotic vibration mode in a space with greater dimension using the classical Lyapunov stability definition and some calculating procedures. Instability of phase trajectories is used as a criterion of the chaotic behavior in dynamical systems. Trajectories with very close initial values are compared. Use of the Lyapunov stability definition shows mutual stability/instability of the trajectories. Calculations permit to observe an appearance and enlargement of the chaotic behavior regions. Specific results are obtained for the nonautonomous Duffing equation and pendulum system.

Chaos control in a nonlinear pendulum using a semi-continuous method

Chaotic behavior of dynamical systems offers a rich variety of orbits, which can be controlled by small perturbations in either a specific parameter of the system or a dynamical variable. Chaos control usually involves two steps. In the first, unstable periodic orbits (UPOs) that are embedded in the chaotic set are identified. After that, a control technique is employed in order to stabilize a desirable orbit. This contribution employs the close-return method to identify UPOs and a semi-continuous control method, which is built up on the OGY method, to stabilize some desirable UPO. As an application to a mechanical system, a nonlinear pendulum is considered and, based on parameters obtained from an experimental setup, analyses are carried out. At first, it is considered signals generated by numerical integration of the mathematical model. After that, the analysis is done from scalar time series and therefore, it is important to evaluate the effect of state space reconstruction. Delay coordinates method is employed with this aim. Finally, an analysis related to the effect of noise in controlling chaos is of concern. Results show situations where these techniques may be used to control chaos in mechanical systems.

Chaos for some infinite‐dimensional dynamical systems

AbstractThis paper is devoted to the problem of chaotic behaviour of infinite‐dimensional dynamical systems. We give a survey of different approaches to study of chaotic behaviour of dynamical systems. We mainly discuss the ergodic‐theoretical approach to chaos which bases on the existence of invariant measures having strong analytic and mixing properties. This method is applied to study chaotic behaviour of semiflows generated by semilinear partial differential equations and linear transformations. Copyright © 2004 John Wiley & Sons, Ltd.

Stabilization of the chaotic behavior of dynamical systems

Stabilization of the chaotic behavior of dynamical systems

ON SOME NONAUTONOMOUS CHAOTIC SYSTEM ON THE PLANE

We prove the existence of the chaotic behavior in dynamical systems generated by some class of time periodic nonautonomous equations on the plane. We use topological methods based on the Lefschetz Fixed Point Theorem and the Ważewski Retract Theorem.

Analysis of time behavior in the breakdown of the integral quantum Hall effect.

The breakdown of the integral quantum Hall effect at high current is studied in wide high-quality GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As heterostructures, on the i=2 and i=4 Hall plateaus. The striking features of the phenomenon are pronounced steps in longitudinal resistance with rich time-dependent structures, which are analyzed with techniques normally applied to the study of chaotic behavior in dynamical systems. A number of characteristic parameters of the signal are obtained, showing that the underlying dynamics may be of a chaotic kind.

Chaos and gyrotron-type of interaction

The theory of chaotic behaviour of dynamical systems has been used to investigate the basic features of interaction in fast-wave devices such as gyrotrons. A simple model of the interaction process seems to indicate that gyrotrons readily resist the onset of chaotic behaviour even at large amplitudes of oscillations. >

Chaotic motion versus stochastic excitation

Attention is focused on the chaotic behaviour of dynamical systems under stochastic excitation. Characterization techniques associated with Poincare sections, Lyapunov exponents, capacity and information dimensions, power spectra and probability densities are used for a non-linear single-degree-of-freedom system. It is shown that it is virtually impossible to distinguish between chaotic and non-chaotic stochastic motion when a relatively high intensity of the external excitation is involved. While looking for a transition criterion, the algorithmic experience on the subject is increased.

Celestial mechanics since Newton

With Newton's law of gravitation and laws of motion the science of celestial mechanics obtained its beginning and its fundamental principles and rules. The challenges were presented by Poincaré 200 years later with the principle of non-integrability of the gravitational problem of three or more bodies. Most of the advances in our field belong to one of the following categories: 1. (1)|formulation of problems and solution techniques; 2. (2)|direct and inverse approaches; 3. (3)|stability. The analytical approaches, also known as general perturbation methods appear already in the Principia, often hidden behind geometrical presentations. Hamiltonian dynamics, transformations, topological techniques, bifurcation and mapping are today parts of analytical celestial mechanics. The significant advances in numerical techniques reached the level of brain-computer interfaces, algebraic manipulations and the demonstration of chaotic behaviour of dynamical systems. Newton was concerned with describing the field of gravitation and then establishing orbits in this field. Following him, the problems of inverse dynamics became considerably more refined when higher order gravitational coefficients were being established by satellite orbit observations. The direct problems are today culminating in meaningful orbit predictions of the outer solar system for 10 8 years. With increased observational accuracy, the formulation of the models used has changed considerably and cosmology, stellar dynamics, and dissipative systems became subjects of celestial mechanics. Successful predictions of the qualitative and quantitative behaviour of dynamical systems involved in celestial mechanics depend significantly on the stability characteristics of the problems studied. Recognition of the uncertainties and limited validity of our predictions is probably one of the greatest advances in today's celestial mechanics. This paper is concluded with a short historical review of our giants and of their contributions to pre- and post-Newtonian celestial mechanics.