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2018 | Buch

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Regularity and Stochasticity of Nonlinear Dynamical Systems

herausgegeben von: Dimitri Volchenkov, Xavier Leoncini

Verlag: Springer International Publishing

Buchreihe : Nonlinear Systems and Complexity

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book presents recent developments in nonlinear dynamics and physics with an emphasis on complex systems. The contributors provide recent theoretic developments and new techniques to solve nonlinear dynamical systems and help readers understand complexity, stochasticity, and regularity in nonlinear dynamical systems. This book covers integro-differential equation solvability, Poincare recurrences in ergodic systems, orientable horseshoe structure, analytical routes of periodic motions to chaos, grazing on impulsive differential equations, from chaos to order in coupled oscillators, and differential-invariant solutions for automorphic systems, inequality under uncertainty.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Solvability of Some Integro-Differential Equations with Anomalous Diffusion

Abstract

The paper deals with the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the Laplace operator in a fractional power. The proof of existence of solutions relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used.

Vitali Vougalter, Vitaly Volpert

Chapter 2. Poincaré Recurrences in Ergodic Systems Without Mixing

Abstract

We study numerically the statistics of recurrences in ergodic sets of the circle map type by using the multifractality analysis. We consider the standard circle map as well as the sets generated in stroboscopic sections of phase trajectories in a nonautonomous van der Pol oscillator and in a harmonically driven conservative nonlinear oscillator. The universal properties of the recurrence time dependence on the size of a return region are established. The numerical results demonstrate a full correspondence with the theoretical data obtained by Valentin Afraimovich.

Vadim Anishchenko, Nadezhda Semenova, Elena Rybalova, Galina Strelkova

Chapter 3. Success, Hierarchy, and Inequality Under Uncertainty

Abstract

Domains with a lot of uncertainty have the highest likelihood of skilled people failing. Those that succeed the most under uncertainty are often simply those that tried harder and whose early luck compounded. By fostering hierarchical organization in a group, uncertainty ultimately leads to inequality. Wealth inequality in a population arises from risky decisions being taken under uncertainty by the vital few: The more adventurous traders are, the greater their fortune, and the fewer lucky ones there are. Scarcity also promotes inequality by necessitating competition and fueling conspicuous consumption. Existing econometric data suggest that rising income inequality is a global phenomenon, occurring whenever the national economy is out of step with the world average. Rampant inequality may transform the uncertainty of national economic development into uncertainty of international relations.

Dimitri Volchenkov

Chapter 4. Functional Differential Equations with Piecewise Constant Argument

Abstract

We introduce a new class of functional differential equations with functional response on piecewise constant argument, \({ FDEPCA}\). It contains functional differential equations with continuous time [21, 25, 28, 31] as well as differential equations with piecewise constant argument [1, 1‐17, 22, 22, 23]. We concentrate only on retarded equations, but one can easily extend the discussion to any type of piecewise constant argument and functional differential equations. Nonlinear and quasilinear systems are under consideration. At the end of the chapter, we suggest how one can apply the systems for solution of real-world problems, provided more general systems for future investigations.

M. U. Akhmet

Chapter 5. Grazing in Impulsive Differential Equations

Abstract

Discontinuous dynamical systems with grazing solutions are discussed. A variational system around a grazing solution which depends on near solutions is constructed. Orbital stability of grazing cycles is examined by linearization. Small parameter method is extended for analysis of neighborhoods of grazing orbits, and grazing bifurcation of cycles is observed in an example. Linearization around an equilibrium grazing point is discussed. In the another part of the chapter, non-autonomous systems with stationary impulses are considered. A concise review on sufficient conditions for the linearization and stability is presented. The existence and stability of periodic solutions are considered under the circumstance of the regular perturbation. Examples are presented through which the theoretical results are illustrated. Appropriate illustrations with grazing limit cycles and bifurcations are depicted to support the theoretical results.

M. U. Akhmet, A. Kıvılcım

Chapter 6. On Local Topological Classification of Two-Dimensional Orientable, Non-Orientable, and Half-Orientable Horseshoes

Abstract

Smale horseshoes of new types, the so called half-orientable horseshoes, have been found in Gonchenko et al. (Int. J. Bifurc. Chaos, 18(10):3029–3052, 2008, [1]). Such horseshoes can exist as invariant sets for endomorphisms of the disk and for diffeomorphisms of non-orientable two-dimensional manifolds as well. They have many features different from those of the classical orientable and non-orientable horseshoes.

S. V. Gonchenko, A. S. Gonchenko, M. I. Malkin

Chapter 7. From Chaos to Order in a Ring of Coupled Oscillators with Frequency Mismatch

Abstract

In this chapter we describe the route to synchronization in a ring of three unidirectionally in the presence of a mismatch between their natural frequencies. Three coupled oscillators is a simplest network motif where each oscillator is nothing more than a node. Network motifs repeat themselves in a specific network or even among various networks, and can be responsible for particular functions. On the route to synchronization the oscillators pass through intermittent phase synchronization, phase synchronization, lag or anticipating synchronization with respect to the coupling strength and frequency mismatch. When the oscillators’ natural frequencies are very close to each other, they are chaotic for any coupling strength, whereas for larger mismatch and strong coupling they exhibit regular dynamics. The results of numerical simulations are in good agreement with electronic experiments.

Alexander N. Pisarchik, Mariano Alberto García-Vellisca

Chapter 8. Dynamics of Some Nonlinear Meromorphic Functions

Abstract

In order to introduce the dynamics of the class of transcendental meromorphic functions with at least one pole which is not omitted, denoted by \(\mathscr {M}\), we introduce the class Mer. The main definitions and properties concerning the Stable and Chaotic sets of functions in class Mer are presented. We define the escaping set, of a function in class Mer, which is useful to program the computer in order to get dynamical planes for some functions in three families given in Sect. 8.4. We also explain some dynamical differences between classes of functions contained in Mer.

P. Domínguez, M. A. Montes de Oca, G. J. F. Sienra

Chapter 9. Dynamics of Oscillatory Networks with Pulse Delayed Coupling

Abstract

The chapter is devoted to the dynamics of networks of oscillators with pulse time-delayed coupling. We develop a mathematical technique that allows to reduce the dynamics of such networks to multi-dimensional maps. With the help of these maps we consider networks of various configurations: a single oscillator with feedback, a feed-forward ring, a pair of oscillators with mutual coupling, a small network with heterogeneous delays, and a large network with all-to-all coupling. In all these examples we show that the role of the delay is significant and leads to the modification of the existing dynamical regimes and the emergence of new ones.

Vladimir Klinshov, Dmitry Shchapin, Serhiy Yanchuk, Vladimir Nekorkin

Chapter 10. Bifurcation Trees of Period-3 Motions to Chaos in a Time-Delayed Duffing OscillatorTime-delayed duffing oscillator

Abstract

The time-delayed Duffing oscillator is extensively applied in engineering and particle physics. Determination of periodic motions in such a system is significant. Thus, here in, period motions in the time-delayed Duffing oscillator are discussed through a semi-analytical method. The semi-analytical method is based on the implicit mappings constructed by discretization of the corresponding differential equation. Complex period-3 motions are predicted, and the corresponding stability and bifurcation analysis are completed.

Albert C. J. Luo, Siyuan Xing

Chapter 11. Travelable Period-1 Motions to Chaos in a Periodically Excited Pendulum

Abstract

In this chapter, the analytical bifurcation trees of travelable period-1 motions to chaos in a periodically excited pendulum will be presented with varying excitation amplitude. The analytical prediction is based on the implicit discrete maps obtained from the midpoint scheme of the corresponding differential equation. Using the discrete maps, mapping structures will be developed for various periodic motions, and analytical bifurcation trees of periodic motions to chaos can be obtained.

Yu Guo, Albert C. J. Luo

Chapter 12. Automorphic Systems and Differential-Invariant Solutions

Abstract

The notion of differential-invariant solutions is a generalization notions of invariant and partially invariant solutions. It is closely related to the notion of automorphic systems. The system of differential equations is called an automorphic with respect to the Lie group \(G\), if there is any solution of this system is obtained one fixed solution through action group of transformations \(G\). In contrast to the invariant and partially invariant, making use of the differential-invariant solutions has not yet received equally widespread. In this chapter, it is shown that if Lie group \(G\) is a finite-dimensional group, then some finite (as a rule of zero order) prolongation of a automorphic system is completely integrable system. This fact is useful in constructing automorphic systems. Also, we prove theorems on integration and on order reducing for the automorphic systems.

A. A. Talyshev

Backmatter

Titel: Regularity and Stochasticity of Nonlinear Dynamical Systems
herausgegeben von: Dimitri Volchenkov
Xavier Leoncini
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-58062-3
Print ISBN: 978-3-319-58061-6
DOI: https://doi.org/10.1007/978-3-319-58062-3