Introduction to the focus issue: Fifty years of chaos: Applied and theoretical

Abstract

The discovery of deterministic chaos in the late nineteenth century, its subsequent study, and the development of mathematical and computational methods for its analysis have substantially influenced the sciences. Chaos is, however, only one phenomenon in the larger area of dynamical systems theory. This Focus Issue collects 13 papers, from authors and research groups representing the mathematical, physical, and biological sciences, that were presented at a symposium held at Kyoto University from November 28 to December 2, 2011. The symposium, sponsored by the International Union of Theoretical and Applied Mechanics, was called 50 Years of Chaos: Applied and Theoretical. Following some historical remarks to provide a background for the last 50 years, and for chaos, this Introduction surveys the papers and identifies some common themes that appear in them and in the theory of dynamical systems.

Antimonotonicity, Chaos and Multidirectional Scroll Attractor in Autonomous ODEs Chaotic System

“…(2) r [0,1] In Fig.4 (b), along with increase of r, the orbits transform from chaos to coexisting period-14 to chaos and to When a = c = 0.6, p = r = 2, q = 1.5, b = 0.4, coexisting bifurcation modes appear, and the coexisting attractors are shown in Fig. 5 and Fig.…”

“…Chaos, as one of the most important topics, has attracted many scholars to focus on the study and analysis in this field [1]. During the past decades, the research on chaos has progressed starting from theoretical research [2] and chaos control and anti-control [3]- [5], chaotic circuits [6] effectively.…”

Generation and Dynamics Analysis of N-Scroll Existence via Four Different Functions in Translation Chaotic System

“…When A=1, a=0. 1,r=2,p=q=1,b=1.2,c=0.6,m=0.2,N=1(NϵR), the system can display 3-scroll attractors, and N=3, the system can display 7-scroll attractors, as shown in Figure 13.…”

“…Chaos theory, with relativity and quantum mechanics, is called one of the greatest discoveries of the twentieth Century. Earlier, scientists had realized that some deterministic dynamical systems involving only a few variables could exhibit complexity reminiscent of many-particle systems if the dynamics is chaotic, as can be quantified by the existence of a positive Lyapunov exponent, Hopf bifurcation and Poincaré map [1][2][3][4][5]. More, topological mixing [6] and periodicity [7] are also important factors from popular accounts of chaos, which equate chaos with sensitivity to initial conditions.…”

Bounded Orbits and Multiple Scroll Coexisting Attractors in a Dual System of Chua System

“…By and large, the research on chaos, various types of strange attractors, coexisting attractors, synchronization control and their applications have became the hotspots [1][2][3][4][5]. Their interesting and complex dynamical phenomena are applied in various fields, for example chaos prediction in nonlinear viscoelastic plates [6], nonlinear resonances and multi-stability in simple neural circuits [7], synchronization control [8,33], grid multi-scroll chaotic attractors [4,5,9,30], multi-wing hyper chaotic system [10,30], electronic circuits [11][12][13][22][23][24], image encryption application [25,34], oscillator of memory elements [14,35,36,41], embedded implementation for digital systems [37][38][39][40] et al As well as we known, the chaotic application in electronic circuits began Chua circuit and generation of multiple scroll attractors started from Suykens and Vandewalle, beyond the double scroll [1]. Up to now, the main work and research on chaos theory, analysis methods and application fields are as follows: (i) chaotic systems and scroll attractors seem to have reached their maturity, i.e.…”