Introduction to the focus issue: fifty years of chaos: applied and theoretical.

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.

[1]  Stefano Lenci,et al.  Controlling practical stability and safety of mechanical systems by exploiting chaos properties. , 2012, Chaos.

[2]  T Kapitaniak,et al.  Synchronous rotation of the set of double pendula: experimental observations. , 2012, Chaos.

[3]  N Sri Namachchivaya,et al.  Particle filtering in high-dimensional chaotic systems. , 2012, Chaos.

[4]  Vladimir I. Arnold,et al.  Instability of Dynamical Systems with Several Degrees of Freedom , 2020, Hamiltonian Dynamical Systems.

[5]  J. E. Littlewood,et al.  On Non‐Linear Differential Equations of the Second Order: I. the Equation y¨ − k(1‐y2)y˙ + y = bλk cos(λl + α), k Large , 1945 .

[6]  T Kapitaniak,et al.  The three-dimensional dynamics of the die throw. , 2012, Chaos.

[7]  S. Smale.  Diffeomorphisms with Many Periodic Points , 1965 .

[8]  N. Levinson,et al.  A Second Order Differential Equation with Singular Solutions , 1949 .

[9]  K. M. Carroll,et al.  Cartography of high-dimensional flows: a visual guide to sections and slices. , 2012, Chaos.

[10]  Philip Holmes,et al.  Ninety Plus Thirty Years of Nonlinear Dynamics: Less is More and More is Different , 2005, Int. J. Bifurc. Chaos.

[11]  Miguel A F Sanjuán,et al.  Dynamics of partial control. , 2012, Chaos.

[12]  George D. Birkhoff,et al.  Proof of Poincaré’s geometric theorem , 1913 .

[13]  B. V. D. Pol,et al.  Frequency Demultiplication , 1927, Nature.

[14]  M. Gameiro,et al.  Combinatorial-topological framework for the analysis of global dynamics. , 2012, Chaos.

[15]  Kazuyuki Aihara,et al.  β-expansion attractors observed in A/D converters. , 2012, Chaos.

[16]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[17]  Norio Akamatsu,et al.  ON THE BEHAVIOR OF SELF-OSCILLATORY SYSTEMS WITH EXTERNAL FORCE , 1974 .

[18]  G. D. Birkhoff.  Sur quelques courbes fermées remarquables , 1932 .

[19]  B. Eckhardt,et al.  Periodic orbits near onset of chaos in plane Couette flow. , 2012, Chaos.

[20]  H. Poincaré,et al.  Erratum zu: Etude des surfaces asymptotiques , 1890 .

[21]  Kazuyuki Aihara,et al.  Chaos in neurons and its application: perspective of chaos engineering. , 2012, Chaos.

[22]  C. E. Puente,et al.  The Essence of Chaos , 1995 .

[23]  David Aubin,et al.  Writing the History of Dynamical Systems and Chaos: Longue Durée and Revolution, Disciplines and Cultures , 2002 .

[24]  M. Bernhard.  Introduction to Chaotic Dynamical Systems , 1992 .

[25]  I. Mezić,et al.  Applied Koopmanism. , 2012, Chaos.

[26]  Kenichi Arai,et al.  Noise amplification by chaotic dynamics in a delayed feedback laser system and its application to nondeterministic random bit generation. , 2012, Chaos.

[27]  A. Andronov.  CHAPTER III – NON-CONSERVATIVE SYSTEMS , 1966 .