A nonlinear dynamical system exhibits chaotic hysteresis if it simultaneously exhibits chaotic dynamics (chaos theory) and hysteresis. As the latter involves the persistence of a state, such as magnetization, after the causal or exogenous force or factor is removed, it involves multiple equilibria for given sets of control conditions. Such systems generally exhibit sudden jumps from one equilibrium state to another (sometimes amenable to analysis using catastrophe theory). If chaotic dynamics appear either prior to or just after such jumps, or are persistent throughout each of the various equilibrium states, then the system is said to exhibit chaotic hysteresis. Chaotic dynamics are irregular and bounded and subject to sensitive dependence on initial conditions.
Background and applications
The term was introduced initially by Ralph Abraham and Christopher Shaw (1987), but was modeled conceptually earlier and has been applied to a wide variety of systems in many disciplines. The first model of such a phenomenon was due to Otto Rössler in 1983, which he viewed as applying to major brain dynamics, and arising from three dimensional chaotic systems. In 1986 it was applied to electric oscillators by Newcomb and El-Leithy, perhaps the most widely used application since (see also Pecora and Carroll, 1990).
The first to use the term for a specific application was J. Barkley Rosser, Jr. in 1991, who suggested that it could be applied to explaining the process of systemic economic transition, with Poirot (2001) following up on this in regard to the Russian financial crisis of 1998. Empirical analysis of the phenomenon in the Russian economic transition was done by Rosser, Rosser, Guastello, and Bond (2001). While he did not use the term, Tönu Puu (1989) presented a multiplier-accelerator business cycle model with a cubic accelerator function that exhibited the phenomenon.
Other conscious applications of the concept have included to Rayleigh-Bénard convection rolls, hysteretic scaling for ferromagnetism, and a pendulum on a rotating table (Berglund and Kunz, 1999), to induction motors (Súto and Nagy, 2000), to combinatorial optimization in integer programming (Wataru and Eitaro, 2001), to isotropic magnetization (Hauser, 2004), to bursting oscillations in beta cells in the pancreas and population dynamics (Françoise and Piquet, 2005), to thermal convection (Vadasz, 2006), and to neural networks (Liu and Xiu, 2007).
- Ralph H. Abraham and Christopher D. Shaw. “Dynamics: A Visual Introduction.” In F. Eugene Yates, ed., Self-Organizing Systems: The Emergence of Order. New York: Plenum Press, pp. 543-597, 1987.
- Otto E. Rössler. “The Chaotic Hierarchy.” Zeitschrift für Natuforschung 1983, 38a, pp. 788-802.
- R.W. Newcomb and N. El-Leithy. “Chaos Generation Using Binary Hysteresis.” Circuits, Systems, and Signal Processing September 1986, 5(3), pp. 321-341.
- L.M. Pecora and T.L. Carroll. “Synchronization in Chaotic Systems.” Physical Review Letters February 19 1990, 64(8), pp. 821-824.
- J. Barkley Rosser, Jr. From Catastrophe to Chaos: A General Theory of Economic Discontinuities. Boston/Dordrecht: Kluwer Academic Publishers, Chapter 17, 1991.
- Clifford S. Poirot. “Financial Integration under Conditions of Chaotic Hysteresis: The Russian Financial Crisis of 1998.” Journal of Post Keynesian Economics' Spring 2001, 23(3), pp. 485-508.
- J. Barkley Rosser, Jr., Marina V. Rosser, Stephen J. Guastello, and Robert W. Bond, Jr. “Chaotic Hysteresis and Systemic Economic Transformation: Soviet Investment Patterns.” Nonlinear Dynamics, Psychology, and Life Sciences October 2001, 5(4), pp. 545-566.
- Tönu Puu. Nonlinear Economic Dynamics. Berlin: Springer-Verlag, 1989.
- N. Berglund and H. Kunz. “Memory Effects and Scaling Laws in Slowly Driven Systems.” Journal of Physics A: Mathematical and General January 8, 1999, 32(1), pp. 15-39.
- Zoltán Súto and István Nagy. “Study of Chaotic and Periodic Behaviours of a Hysteresis Current Controlled Induction Motor Drive.” In Hajime Tsuboi and István Vajda, eds., Applied Electromagnetics and Computational Technology II. Amsterdam: IOS Press, pp. 233-243.
- Murano Wataru and Aiyoshi Eitaro. “Opening Door toward 21st Century. Integer Programming by the Multi-Valued Hysteresis Machines with the Chaotic Properties.” Transactions of the Institute of Electrical Engineers of Japan C 2001, 121(1), pp. 76-82.
- Hans Hauser. “Energetic Model of Ferromagnetic Hysteresis: Isoptropic Magentization.” Journal of Applied Physics September 1, 2004, 96(5), pp. 2753-2767.
- J.P. Françoise and C. Piquet. “Hysteresis Dynamics, Bursting Oscillations and Evolution to Chaotic Regimes.” Acta Biotheoretica 2005, 53(4), pp. 381-392.
- P. Vadasz. “Chaotic Dynamics and Hysteresis in Thermal Convection.” Journal of Mechanical Engineering Science “ 2006, 220(3), pp. 309-323.
- Xiangdong Liu and Chunko Xiu. “Hysteresis Modeling Based on the Hysteretic Chaotic Neural Network.” Neural Computing Applications online October 30, 2007: http://www.springerlink.com/content/x76777476785m48.
Wikimedia Foundation. 2010.
Look at other dictionaries:
List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… … Wikipedia
J. Barkley Rosser, Jr. — J. Barkley Rosser, Jr. Born 1948 Ithaca, New York, USA Nationality American Field Economics, Complexity Alma mater UW Madison, Wisconsin; USA John Barkley Rosser, Jr. (born 12 April 1948) is a mathematical economist and Profess … Wikipedia
Cellular neural network — Cellular neural networks (CNN) are a parallel computing paradigm similar to neural networks, with the difference that communication is allowed between neighbouring units only. Typical applications include image processing, analyzing 3D surfaces,… … Wikipedia
Lau Wai Shing — Wai Shing Lau (simplified Chinese name: 刘偉成, born July 29, 1955 in Hong Kong) is also known as Lau Wai Shing. The family name of Lau is sometimes spelled as Liu like Liu Bang (founder of the Han dynasty) or Liu Shaoqi or Liu Bocheng. This is… … Wikipedia
Positive feedback — Alarm or panic can spread by positive feedback among a herd of animals to cause a stampede. Positive feedback is energy taken from the output of a system and reapplied to the input, which is phase congruent with the input signal. A system… … Wikipedia
Complex system — This article largely discusses complex systems as a subject of mathematics and the attempts to emulate physical complex systems with emergent properties. For other scientific and professional disciplines addressing complexity in their fields see… … Wikipedia
Duffing equation — A Poincaré section of the forced Duffing equation suggesting chaotic behaviour The Duffing equation is a non linear second order differential equation used to model certain damped and driven oscillators. The equation is given by … Wikipedia
Rayleigh–Bénard convection — Bénard cells. Rayleigh–Bénard convection is a type of natural convection, occurring in a plane of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as Bénard cells. Rayleigh–Bénard convection is one… … Wikipedia
Bénard cells — are convection cells that appear spontaneously in a liquid layer when heat is applied from below. They can be obtained using a simple experiment first conducted by Henri Bénard, a French physicist, in 1900. The experiment illustrates the theory… … Wikipedia
Robijn Bruinsma — Robijn F. Bruinsma (b. May 15, 1953, Haarlem, The Netherlands) is a theoretical physicist and is Professor of Physics at the University of California at Los Angeles and Chair of the Department of Theoretical Physics for the Life Sciences at… … Wikipedia