Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Texas can cause a tornado in Brazil.
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.
In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum.
In mathematics, Sharkovskii's theorem, named after Oleksandr Mykolayovych Sharkovsky, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.
Sir Michael Victor Berry,, is a mathematical physicist at the University of Bristol, England.
Ashoke Sen FRS is an Indian theoretical physicist and distinguished professor at the International Centre for Theoretical Sciences (ICTS), Bangalore. A former distinguished professor at the Harish-Chandra Research Institute, Allahabad, He is also an honorary fellow in National Institute of Science Education and Research (NISER), Bhubaneswar, India he is also a Morningstar Visiting professor at MIT and a distinguished professor at the Korea Institute for Advanced Study. His main area of work is string theory. He was among the first recipients of the Breakthrough Prize in Fundamental Physics "for opening the path to the realization that all string theories are different limits of the same underlying theory".
In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory—the new one having double the period of the original. With the doubled period, it takes twice as long for the numerical values visited by the system to repeat themselves.
Rey Juan Carlos University is a Spanish public research university located in the southern area of the Community of Madrid (Spain), with five campuses at Móstoles, Alcorcón, Vicálvaro, Aranjuez and Fuenlabrada.
The Lorenz system is a system of ordinary differential equations first studied by mathematician and meteorologist Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. In popular media the "butterfly effect" stems from the real-world implications of the Lorenz attractor, namely that several different initial chaotic conditions evolve in phase space in a way that never repeats, so all chaos is unpredictable. This underscores that chaotic systems can be completely deterministic and yet still be inherently unpredictable over long periods of time. Because chaos continually increases in systems, we cannot predict the future of systems well. E.g., even the small flap of a butterfly’s wings could set the world on a vastly different trajectory, such as by causing a hurricane. The shape of the Lorenz attractor itself, when plotted in phase space, may also be seen to resemble a butterfly.
John Cromwell Mather is an American astrophysicist, cosmologist and Nobel Prize in Physics laureate for his work on the Cosmic Background Explorer Satellite (COBE) with George Smoot.
Tien-Yien Li (李天岩) was a University Distinguished Professor of Mathematics and University Distinguished Professor Emeritus at Michigan State University. There, he spent 42 years and supervised 26 Ph.D. dissertations.
Leon Ong Chua is an American electrical engineer and computer scientist. He is a professor in the electrical engineering and computer sciences department at the University of California, Berkeley, which he joined in 1971. He has contributed to nonlinear circuit theory and cellular neural network theory.
Lennart Ljung is a Swedish professor in the Chair of Control Theory at Linköping University since 1976. He is known for his pioneering research in system identification, and is regarded as a leading researcher in control theory.
Miguel Angel Fernández Sanjuán is a Spanish Theoretical Physicist from Leon, Spain. He is known for his contributions in nonlinear dynamics, chaos theory, and control of chaos, and has published several scientific papers and popular news articles. He has supervised around 20 PhD students in Nonlinear Dynamics, Chaos and Complex Systems.
Guanrong Chen (陈关荣) or Ron Chen is a Chinese mathematician who made contributions to Chaos theory. He has been the chair professor and the founding director of the Centre for Chaos and Complex Networks at the City University of Hong Kong since 2000. Prior to that, he was a tenured full professor at the University of Houston, Texas. Chen was elected Member of the Academy of Europe in 2014, elected Fellow of The World Academy of Sciences in 2015, and elected IEEE Fellow in 1997. He is currently the editor-in-chief for the International Journal of Bifurcation and Chaos.
Zeev Rudnick or Ze'ev Rudnick is a mathematician, specializing in number theory and in mathematical physics, notably quantum chaos. Rudnick is a professor at the School of Mathematical Sciences and the Cissie and Aaron Beare Chair in Number Theory at Tel Aviv University.
Celso Grebogi is a Brazilian theoretical physicist who works in the area of chaos theory. He is one among the pioneers in the nonlinear and complex systems and chaos theory. Currently he works at the University of Aberdeen as the "Sixth Century Chair in Nonlinear and Complex Systems". He has done extensive research in the field of plasma physics before his work on the theory of dynamical systems. He and his colleagues have shown with a numerical example that one can convert a chaotic attractor to any one of numerous possible attracting time-periodic motions by making only small time-dependent perturbations of an available system parameter. This article is considered as one among the classic works in the control theory of chaos and their control method is known as the OGY method. He was listed in the 2016 Thomson Reuters Citation Laureates.
Edward Ott is an American physicist most noted for his contributions to the development of chaos theory.
Muthusamy Lakshmanan is an Indian theoretical physicist currently working as Professor of Eminence at the Department of Nonlinear Dynamics of Bharathidasan University. Presently he is the DST-SERB National Science Chair awarded by Science and Engineering Research Board, Department of Science and Technology. He has held several research fellowships which included Raja Rammanna fellowship of Department of Atomic Energy, Alexander von Humboldt fellowship, Japan Society for the Promotion of Science fellowship, Royal Society Nuffield Foundation fellowship, and NASI-Senior Scientist Platinum Jubilee Fellowship. In the year 2021, on August 15, he was conferred with Dr. A. P. J Abdul Kalam Award by the Government of Tamil Nadu.
Michael C. Mackey is a Canadian-American biomathematician and Professor in the Department of Physiology of McGill University in Montreal, Quebec, Canada who holds the Joseph Morley Drake Emeritus Chair.
