In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state.
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These 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 Brazil can cause a tornado in Texas.
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, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of bifurcation theory. In the context of discrete-time dynamical systems, the diagram is also called orbit diagram.
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed.
Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a Cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems and discrete systems.
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. The term "butterfly effect" in popular media may stem 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, it is impossible to predict the future of systems well. For instance, 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.
Denis Louis Blackmore was an American mathematician and a full professor of the Department of Mathematical Sciences at New Jersey Institute of Technology. He was also one of the founding members of the Center for Applied Mathematics and Statistics at NJIT. Dr. Blackmore was mainly known for his many contributions in the fields of dynamical systems and differential topology. In addition to this, he had many contributions in other fields of applied mathematics, physics, biology, and engineering.
Geomathematics is the application of mathematical methods to solve problems in geosciences, including geology and geophysics, and particularly geodynamics and seismology.
Valentin Afraimovich was a Soviet, Russian and Mexican mathematician. He made contributions to dynamical systems theory, qualitative theory of ordinary differential equations, bifurcation theory, concept of attractor, strange attractors, space-time chaos, mathematical models of non-equilibrium media and biological systems, travelling waves in lattices, complexity of orbits and dimension-like characteristics in dynamical systems.
Morris William Hirsch is an American mathematician, formerly at the University of California, Berkeley.
In the fields of dynamical systems and control theory, a fractional-order system is a dynamical system that can be modeled by a fractional differential equation containing derivatives of non-integer order. Such systems are said to have fractional dynamics. Derivatives and integrals of fractional orders are used to describe objects that can be characterized by power-law nonlocality, power-law long-range dependence or fractal properties. Fractional-order systems are useful in studying the anomalous behavior of dynamical systems in physics, electrochemistry, biology, viscoelasticity and chaotic systems.
Edward Ott is an American physicist most noted for his contributions to the development of chaos theory.
Jack Kenneth Hale was an American mathematician working primarily in the field of dynamical systems and functional differential equations.
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 the Science and Engineering Research Board, Department of Science and Technology. He has held several research fellowships which included Raja Ramanna fellowship of the 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 the Dr. A. P. J Abdul Kalam Award by the Government of Tamil Nadu.
Annalisa Crannell is an American mathematician, and an expert in the mathematics of water waves, chaos theory, and geometric perspective. She is a professor of mathematics at Franklin & Marshall College.
The Malkus waterwheel, also referred to as the Lorenz waterwheel or chaotic waterwheel, is a mechanical model that exhibits chaotic dynamics. Its motion is governed by the Lorenz equations. While classical waterwheels rotate in one direction at a constant speed, the Malkus waterwheel exhibits chaotic motion where its rotation will speed up, slow down, stop, change directions, and oscillate back and forth between combinations of such behaviours in an unpredictable manner.
Laura Gardini is an Italian mathematician who studies chaos in dynamical systems, with applications in mathematical finance. She is professor in mathematics for economic applications at the University of Urbino.
