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Devaney is known for formulating a simple and widely used definition of chaotic systems, one that does not need advanced concepts such as measure theory. [8] In his 1989 book An Introduction to Chaotic Dynamical Systems, Devaney defined a system to be chaotic if it has sensitive dependence on initial conditions, it is topologically transitive (for any two open sets, some points from one set ...
Chaotic maps often occur in the study of dynamical systems. Chaotic maps and iterated functions often generate fractals . Some fractals are studied as objects themselves, as sets rather than in terms of the maps that generate them.
Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems.Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does ...
Deterministic system (mathematics) Linear system; Partial differential equation; Dynamical systems and chaos theory; Chaos theory. Chaos argument; Butterfly effect; 0-1 test for chaos; Bifurcation diagram; Feigenbaum constant; Sharkovskii's theorem; Attractor. Strange nonchaotic attractor; Stability theory. Mechanical equilibrium; Astable ...
The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions. Chaos theory (or chaology [1]) 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.
A computer-assisted proof of chaotic behavior (more precisely, of positive topological entropy) in Chua's circuit was published in 1997. [6] A self-excited chaotic attractor , known as " the double scroll " because of its shape in the ( x , y , z ) space, was first observed in a circuit containing a nonlinear element such that f ( x ) was a 3 ...
The system becomes more chaotic as dynamical symmetries are broken by increasing the quantum defect; consequently, the distribution evolves from nearly a Poisson distribution (a) to that of Wigner's surmise (h). Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems.
The Duffing equation is an example of a dynamical system that exhibits chaotic behavior. Moreover, the Duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour.
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