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Strange attractors and continuous-time chaotic systems
Published in Arturo Buscarino, Luigi Fortuna, Mattia Frasca, ®and Laboratory Experiments, 2017
Arturo Buscarino, Luigi Fortuna, Mattia Frasca
Moreover, since in deterministic systems state space trajectories have no common points, aperiodicity in autonomous continuous-time systems is only possible if the number of state variables is at least three. In the case of continuous-time systems, attractors may also be strange. The term refers to the fact that the orbit is bounded, but not periodic or convergent; on the contrary, it has a complex fractal structure. Chaotic systems display strange attractors, that is, the trajectories are confined in a limit set, in which an infinite number of trajectories approach each other without intersecting one another. Given any point of a trajectory, at some time the trajectory will return arbitrarily close to that point, but will never pass again through the same point, thus forming a dense set of points in a specific geometric structure. Despite the high sensitivity to initial conditions that makes each trajectory unique, the unfolding of the trajectory in the phase space is a geometric structure that is qualitatively the same for each initial condition.
Matrix Calculus for Machine Learning
Published in Richard M. Golden, Statistical Machine Learning, 2020
A sequence of points in Rd may be denoted as either: x(0), x(1), x(2), …,{x(t)}t=0∞, or{x(t)}.Point ConvergenceLet x(1), x(2), … be a sequence of points in a metric space (E, ρ). Let x* ∈ E. Assume that for every positive real number ε there exists a positive integer T(ε) such that ρ(x(t), x*) < ε whenever t > T(ε). Then, the sequence x(1), x(2), … is said to converge or approach the limit pointx*. A set of limit points is called a limit set. The sequence x(1), x(2), … is called a convergent sequence to x*.
Bifurcation and Chaos
Published in Wai-Kai Chen, Feedback, Nonlinear, and Distributed Circuits, 2018
Michael Peter Kennedy, Vandenberghe Lieven
If trajectories starting from states close to a limit set converge to that steady state, the limit set is called an attracting limit set. If, in addition, the attracting limit set contains at least one trajectory that comes arbitrarily close to every point in the set, then it is an attractor. If nearby points diverge from the limit set, it is called a repellor.
A partially degenerate reaction–diffusion cholera model with temporal and spatial heterogeneity
Published in Applicable Analysis, 2023
Let be the omega limit set of the orbit . For any , one immediately finds and for all . Therefore, . Consequently, for any . Since system (2) restricted on is a monotone system, is locally Lyapunov stable for Q in due to [34, Lemma 2.2.1]. So claim 1 is established.
Dimensions in infinite iterated function systems consisting of bi-Lipschitz mappings
Published in Dynamical Systems, 2020
Since ρ is topologically equivalent to the Euclidean metric, is a compact metric space in and is a family of contractions on . For , we denote the diameter of E by . For each , the compact sets , , are decreasing and their diameters converge to zero. In fact, by (1), This implies that the set is a singleton and therefore this formula defines a map , called the coding map, which, in view of (2), is continuous. Following [7,13], we define the limit set and attractor of an IFS as follows.
Attractors of sequences of iterated function systems
Published in Dynamical Systems, 2019
If D is a finite union of closed intervals, then any family of continuous, contracting functions f1, …, fk: D → D with disjoint images defines a function on the compact subsets of D via .1We define to be the compact set . This is the dynamically defined Cantor set associated to . We also call an iterated function system, or IFS, and refer to as the limit set of . If and are two IFSs and , then we write . We will also write Cn(W) = (W1○⋅⋅⋅○Wn)(D) and . We say that is a C1 + α-diffeomorphic IFS if every function in is a C1 + α diffeomorphism.