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Describing chaotic systems
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abstract
In this paper, we discuss the Lyapunov exponent definition of chaos and how it can be used to quantify the chaotic behavior of a system. We derive a way to practically calculate the Lyapunov exponent of a one-dimensional system and use it to analyze chaotic behavior of the logistic map, comparing the $r$-varying Lyapunov exponent to the map's bifurcation diagram. Then, we generalize the idea of the Lyapunov exponent to an $n$-dimensional system and explore the mathematical background behind the analytic calculation of the Lyapunov spectrum. We also outline a method to numerically calculate the maximal Lyapunov exponent using the periodic renormalization of a perturbation vector and a method to numerically calculate the entire Lyapunov spectrum using QR factorization. Finally, we apply both these methods to calculate the Lyapunov exponents of the H\'enon map, a multi-dimensional chaotic system.
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Cited by 2 Pith papers
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Asymmetric coupling of nonchaotic Rulkov neurons: Fractal attractors, quasimultistability, and final state sensitivity
In an asymmetrically coupled pair of nonchaotic Rulkov neurons, coexisting spiking and chaotic bursting dynamics form a short-lived quasimultistability with fractal basin boundaries and extreme final-state sensitivity.
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Chaotic Dynamics and Fractal Geometry in Ring Lattice Systems of Nonchaotic Rulkov Neurons
Electrically coupling nonchaotic Rulkov neurons in a ring produces chaotic spiking, synchronized bursting, and hyperchaos, with Lyapunov dimensions reaching 45 of 60 state-space dimensions.
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