Rich dynamics and anticontrol of extinction in a prey–predator system

Abstract

This paper reveals some new and rich dynamics of a two-dimensional prey-predator system and to anticontrol the extinction of one of the species. For a particular value of the bifurcation parameter, one of the system variable dynamics is going to extinct, while another remains chaotic. To prevent the extinction, a simple anticontrol algorithm is applied so that the system orbits can escape from the vanishing trap. As the bifurcation parameter increases, the system presents quasiperiodic, stable, chaotic and also hyperchaotic orbits. Some of the chaotic attractors are Kaplan-Yorke type, in the sense that the sum of its Lyapunov exponents is positive. Also, atypically for undriven discrete systems, it is numerically found that, for some small parameter ranges, the system seemingly presents strange nonchaotic attractors. It is shown both analytically and by numerical simulations that the original system and the anticontrolled system undergo several Neimark-Sacker bifurcations. Beside the classical numerical tools for analyzing chaotic systems, such as phase portraits, time series and power spectral density, the '0-1' test is used to differentiate regular attractors from chaotic attractors.The parameter b does not influence the system dynamics. Therefore, hereafter in all numerical experiments, set b = 0.2 and, unless specified, d = 3.5.Local (finite-time) Lyapunov exponents (LEs), are determined numerically from the system equations. Except for two different attraction basins, which appear for some range of the bifurcation parameter a within the existence domain, the values of the local LEs are approximatively the same. Hereafter, the local LEs are simply called LEs and the spectrum is denoted Λ = {λ 1 , λ 2 }, with λ 1 > λ 2 .Unless specified, the iteration number, necessarily to obtain meaningful numerical results, is set to n = 5e5. Denote by P i , i = 1, 2, ..., some most important points in the partition of the parameter range a ∈ [2, 4] and by N S the point corresponding to the Neimark-Sacker bifurcation. Zero LEs are considered having at least 4 zero decimals, i.e. with error less than 1e − 5.In order to analyze the qualitative changes of the system and to follow the changes in the system dynamics, consider the bifurcation diagram on the plane (a, x), together with Lyapunov spectrum Λ, for a ∈ [2, 4], where the most important dynamics are shown in Fig. 2. Note that the particular shape of the maximal LE begining from P 1 and P 2 , resembles the existence of SNAs, a notion described first by Grebogi et al. in 1984 [12]. Supplementarily, the binary '0-1' test (see Apendix D) is utilized to distinguish clearly regular attractors from chaotic attractors. This test indicates a value close to 0 for regular dynamics, and a value that tends to 1 for chaos.To study the nature of the attractors, phase plots, time series, LEs, normalized power spectral density (PSD) and '0-1' test are utilized. As the time series are real, PSD is two-sided symmetric and, therefore, only the left-side is discussed here.The PSD is used to ...

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