Multistability in a Periodically Forced Brusselator
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STATISTICAL
Multistability in a Periodically Forced Brusselator Paulo C. Rech1 Accepted: 5 October 2020 © Sociedade Brasileira de F´ısica 2020
Abstract In this paper, we report on multistability in a periodically forced Brusselator, which is modeled by a nonlinear nonautonomous system of two first-order ordinary differential equations. Multistability regions are detected in a cross section of the four-dimensional parameter space of the model, namely the (ω, F ) parameter plane, where ω and F are respectively angular frequency and amplitude of an external forcing. Lyapunov exponents spectra are used to characterize the dynamical behavior of each point in the abovementioned parameter plane. Moreover, basins of attraction, bifurcation diagrams, and phase-space portraits are used to illustrate the coexistence of periodic and chaotic behaviors. Keywords Forced Brusselator · Multistability · Lyapunov exponents spectrum · Chaos
1 Introduction The Brusselator mathematical model given by: x˙ = α − (1 + β) x + x 2 y, y˙ = βx − x 2 y,
(1)
was proposed by Prigogine and Lefever [1], being a model for chemical reactions with oscillations, therefore maintaining a prolonged non-equilibrium state. For a complete description of the x, y dimensionless variables and the α, β positive control parameters, we suggest consulting Hannon and Ruth [2]. As explicitly shown in Epstein and Pojman [3], the Brusselator is a four-step chemical reaction which represents for example the Belousov-Zhabotinsky reaction [4], besides several others. In this paper, we deal with a sinusoidally forced Brusselator given by: x˙ = α − (1 + β) x + x 2 y + F cos ωt, y˙ = βx − x 2 y,
Paulo C. Rech
[email protected] 1
Departamento de F´ısica, Universidade do Estado de Santa Catarina, 89219-710, Joinville, Brazil
(2)
where F and ω are respectively amplitude and angular frequency of an external periodic forcing. System (2) was recently investigated by Luo and Guo [5], from the point of view of analytical solutions of some periodic evolutions. Reference [5] presents for example, the bifurcation tree of period 1 to period 8 evolutions through frequencyamplitude characteristics for system (2), among other equally important results. Our main goal in this paper is to investigate numerically the (ω, F ) parameter plane of system (2), with α = 0.4, β = 1.2 kept fixed, in the search for regions of multistability [6]. Therefore, our contribution to advance knowledge of the forced Brusselator considers (ω, F ) parameter planes, which are cross sections of the (α, β, ω, F ) fourdimensional parameter space of system (2). As we will see in detail in the next section, each (ω, F ) parameter plane is constructed by considering a mesh of one million points. The characterization of the dynamical behavior of each of these points, chaos or regularity, in each parameter plane, will be done by the respective Lyapunov exponents spectrum, computed by the algorithm described in Wolf et al. [7]. From there, it will be possible to identify, by a simple visual inspection, the ex
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