Abstract
For the wide class of 3D autonomous quadratic dynamical systems depending on parameters the sucient conditions of boundedness of solutions of any system from this class are found. A connection between change of one of the parameters and a recurrence plot structure, which was built on the time series for any system of this class, is determined. Due to this connection it is possible to nd bifurcation values of the parameter of any system from the considered class only on its time series without knowledge of dierential equations of this system. Examples are given.
Highlights
Let x0 = x(t0), x1 = x(t1), ..., xn = x(tn) (1.1)be a finite sequence of numerical values of some scalar dynamical variable x(t) measured with the constant time step ∆t in the moments ti = t0 +i∆t; xi = x(ti); i = 0, 1, ..., n
As a large body of literature exists on applying the technique to the time series from chaotic attractors [4] – [8], a relatively unexplored issue is its applicability to dynamical systems depending on parameters
Our focus will be concentrated on the analysis of influence of a period doubling bifurcation on behavior of the recurrence plot structure for the time series (1.1)
Summary
Sequence (1.1) is called a time series [1] – [3]. A common practice in chaotic time series analysis has been to reconstruct the phase space by utilizing the delay-coordinate embedding technique, and to compute dynamical invariant quantities of interest such as unstable periodic orbits, the fractal dimension of the underlying chaotic set, and its Lyapunov spectrum. As a large body of literature exists on applying the technique to the time series from chaotic attractors [4] – [8], a relatively unexplored issue is its applicability to dynamical systems depending on parameters. Our focus will be concentrated on the analysis of influence of a period doubling bifurcation on behavior of the recurrence plot structure for the time series (1.1).
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