Abstract
The existence of periodic windows in the parameter space is a common feature of nonlinear systems capable to produce chaotic behavior. Detection of positions of periodic windows is important both from the theoretical and practical points of view. In this work, a systematic method to detect periodic windows in bifurcation diagrams is proposed. The search method is a combination of the trajectory monitoring approach to find unstable periodic orbits and the continuation method to calculate positions of periodic windows. The method is applied to the Chua's circuit with a smooth nonlinearity. It is shown that the proposed method outperforms standard methods to find periodic windows in bifurcation diagrams.
Highlights
C ONSTRUCTION of bifurcation diagrams is one of the main tools to study the dynamics of nonlinear systems under parameter variation and to detect various bifurcation types [1]–[6].Single parameter bifurcation diagrams are constructed by plotting steady state trajectories versus the value of a selected system’s parameter called the bifurcation parameter
Two-parameter and three-parameter bifurcation diagrams are obtained by finding steady state solutions in a two- or three-dimensional parameter space and plotting regions in the parameter space with different types of steady state behavior using different colors [7], [8]
In the following n denotes the number of test parameter values selected from a given interval, N denotes the number of test parameter values for which the steady state trajectory is periodic, and W denotes the number of periodic windows detected
Summary
C ONSTRUCTION of bifurcation diagrams is one of the main tools to study the dynamics of nonlinear systems under parameter variation and to detect various bifurcation types [1]–[6]. We consider the Chua’s circuit with a cubic nonlinearity and study the existence of periodic windows for this system. Preliminary version of the results presented in this work are described in [30], where the study of periodic windows in a neighborhood of the point in the parameter space where the spiral attractor exists is carried out. A more detailed description of the search method is presented, the Chua’s circuit with a different bifurcation parameter is considered, and a more challenging case of the double-scroll attractor is investigated. The results obtained are compared with the result produced by the standard method based on finding steady state behavior for selected points in the parameter space. It is shown that the standard method fails to find narrow windows and the reasons for its failure are explained
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