Bifurcation structures within robust chaos

Both chaotic attractors as well as bifurcation scenarios formed by several attractors belong to key concepts of nonlinear dynamics. Nevertheless, bifurcation scenarios formed by chaotic attractors only were barely investigated in the past. Therefore, the presented work is motivated by a seemingly simple question, namely which generic bifurcation structures may occur in the case that a dynamical system shows chaotic behavior without periodic inclusions? A major reason why this issue was not thoroughly investigated until now, was the lack of adequate numerical methods. Hence, the first step of this work is given by the development of reliable and efficient numerical methods suitable for investigation of bifurcation structures in the chaotic domain. It is shown that depending on certain properties of considered models, several investigation methods can be used which are different with respect to their efficiency. As an important application example, some bifurcation structures occurring in piecewise-linear systems will be considered. The developed methods allow us to describe two generic bifurcation patterns which organize the chaotic domain in these systems. Based on the obtained numerical results, the detected bifurcation structures will be investigated analytically. This confirms the reliability of the developed methods and explains the origins of numerically detected bifurcation structures. Further, the connection between bifurcation structure in the chaotic domain and bifurcation structures in the adjacent periodic domain will be established. Due to this connection, we can demonstrate the self-similarity of the chaotic domain. The robustness of the numerical approaches is confirmed by the fact that the bifurcation structures can be detected also in the regions of parameter space where the dynamics is not benign from the numerical point of view. The work will be concluded with a discussion which demonstrate the relevance of the obtained results for engineering applications.