Dangerous Bifurcations in Piecewise Smooth Maps

Saha, Arindam (2015) Dangerous Bifurcations in Piecewise Smooth Maps. Masters thesis, Indian Institute of Science Education and Research Kolkata.

PDF (MS dissertation of Arindam Saha (10MS02)) Arindam_Saha-10MS02.pdf - Submitted Version Restricted to Repository staff only Download (3MB)

Abstract

Piecewise smooth maps have gained wide attention of researchers in recent years due their wide applicability in dynamical systems and rich dynamical properties. Not only are they used to describe piecewise smooth systems like switching circuits, impacting mechanical systems, neural dynamics, cardiac rhythms and walking robots, they also show a range of different bifurcation phenomena. Many interesting phenomena arise due to border collision like vanishing of the periodic orbit, a fixed point directly bifurcating into a chaotic orbit or a fixed point giving rise to multiple periodic, quasi-periodic and chaotic orbits have been recently reported. It has been shown that the map exhibits robust chaos and existence of infinitely many co-existing attractors. An interesting case of such a bifurcation is the dangerous bifurcation. Dangerous bifurcations in piecewise smooth continuous maps have been defined to occur when a stable fixed point occurs before and after the bifurcation, and yet the basin of attraction shrinks to zero size at the bifurcation point. This results in all trajectories diverging off to infinity at the bifurcation point despite eigenvalues lying within the unit circle. This results in a potentially dangerous situation where systems behave contrary to what is predicted by the eigenvalues of the system. Despite the presence of an attractor, the physical system diverges, entering in to an extremely high energy state. This might result in a total breakdown of the system. In this thesis we generalize the definition to one where the attractor whose basin size shrinks, might be of any periodicity. Various parameter regimes are explored numerically to witness the various dynamical features of bifurcation. The exploration reveals many interesting observations. In addition to a richer basin structure, we were able to see many other bifurcation phenomena in conjugation with dangerous bifurcations. The analysis shows that occurrence of multiple attractors or robust chaos together with dangerous bifurcations might compound the problems for systems working in those regimes. Our analysis reminds us of the fact that presence of stable and unstable periodic orbits plays a vital role in determining the dynamical properties of a system. While presence of stable periodic orbits form the basins of attraction, the unstable periodic orbits play a crucial role in forming the basin boundaries of the attractors. In fact appearance, disappearance or change in stability of these periodic orbits are responsible for the numerous bifurcation phenomena listed above. Hence an important component in characterising a given system is determining the parameters for which a particular periodic orbit might exist. Hence in order to address a wider range of phenomena, we turn our attention to existence and stability criteria of periodic orbits. Although some work has been done to find out these conditions earlier, the methods used were limited to specific types of periodic orbits in lower dimensional space. We focus on finding out a generic method that would span a wide range of periodic orbits in an arbitrary dimensional space. To do this, we use and extend a technique developed by Russian mathematician Leonov. On careful observation of the conditions, we realise that a significant role is played by the powers of matrices involved in the normal forms of the piecewise smooth maps. We also develop a technique which computes those powers in terms of sequences. The method proposed is a faster and more elegant way of obtaining those conditions without iterating the complete map. The method revolves around direct computation of higher powers of matrices without computing the lower ones and is applicable on any dimension of the phase space. In the later part of the thesis, we also illustrate the use of this method in computing the regions of existence and stability of a particular class of periodic orbits in two and three dimensions.

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