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01/11/07 Colloquium
Dr. Kourosh Zarringhalam
University of New Hampshire
University of New Hampshire
Chaotic Unstable Periodic Orbits: Theory and Applications
Abstract: Recent theoretical work suggests that periodic orbits of chaotic systems are a rich source of qualitative information about the dynamical system. The presence of unstable periodic orbits located densely on the attractor is a typical characteristic of chaotic systems. This abundance of unstable periodic orbits can be utilized in a wide variety of theoretical and practical applications.In the first part of this talk, we will present a control scheme for stabilizing the unstable periodic orbits of chaotic systems. The technique allows for creation of thousands of periodic orbits. These approximated chaotic unstable periodic orbits are called cupolets (Chaotic Unstable Periodic Orbit-lets).
In the second part of the talk, we will show that the cupolet transformation can be considered as an alternative to Fourier and wavelet transformations. In fact, this new framework provides a continuum between Fourier and wavelet transformations and can be used in variety of applications such as data and music compression, as well as image and video processing.
Finally, we will present a general purpose shadowing theorem for orbits of ordinary differential equations. The theorems provide a way to establish the existence of true orbits near approximated ones.
