CUPOLETS: Chaotic unstable periodic orbits theory and applications
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 [19]. In particular, chaotic communication techniques and methods of controlling chaos depend on this property of chaotic attractors [12, 13]. In the first part of this thesis, a control scheme for stabilizing the unstable periodic orbits of chaotic systems is presented and the properties of these orbits are investigated. The technique allows for creation of thousands of periodic orbits. These approximated chaotic unstable periodic orbits are called cupolets (C&barbelow;haotic U&barbelow;nstable P&barbelow;eriodic O&barbelow;rbit- lets). We show that these orbits can be passed through a phase transformation to a compact cupolet state that possesses a wavelet-like structure and can be used to construct adaptive bases. The cupolet transformation can be regarded 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. The key point in this method is that all of these different dynamical behaviors are easily accessible via small controls. This technique is implemented in order to produce cupolets which are essentially approximate periodic orbits of the chaotic system. The orbits are produced with small perturbations which in turn suggests that these orbits might not be very far away from true periodic orbits. The controls can be considered as external numerical errors that happen at some points along the computer generated orbits. This raises the question of shadowability of these orbits. It is very interesting to know if there exists a true orbit of the system with a slightly different initial condition that stays close to the computer generated orbit. This true orbit, if it exists, is called a shadow and the computer generated orbit is then said to be shadowable by a true orbit. We will present two general purpose shadowing theorems for periodic and nonperiodic orbits of ordinary differential equations. The theorems provide a way to establish the existence of true periodic and non-periodic orbits near the approximated ones. Both theorems are suitable for computations and the shadowing distances, i.e., the distance between the true orbits and approximated orbits are given by quantities computable form the vector field of the differential equation.
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