A class of piecewise C2 transformations from an interval into itself with slopes greater than 1 in absolute value, and having the property that it takes partition points into partition points is shown to have unique absolutely continuous invariant measures. For this class of functions, a central limit theorem holds for all real measurable functions. For the subclass of piecewise linear transformations having a fixed point, it is shown that the unique absolutely continuous invariant measures are piecewise constant.

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On a class of transformations which have unique absolutely continuous invariant measures

We analyze a simple model of deterministic diffusion. The model consists of a one-dimensional array of scatterers with moving point particles. The particles move from one scatterer to the next according to a piecewise linear, expanding, deterministic map on unit intervals. The microscopic chaotic scattering process of the map can be changed by a control parameter. The macroscopic diffusion coefficient for the moving particles is well defined and depends upon the control parameter. We calculate the diffusion coefficent and the largest eigenmodes of the system by using Markov partitions and by solving the eigenvalue problems of respective topological transition matrices. For different boundary conditions we find that the largest eigenmodes of the map match the ones of the simple phenomenological diffusion equation. Our main result is that the diffusion coefficient exhibits a fractal structure as a function of the control parameter. We provide qualitative and quantitative arguments to explain features of this fractal structure.

Simple deterministic dynamical systems with fractal diffusion coefficients.

: Chaotic systems have received much attention in the mathematics and physics communities in the last two decades; and they are receiving increasing attention in various engineering disciplines as well Experimental evidence suggests that these systems may be useful models for a wide variety of physical phenomena, including turbulence, vibrations of buckled elastic structures, and behavior of certain feedback control devices This thesis deals with both the analysis and synthesis of chaotic maps and time-sampled chaotic flows, with a focus on the problems and issues that arise with noise-corrupted orbit segments generated by these maps and flows Both dissipative and nondissipative systems are considered, with both types of systems considered in the context of analysis and the latter type also considered in the context of synthesis With respect to dissipative systems, three probabilistic state estimation algorithms are introduced and applied to three problem scenarios, with the scenarios distinguished by the amount of a priori knowledge of the dynamics of the underlying chaotic system Cramer-Rao, Barankin, and Weiss-Weinstein upper bounds on state estimator performance are derived and both experimentally and qualitatively analyzed The analysis reveals that intrinsic properties of chaotic systems-positive Lyapunov exponents and boundedness of attractors-have a fundamental influence on achievable state estimator performance with these systems

estimation and detection with chaotic systems

If the phase space X of a motion $x_{n + 1} = f(x_n )$ is discretized into a space of states $X_1 , \cdots ,X_N $, then probabilities can be assigned to sample paths in the state space so as to coincide with the ones assigned by a finite Markov chain. Theorems 1 and 2 show how the assignment of such probabilities rests on the properties of $f( \cdot )$ and on the construction of the states. Theorems 3 and 4 extend these results to the case in which $x_{n + 1} = f(x_n ,\omega )$, $\omega \in \Omega $ being a random event. Theorems 5 and 6 indicate certain applications relating to stochastic systems in which a decision-maker applies some control action which is fully or partially determined by the observed state of the system.

S. Sankar Sengupta

Papers

Existence and Control of Markov Chains in Systems of Deterministic Motion