Final State Sensitivity Research Articles

The equality of fractal dimension and uncertainty dimension for certain dynamical systems

[MGOY] introduced the uncertainty dimension as a quantative measure for final state sensitivity in a system. In [MGOY] and [P] it was conjectured that the box-counting dimension equals the uncertainty dimension for basin boundaries in typical dynamical systems. In this paper our main result is that the box-counting dimension, the uncertainty dimension and the Hausdorff dimension are all equal for the basin boundaries of one and two dimensional systems, which are uniformly hyperbolic on their basin boundary. When the box-counting dimension of the basin boundary is large, that is, near the dimension of the phase space, this result implies that even a large decrease in the uncertainty of the position of the initial condition yields only a relatively small decrease in the uncertainty of which basin that initial point is in.

COMPLEX BEHAVIOR IN DIGITAL FILTERS

This paper presents recent results concerning possibilities of complex behavior in extremely simple second-order digital filter structures. Two types of such behavior have been confirmed by simulation experiments and rigorous mathematical analysis: 1) chaotic behavior defined here as the existence of aperiodic, bounded trajectories displaying sensitive dependence to the initial states. Such trajectories often form very complex, self-similar patterns in the state-space. 2) an abundance of oscillatory solutions and final state sensitivity with respect to system parameters. This type of behavior can be identified by an extremely complicated structure of the parameter space (existence of Arnold tongues) and fine structure of changes of dynamic behavior when varying filter parameters (devil’s staircase). The first type of behavior has been discovered in filter sections employing two’s complement adder overflow characteristics and confirmed via symbolic dynamics technique. Properties of the second type have been found in extensive numerical experiments carried out for filter sections employing saturation arithmetic and in which a full confirmation via the mathematical analysis of an associated one-dimensional model of the system has been made.

Relating the various scaling exponents used to characterize fat fractals in nonlinear dynamical systems

Fat fractals (sets with fractal structure, but nonzero measure) are becoming increasingly important in nonlinear dynamical systems. These sets have allowed quantitative analyses of sensitivity to parameters, final-state sensitivity, and quantum chaos, and have played important roles in discussions of nonlinear twist maps, one-dimensional maps with quadratic maxima, Arnol'd tongues in circle maps, and practal basin boundaries. Unfortunately, since fat fractals have nonzero measure, they must have the same (integer) dimension as the underlying space. Therefore, their dimension is insensitive to the fractal structure and does not provide an appropriate means of characterization. Hence, several scaling exponents have been suggested for this purpose, and there has been some debate over which is best. Also, there has been considerable confusion over how these exponents differ and whether they are simply related. We shed light on this issue by examining these exponents and their properties and deriving relationships between them.

The shape of the basin of attraction and transients via transformation to normal form

A method is derived to approximate the basin and investigate the transients of an attracting fixed point with complex multipliers for an analytic invertible mapping of the real plane. It consists of constructing approximate but explicit expressions for the normal transformation about the fixed point. This transformation has the whole basin as domain, and transforms the nonlinear mapping restricted to the basin, to a linear one. With this transformation a non-negative functional for the basin is defined whose value decreases along an orbit. Explicit expressions are derived for a sequence of contours approximating a level line of this functional. At large values of the functional a level line approximates the basin boundary. Calculation of a sequence of level lines yields estimates for that part of the basin where regular (monotonic) convergence to the attracting fixed point is to be expected. The shape of the level lines demonstrates the occurrence of transient-periodic behavior. The way in which the unstable manifold of the saddle, born together with the attractor from a tangent bifurcation, intersects the level lines yields a criterion for a saddle-sink connection. In that case the stable manifold as part of the basin boundary has a simple structure, in contrast to the fractal structure that occurs when there is a homoclinic orbit, which causes final state sensitivity. The calculations are carried out for the Hénon mapping.

Final-state sensitivity for time-optimal control problems

In optimal-control problems where a desired state is to be reached in minimum time, substantial time reductions can be realized by defining the target to be within some small tolerance rather than by specifying the exact value. The minimum time and the optimal control policy as well as the sensitivity information can be readily obtained by applying linear programming techniques whether the system is in the time domain or in the finite-Laplace-transform domain. The computational aspects of the methods of solution in the two domains are compared.

Characterization of fat fractals in nonlinear dynamical systems.

Fat fractals (sets with fractal structure, but nonzero measure) are becoming increasingly important in the study of nonlinear dynamical systems. Since fat fractals are not well characterized by their (integer) dimensions, several scaling exponents have been proposed to characterize these sets, and there has been some debate over which exponent is best. We shed light on this issue by examining these exponents and finding relationships between them. Finally, these results lead naturally to quantitative definitions of the notions of sensitivity to parameters and final-state sensitivity.

Scaling of the uncertainty exponent.

The uncertainty exponent describing the generalized final-state sensitivity of the logistic map at its periodic windows is analyzed numerically. This exponent itself scales with the nonlinearity parameter according to a power law. The universal value of the exponent characterizing this scaling behavior is conjectured. A consequence of the above scaling for dimensions of fractal sets is pointed out.

Generalized final state sensitivity of the logistic map

It is shown numerically that the one-dimensional logistic map displays at its periodic windows a generalized final state sensitivity with respect to initial conditions. The uncertainty exponent characterising this sensitivity is analysed at various values of the control parameter.

Selection between multiple periodic regimes in a biochemical system: Complex dynamic behaviour resolved by use of one-dimensional maps

We analyse a model biochemical system in which two autocatalytic enzyme reactions are coupled in series, in conditions where multiple stable periodic regimes coexist for the same set of parameter values. We determine how the periodic regimes are reached from different initial conditions. The structure of the attraction basins is generally simple in the case of two coexisting limit cycles (birhythmicity). This structure and the associated behaviour may, however, become highly complex. In particular, the system exhibits enhanced sensitivity to initial conditions when the boundaries of the attraction basins are fractal. In the latter case, it becomes difficult to predict the evolution towards either one of two limit cycles, a phenomenon known as final state sensitivity. We show how these complex phenomena can be explained in a unified and simple manner by means of one-dimensional return maps derived from the time evolution of the model and from fifth degree polynomial equations. We suggest experimental tests of the sensitivity to initial conditions in chemical systems presenting birhythmicity. The physiological significance of the results is discussed with respect to the sensitivity of regulatory systems admitting multiple stable biological rhythms.

Multiple periodic regimes and final state sensitivity in a biochemical system

We analyze a model biochemical system governed by three nonlinear differential equations, under conditions where multiple stable periodic regimes coexist. We determine the structure of their basins of attraction and show that final state sensitivity is obtained when two stable oscillations are separated by unstable chaos. The effect of fractal basin boundaries on the response of the oscillatory system to perturbations is discussed.

Final state sensitivity: An obstruction to predictability

It is shown that nonlinear system with multiple attractors commonly require very accurate initial conditions for the reliable prediction of final states. A scaling exponent for the final-state-uncertain phase space volume dependence on uncertainty in initial conditions is defined and related to the fractal dimension of basin boundaries.