Modeling the Invariant Density Curve of a Noisy Chaotic Map

The goal of the research is to analytically determine the invariant density curve for a given chaotic map altered by noise and to compare it to a numerically generated invariant density curve for the same map. The existence of an analytic solution allows one to predict the overall behavioral trend, if not the exact orbit, of a map and is potentially tremendously useful in the modeling of any system for which maps are used. I will consider both additive and parametric noise, focusing primarily on the latter. Introduction A goal of science is to accurately model the surrounding world and confirm testable hypotheses by comparing predictions from a theory with the results of nature. To make such predictions the scientist connects the relevant fundamental laws to a real world system through the creation of a mathematical model. Mathematicians have found use in describing such systems in discrete time using difference equations, or iterative functions, where each time step of the function is determined relative to the previous step, and only the previous step (e.g. there is no explicit dependence upon the value of the function two iterations ago) [1, 2]. The general form of an iterative function is as follows: This type of function requires knowledge of the initial state of the physical (or mathematical) system in order to be implemented. Since each step of an iterative function relies on the previous step, one finds that the paths of certain difference equations are heavily dependent on the initial value of the variable(s) [1]. This sensitive dependence occurs if the iterative function is chaotic. If chaotic the orbit never quite reaches the initial position and the system is characteristically non-conservative [1]. The measure of the chaotic nature of a difference equation is the Lyapunov exponent, λ, which is a measure of the difference in the orbit and the initial case after a time t – t0. For two trajectories with the initial separation distance δ0 at time t0 the separation distance at time t is approximately given by: [1]. By definition, in a chaotic system λ ≥ 0. Because difference functions require the computation of many iterations a computer is indispensible to the analysis of such functions. Unfortunately the output of a chaotic iterative function, due to its high level of sensitivity on initial conditions, cannot be very well predicted. For this reason one may opt to view the outputs of the function by plotting a histogram over the defined domain of the function and binning the elements that fall within intervals of a certain size. The size of the bins is given by the following: The histogram plot is important because in the infinite limit of the number of intervals and iterations the peaks of the histogram combine to become a smooth curve with does not depend on the particular x0 of a given orbit. The normalized version of this curve is called the invariant density of the system which is equivalent to a probability density, thus if the area under the curve of an interval on the x axis is equivalent to the probability that an iteration will happen within the interval. This is, in other words, the fraction of the total time spent in that interval by the orbit (note that over all space this area, or integral, is equal to unity). The invariant density is a useful object of study since one can plot the invariant density for one map using any initial condition and find that the same curve is reproduced regardless of the initial condition(s). Changing the parameters of the map will change the invariant density curve so, using this method, one is allowed to compare chaotic maps of different parameters without having to bother about the map’s sensitivity to initial conditions. A common problem that arises when one models a physical system is that the surroundings often have unpredictable effects upon the system of study. One cannot completely isolate the system of study from this noise and one may not be able to quantify or measure these unpredictable effects, the presence of which make the model, and analysis of the model, inaccurate [1, 4]. However, it is possible to mirror these effects with an additive noise term and attempt to analyze its affect on the system by the comparison of the resultant invariant density curves of the noise and no-noise mapping. The map of the iterative function including the noise term (where ε is a constant magnitude term and rn is a randomly generated number uniformly between -1 and 1) is the following: When noise is added to a system it is important that the magnitude of the noise is chosen appropriate to the magnitude of the output of the function so that the noise affects the curve but does not dominate it. Once this noise has been added one must ask an important question: is the original function still valid? If the invariant density curve is still retained as the magnitude of the noise (ε) goes to zero then the original function is an accurate portrayal of reality [4]. Throughout the paper I will refer to this additive noise as “noise in dynamics”. Maps have been used in several instances with significant success. Perhaps the most famous is logistic map which has been used to model population growth. It is given below where r is a birth or growth rate and is positive (I should mention that this model has found success in modeling certain populations, such as in animal groups, but limited or no success in other, more complicated, populations such as bacterial growth.) [2] Another way to use noise to mimic unpredictability is to use another type of noise, parametric noise, or parameters which are time dependant. In the example of the logistic map this would mean that the parameter, r, would change its value randomly every iteration, becoming rn. A change in parameter is a realistic thing to study since (again in the logistic map example) it might mean an unexpected drought or catastrophe which could alter the growth rate of a population. When compared to an analytic treatment of the noise, if the affect of the parametric noise on the shape of the function goes to zero as the variation of the noise tends towards zero then one can say that the analytical function accurately portrays the behavior of the map. I will refer to the use of parametric noise as “noise in parameters” throughout the course of this paper. Methods The primary source of useful information on the behavior of an iterative function is the invariant density curve which is not dependent upon initial conditions and thus provides information on the general behavior of the map as opposed to specific and wildly varying individual orbits. The study includes the numerical generation of curves for both the noise in dynamics and noise in parameters. The noise in parameters case will also be analyzed analytically in order to generate a point of comparison between the numerically generated curve and a theoretical curve. As I mentioned in the introduction the invariant density curve can be seen as the curve of a histogram as the number of iterations of the function tends to infinity and the size of the intervals over the domain of the histogram bars becomes infinitesimal. For clarity in comparison my graphs generate a curve by connecting the center points of the bars of this histogram to make a smooth curve. This, of course, is an approximation since I cannot iterate the function an infinite number of times but since, with enough iterations, the curves look smooth, this is clearly a decent approximation. In order to generate these curves I have written a program using Matlab which simply calls on a function from another file and iteratively runs it using a random or inputted initial condition (if you recall in my introduction the curve I am trying to generate does not depend on the initial conditions which is its advantage) as well as inputted magnitudes of noise and parameters and generates an array with a counter which increases for each time the output of the function falls within a certain interval of the array. This array is plotted “sideways” so that the size of the counter increases the height of the graph at each interval. A line is then drawn from the center point of each interval to plot the smooth invariant density curve. Two things should be mentioned with regards to the program. The noise can push the output of a function past the domain over which the function is defined. To counter this problem the program has built in controls to reflect this value back into the domain before the calculation of the next iteration can be done. It should also be mentioned that the number of iterations or size of bins that one uses change the height of the curve which can make it difficult to compare two curves which were generated using different values of these two. To fix this problem I do not just plot the height of the array but rather: to normalize it, so that the plot is a true invariant density curve. I found that my home computer could handle about 5,000,000-10,000,000 iterations depending on the complexity of the function in a reasonable amount of time. In the following sections these curves are plotted in this manner.