Iterated map

Substituting A/ = 9 into equation 7.92, we obtain the following iterative map for the density p ... 

For small enough values of p so that pf p) < p for all 0 < p < 1, p = 0 will be the only fixed point. As p increases, there will eventually be some density p for which pf p ) > p in this case, we can expect there to be nonzero fixed point densities as well. Qualitatively, the mean-field-predicted behaviors will depend on the shape of the iterative map. If / has a concave downward profile, for example (i.e. if/" < 0 everywhere), then, as p decreases, Poo decreases continuously to zero at some critical value of p = Pc- Note also that the iterative map /jet for the deterministic rule associated with its minimally diluted probabilistic counterpart is given by /jet = //p-... 

Table 7.3 lists the four rules in this minimally-diluted rule-family, along with their corresponding iterative maps. Notice that since rules R, R2 and R3 do not have a linear term, / (p = 0) = 0 and mean-field-theory predicts a first-order phase transition. By first order we mean that the phase transition is discontinuous there is an abrupt, discontinuous change at a well defined critical probability Pc, at which the system suddenly goes from having poo = 0 as the only stable fixed point to having an asymptotic density Poo 7 0 as the only stable fixed point (see below). 

Table 7.3 The four rules in the family of two-state/ one-absorbing-state probabilistic PCA defined in the text, along with their corresponding iterative maps (see equation 7.98).

Figure 7.8 shows a plot of the iterative map /2(p) for rule R2 as a function of p for four different values of p p = 1 (top curve), p > Pc, P = Pc and p < Pc, where Pc 0.5347. Notice that all four curves have zero first and second derivatives at the origin. This ensures the existence of some critical value Pc such that for all p < Pc, p t + 1) < p t) and thus that limt->oo p t) = 0. In fact, for all 0 < p < Pc the origin is the only stable fixed point. At p = Pc, another stable fixed point ps 0.373 appears via a tangent bifurcation. For values of p greater than Pc, /2 undergoes a... 

Fig. 7.8 Plot of the iterative map /2(p) as a function of p for four different values ofp see text.

See chapter 4 for a general discussion of period-doubling and chaos in one-control-parameter iterative maps. 

Intermlttency Manneville [mann80] showed that for the special case of a generic intermittency threshold in which the tangent point lies at the endpoint of the interval (in the case of a one dimensional iterative map of an interval to itself), the resulting chaotic dynamics has a power spectrum S f) 1/ (/(log/) ) for low /. Miracky, et. al. were able to modify the map to obtain an exact 1// behavior [mirack87]. Because the result depends on the fine-tuning of an external parameter, however, it does not so mucdi explain the generic appearance of flicker noise phenomena as beg the obvious question, why do systems typically sit at whatever... 

Collet, P. and Eckmann, J.-P., 1980, Iterated Maps on the Interval as Dynamical Systems, Progr. Phys. Vol. 1. Birkhauser, Boston. 

Within a model-based framework, Lim and Oppenheim [Lim and Oppenheim, 1978] studied noise reduction using an autoregressive signal model, deriving iterative MAP and ML procedures. These methods are computationally intensive, although the signal estimation part of the iteration is shown to have a simple frequency-domain... 

The RIs display a number of interesting mathematical and physical properties. To briefly introduce these properties, we define the RIs generated from the same branch (either the stable or the unstable branch) to be of the same family. We also denote by RI +i the reactive island generated from a further iteration of the reactive island RIy. That is, the area of RIy wiU, upon one positive or negative time iteration, map onto RIy+i or RIy-i within the same family. It... 

J. Milnor and W. Thurston, On Iterated Maps of the Interval, Dynamical Systems, College Park, MD, 1986-87, J. C. Alexander ed. Lecture Notes in Math. Vol. 1342 (Springer-Verlag,... 

There are two main types of dynamical systems differential equations and iterated nuips (also known as difference equations). Differential equations describe the evolution of systems in continuous time, whereas iterated maps arise in problems where time is discrete. Differential equations are used much more widely in science and engineering, and we shall therefore concentrate on them. Later in the book we will see that iterated maps can also be very useful, both for providing simple examples of chaos, and also as tools for analyzing periodic or chaotic solutions of differential equati ons. 

By this ingenious trick, Lorenz was able to extract order from chaos. The function = /(z ) shown in Figure 9.4.3 is now called the Lorenz map. It tells us a lot about the dynamics on the attractor given we can predict z, by z, = /(Zq), and then use that information to predict Zj = /(zi), and so on, bootstrapping our way forward in time by iteration. The analysis of this iterated map is going to lead us to a striking conclusion, but first we should make a few clarifications. 

Second, the Lorenz map may remind you of a Poincare map (Section 8.7). In both cases we re trying to simplify the analysis of a differential equation by reducing it to an iterated map of some kind. But there s an important distinction To construct a Poincare map for a three-dimensional flow, we compute a trajectory s successive intersections with a two-dimensional surface. The Poincare map takes a point on that surface, specified by two coordinates, and then tells us how those two coordinates change after the first return to the surface. The Lorenz map is different because it characterizes the trajectory by only one number, not two. This simpler approach works only if the attractor is very flat, i.e., close to two-dimensional, as the Lorenz attractor is. 

This chapter deals with a new class of dynamical systems in which time is discrete, rather than continuous. These systems are known variously as difference equations, recursion relations, iterated maps, or simply maps. 

Solution Our analysis follows a strategy that is worth remembering To analyze the stability of a cycle, reduce the problem to a question about the stability of a fixed point, as follows. Both p and q are solutions of / (x) = x, as pointed out in Example 10.3.2 hence p and q are fixed points of the second-iterate map / (x). The original 2-cycle is stable precisely if p and q are stable fixed points for... 

The third-iterate map f (x) is the key to understanding the birth of the period-3 cycle. Any point p in a period-3 cycle repeats every three iterates, by definition, so such points satisfy p = f p) and are therefore fixed points of the third-iterate map. Unfortunately, since / (x) is an eighth-degree polynomial, we cannot solve for the fixed points explicitly. But a graph provides sufficient insight. Figure 10.4.1 plots p (x) for r = 3.835. 

This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors. 

One lesson of chaos theory is that simple, nonlinear equations can have complicated solutions. The solutions are most interesting if they Involve at least two variables, X and y, that can be used to represent horizontal and vertical positions, and if the variables are advanced step by step in an iterative process. The simplest non-linearities are quadratic (p or y). The most general two-dimensional quadratic iterated map is ... 

To address these questions and prove the validity of our analysis for spiking neuron models we have built a more realistic network model of an insect olfactory system using map-based spiking neurons (Gazelles et al. 2001 Rulkov 2002). These neurons are defined by a simple iterative map. 

Figure 1 Maximum entropy image consistency checks The top image diows ttie skymap after the 5tfa iteration, the centre image after 30 iterations. In the bottom image, Gaussian smoothing witfi the instrumental resolution has been applied to remove the s ier-resolution spikiness of the 30-iteration map, which then is demonstrated to show emission structures very smular to the 5th iteration map.

Figure 3. Return iterate map illustrating the hysteresis route to chaos, taken after (49). Circled regions represent transients.

What is important for us is that certain qualitative properties of multidimensional differential equations can be studied by a one-dimensional difference equation, or an iterated map, in modern terminology. Iterated... 

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