Circle maps

Let us consider the same Laplacian coupled CML as in the previous sections, but with / equal now equal to the one-dimensional circle map i.e. consider... 

At the risk of oversimplifying, there are essentially three different dynamical regimes of the one-dimensional circle map (we have not yet formed our CML) (I) j A < 1 - for which we find mode-locking within the so-called AmoW Tongues (see section 4.1.5) and the w is irrational (11) k = 1 - for which the non mode-locked w intervals form a self-similar Cantor set of measure zero (111) k > 1 - for which the map becomes noninvertible and the system is, in principle, ripened for chaotic behavior (the real behavior is a bit more complicated since, in this regime, chaotic and nonchaotic behavior is actually densely interwoven in A - w space). 

DeBrouwer, S., Edwards, D., and Griffith, T., Simplification of the quasi-periodic route to chaos in agonist-induced vasomotion by iterative circle maps, American Journal of Physiology, Vol. 274, No. 4(2), 1998, pp. H1315-1326. 

Q is the instantaneous volumetric flow rate downward in the negative z direction which intersects the circle mapped ont by one end of a vector that rotates completely around the z axis while the other end is pinned to the z axis at point O. The coordinates of the following points are of interest in developing relations between Vr and vg and the stream function ... 

Rotate vectors OR and OP completely around the z axis and calculate the differential volumetric flow rate downward between the two circles mapped out by... 

Now, rotate vectors OR and OW completely around the z axis and calculate the differential volumetric flow rate downward between the two circles mapped out by points R and W when 9 < rcjl. The velocity component of interest is —Vr, and sin9 d9 is the cross-sectional area for flow. In this case,... 

Figure 5.2 Ford-circle mapping of the periodic table...

A circle map can be easily defined in terms of the Poincare section of the torus (see Figure 28). The elliptical curve in the Poincare section is actually a large number of individual points, each corresponding to the place the quasi-periodic trajectory pierces through the surface of section. An arbitrary zero point can be chosen (see Figure 28), and an angle, 6 , can be defined to label each position, that is, the th point, in the section. The Poincare cross section is constructed by allowing an actual trajectory to create a sequence of points that can be numbered in the order in which they appear. Hence, a sequence of... 

Mathematical functions that approximate the form of the experimentally derived circle map shown in the figure have been extensively studied. - Perhaps the most well known of these is referred to as the sine circle map, which is given by the following iterative relation ... 

Rgnre 29 A circle map derived from simulations with the DOP model of die peroxidase-oxidase reaction. (From Ref. 73 used with permission.)... 

The corresponding circle map for this case will consist of P points below the diagonal and Q total points. The accompanying graph [see Figure 30(a)] of the sine circle map for (I = 0.2, fC = 1.0 shows a periodic orbit with P = 1, Q = 8 for a winding number W = Vs. The winding number may also be irrational if the trajectory is quasiperiodic ... 

This definition allows us to consider chaotic behavior, which the sine circle map also displays. In the case of chaos, W will not converge for large n. 

For the sine circle map, there is a tendency for the winding number (W) to lock in to a rational value P/Q over a range of SI values when K is in the range 0 < K s 1. The density of these mode-locked states in a graph of W versus SI is low for low values of K and increases as K approaches 1. At the value K = 1 (the critical point) the mode-locked states fill up the entire fl axis. The graph of... 

In realistic models, of which those arising in chemistry are good otam-ples, the simple dynamics displayed by the sine circle map becomes more complex. It is possible that the sine circle map describes the dynamics of the system very close to and slightly beyond the transition to chaos, but that once one has gone well into the chaotic region in parameter space, this description no longer applies. 

A studyof a fairly simple model of an enzyme reaaion that exhibits chaotic behavior, the peroxidase-oxidase reaction, provides a good illustration of the role of circle map dynamics and mixed-mode oscillations in the transition to chaos. In the peroxidase-oxidase reaction, the peroxidase enzyme from horseradish (which, as its name implies, normally utilizes hydrogen peroxide as the electron acceptor) catalyzes an aerobic oxidation... 

Whereas the fractal torus is difficult to distinguish from the wrinkled torus, the broken torus (stage 4) is immediately recognizable from its surface of section. The transition from wrinkled to fractal torus can, however, be clearly seen in the associated circle map. The circle map develops an inflection point (see Figure 34) at the transition from wrinkled to fractal torus. The existence of an inflection point means that the circle map is no longer invertible, that is, the circle map cannot be derived from a true two-dimensional torus. It also means that chaotic dynamics are now possible. The transition from stage 2 to stage 3 heralds the death of the two-dimensional torus and the transition to the possibility of chaotic dynamics. 

Plots of these rotation numbers as a function of the bifurcation parameter fe, take on a staircase form, although the staircases are much more complex than those seen in the sine circle map. Figure 36 shows a portion of die staircase... 

Staircase, Fractal Dimension, and Universality of Mode-Locking Structure in the Circle Map. 

Interaction of Resonances in Dissipative Systems. I. Circle Maps. 

Figure 1. Experimental conditions (closed circles) mapped onto the jxessure temperature phase diagram of Se from Ref. [7]. Opened symbols on the phase diagram correspond to the ano-maUes of electrical resistance, density, and heat observed in Ref. [7].

Nucleogenesis in the interior of massive stellar objects yields 100 natural elements of composition Zj A - Z) = 1. Because of radioactive decay at reduced pressure in intergalactic space, the stability ratio converges as a function of mass number to a value of t at yl = 267 = (A — Z ) t> = Z. As a result, only 81 stable elements survive in the solar system as a periodic array conditioned by r. The observed periodicity corresponds to a Ford-circle mapping of the fourth-order unimodular Farey sequence of rational fractions. 

Simulation by number theory is the only known procedure that generates the detailed structure of the periodic table without further assumptions or ad hoc corrections. In its simplest form, the simulation is based on the fact that any atomic nucleus consists of integral numbers of protons (Z) and neutrons N), such that the ratio Z/A is a rational fraction. This ratio converges from unity to the golden ratio (t) with increasing atomic number and yields a distribution commensurate with the periodic table. The detailed structure of the periodic function is contained in the Earey sequence 4 of rational fractions and visuaUzed in its Ford-circle mapping [5]. 

Ford-circle mapping of the periodic table [13], based on this idea, consists of a central circle flanked on both sides by three satellites, which together represent the... 

If the rotation number of the circle map is irrational and is not well approxi-mated by rational numbers then there exists a smooth transformation of variables which brings the map to a rotation with a constant angle ... 

Observe that if the iV-th iteration of the circle map is an identity, then all points on the circle are structurally unstable with a multiplier equal to one. Moreover, all Lyapunov values of each point are equal to zero. This is an infinitely degenerate case. We saw in Sec. 11.3 that to investigate the bifurcations of structurally unstable periodic orbits with A — 1 first zero Lyapunov values it is necessary to consider at least -parameter families. It is now clear that to study bifurcations in this case one has to introduce infinitely many pa rameters. Moreover, it is seen from the proof of Theorem 11.5 that such maps can be obtained by applying a small perturbation to an arbitrary circle map... 

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