Successive near-identity transformations of the original equations yield new equations of simple structure that describe the long time evolution of the motion. By performing sequential near-identity transformations of the original equations, one obtains new equations with a simple structure that describe the long time evolution of the motion. Eliminating secular terms in the transformations determines the shape of the Hamiltonian in the new variables.
The details of Von Zeipel's procedure depend on the Hamiltonian structure of the equations it deals with. In Chapter 3, we study the application of the close identity transformation method to time-dependent problems. Using a two-time method, we derive a nonlinear equation that defines how the spatial variations in the phase of the oscillations evolve in time.
When £ f 0, aR, a~ and ~ are no longer constants of the motion, but vary slowly in time. From this plot we estimate the amplitude and period of the oscillations in the angle.
CHAPTER 2
We need solutions for the Sk and Tk that become bounded as phases e1, e2 become infinite, so that the transformation (r, v). These choices for the Fk and Gk can be integrated to find bounded values for the Sk and Tk. Note that Ro is greater than 2, which is the amplitude of an unperturbed Van der Pol oscillator.
The equation for the phase difference 'I' is decoupled from the amplitude R and can be solved separately. These regions of rapid variations in 'I'(T) correspond to boundary layers connecting the constant values 'I' = 2!. To find the behavior of S in the boundary layer regions, we substitute the approximate form of 'I' in (28) into (33).
We see that long intervals in which R(T) is almost constant are punctuated by short intervals where R(T) undergoes an oscillation.
CHAPTER 3
We allow S and T to depend explicitly on time t, because the equations of motion are explicitly time dependent. Applying the transformation (5) to the original equations (2) and comparing the result with the assumed form (6) of the new equations, we find. We see that S and T will have secular terms proportional to t and e when there are non-zero components to the right of (7) and (8) that are constant along the lines we - t = constant.
With these choices of F and G, equations (7) and (8) can be integrated to find bounded values for S and T. We see that the effect of the resonance is to change the value of .
CHAPTER 4
The two output solutions move away from the saddle with a slope of 1, and the two input solutions enter with a slope of - 1. We now consider the implications of these results for fluctuations in the original variables xk, yk' k = 1,2. Segment AB represents the initial formation of limit cycles, and segment BC a slow departure from the initial limit cycles.
NONLINEAR CHEMICAL OSCILLATIONS IN DISCRETE AND CONTINUOUS SYSTEMS
CHAPTER 5 CHEMICAL WAVES
- Introduction
- The New Variables
Svobodova [3], we begin our study with the investigation of a reaction diffusion process governed by the equations. Before the derivation, we will need specific preliminary results implied by the assumption that (1.2) has a stable, T periodic limit cycle. One of the specific tasks is to account for the experimental observations of Marek and Svobodova [3] about the Belusov-Zhabotinskii reaction.
CHAPTER 6
COUPLED CHEMICAL OSCILLATORS
The Perturbation Theory
It is convenient to change from variables xi, yi(i) variables Ai, 0i(i = 1,2) by means of transformation.
CJAO
Coupled Chemical Oscillators
The connection of the reactors takes place through an exchange of materials through a perforated wall separating the reactors. If the parameters for both reactors are almost identical, so their autonomous frequencies are almost the same, then the phase difference between the oscillations of each reactor tends to a constant value over time. If the parameters of the reactors are changed so that their difference.
Long time intervals of slow variations in phase difference are punctuated by short intervals of fast fluctuations. We account for these observations by studying the time evolution of the phase shift !(•) which is driven by. 10 in which the derivative of the right side is negative are stable and the values of I.
The root ~O with negative slope is stable, and the system will evolve to stable oscillations with constant phase shift .!a·. cation takes place; namely the change from phase locking to rhythm splitting. These regions of rapid variation in Y(T) correspond to boundary layers connecting the constant values I= i + nT.
CHAPTER 7
- A Model of Diffusive Coupling
Assume that the coupling function K(~it) goes to zero when the spatial displacement ~it between the oscillators becomes infinite. A complete solution of the discrete system (1.1), even if it were possible, would contain a lot of irrelevant information. We are not looking for the detailed trajectory of each (xi, yi), but only the overall phase distribution of the IJI oscillator.
In Section 2, we use a continuum approximation to derive an integral-differential equation that describes the time evolution of the phase distribution. In section 4 we study synchronization in systems where K(p - q) has the same constant value for all p, q in the volume containing the oscillators. In the limit where therP is many oscillators per unit volume, we can derive an equation for the distribution of phases 'I' i.
From the form of R we see that (2.7) is a non-linear integral-differential equation for the distribution of phases n('¥, q, T). Since we do not expect the process of synchronization to depend strongly on the detailed form of the limit cycle given in (1.3), we assume a circular limit cycle, with In this last case we note that - n and + ~ are physically indistinguishable, both corresponding to the same point in the cycle of an oscillation.
We study the synchronization process in a system where a certain part of the oscillators are initially synchronized, while the rest have a random phase distribution. In the initial condition (3.16) we take p >> u so that initially the random component dominates the synchronized component. We witness the evolution of the system from an initial state with a nearly uniform phase distribution to a final state where all oscillators are synchronized, with f = 0.
Assume that the oscillators are uniformly distributed in space, and that an oscillator at spatial position q has xi= x(q, t) and y = y(q, t). Oscillators at q = 0 can be considered as stimulating stimuli that initiate a synchronization process in the rest of the medium. To determine the state of B at q = 0, we recall that the phase distribution peaks sharply around ~ 0 as B becomes large and negative.
We see that the even iterates converge to a lower bound function YL(s) and that the odd iterates converge to an upper bound function YR(s). The monotonically increasing nature of f(u) implies that the integrand of (7.8) is non-negative in the region of integration, which has 0 < u < s < v < oo.
TUBE REACTOR CSTR
TRANSITION REGION
SPATIALLY INDEPENDENT OSCILLATIONS IN TIME
