Periodic solutions existence

Our goal then is to show that positive periodic solutions exist under suitable hypotheses. It turns out that it is simpler to state our results in terms of the discrete dynamical system generated by the Poincare map, P Q - Q, rather than in terms of solutions of (3.2) (3.3) is assumed to hold throughout this section. 

The first step is to show that a periodic solution exists. The argument uses a clever idea introduced by Poincare long ago. Watch carefully—this idea will come up frequently in our later work. 

A new periodic orbit with period T + ST, in the vicinity of the above T-periodic solution exists if the Jacobian determinant of the left-hand-side of Equation (58), computed at the initial conditions Xio, is different... 

In the first place, the difference between the (NA) systems and the (A) ones is that for the first there exists always a periodic solution with period 2w (or a rational fraction of 2ir), whereas for the second, the period of oscillation (if it exists) is determined by the parameters... 

If all xx,x2,- have been calculated and are periodic, F is then a known periodic function and if k is not an integer, there exists only one periodic solution of the form... 

I) The existence of a stable singular point of (6-126) is the criterion for the existence of a stable periodic solution motion) of the original system (6-112). 

In this simplified version of the Brusselator model, the trimolecular autocatalytic step, which is a necessary condition for the existence of instabilities, is, of course, retained. However, the linear source-sink reaction steps A—>X—>E are suppressed. A continuous flow of X inside the system may still be ensured through the values maintained at the boundaries. The price of this simplification is that (36) can never lead to a homogeneous time-periodic solution. The homogeneous steady states are... 

The same equations, albeit with damping and coherent external driving field, were studied by Drummond et al. [104] as a particular case of sub/second-harmonic generation. They proved that below a critical pump intensity, the system can reach a stable state (field of constant amplitude). However, beyond the critical intensity, the steady state is unstable. They predicted the existence of various instabilities as well as both first- and second-order phase transition-like behavior. For certain sets of parameters they found an amplitude self-modula-tion of the second harmonic and of the fundamental field in the cavity as well as new bifurcation solutions. Mandel and Erneux [105] constructed explicitly and analytically new time-periodic solutions and proved their stability in the vicinity of the transition points. 

We must also examine the stability of the periodic solution and its limit cycle as it emerges from the bifurcation point. Just as stationary states may be stable or unstable, so may oscillatory solutions. If they are stable they may be observable in practice if they are unstable they will not be directly observable although their existence still has some physical relevance. We will give the recipe for evaluating the stability and character of a Hopf bifurcation in the... 

A numerical solution of the basic equations demonstrated their ability to reproduce concentration oscillations. At the same time, for the systems possessing three and more intermediate products the standard method to prove existence of periodical solutions, using a phase portrait of a system (Section 2.1.1) fails. An additional reduction in a number of differential equations, e.g., using an idea that one of concentrations, say, [BrOj-], serves as a rapid variable and thus the relevant kinetic equation (8.1.5) could be solved as the stationary [10], cannot be always justified due to uncertainty in the kinetic coefficients hi. 

In order for an undamped periodic solution to exist it is necessary that the constant term in this equation and the coefficient of tf> be equal to zero, so that, to the first approximation,... 

This approach needs modification as soon as multiple attracting periodic trajectories exist for a particular set of operating parameters. A conceptually different modification will be necessary to account for attractors which are not simply periodic. Quasi-periodic solutions, characterized by multiple frequencies, are the first type one should expect these are by no means exotic but occur generally in several periodically forced systems. Deterministic chaotic situations, arising from the system nonlinearities (and not the stochastic responses due to random noise) need not be discarded as intractable (Wolf et al., 1986 Shaw, 1981). 

The conditions for the existence of a pT-periodic solution to the forced system can be written in terms of a fixed point equation for the pth iterate of the stroboscopic map... 

Then, for y greater than -y , we find more grey areas, where it seems that the dynamics is chaotic, but there is also a succession of windows where stable periodic solutions are the only ones to be found. The last of these is from y = 3.828 to y = 3.857 where period-three solutons exist. Even the structure of this is beautifully self-similar. And then there s the Charkovsky sequence.. .. But do just let me mention the sensitivity question. 

The most important dynamic bifurcation is Hopf bifurcation. This occurs when Ai and A2 cross the imaginary axis into the right half-plane of C as the bifurcation parameter g changes. At the crossing point both roots are purely imaginary with det(A) > 0 and tr(A) = 0, making Ay2 = i y/det(A). At this value of g, periodic solutions (stable limit cycles) start to exist as depicted in Figures 10 and 11 (A-2). 

Besides the existence of the periodic solutions yq(x) of (53), their stability is also of interest. For this purpose we use as ansatz a superposition of the stationary periodic solution yq(x) and a time-dependent perturbation y/i(x,t)... 

As was shown before, if phase separation is forced by a stationary and spatially periodic temperature modulation, the coarsening dynamics is interrupted above some critical value of the forcing amplitude a and it is locked to the periodicity of the external forcing. However, if this forcing is pulled by a velocity v 0, the traveling periodic solutions of (61) exist only in a certain range of v depending on a. 

Clearly, the maximum velocity vex, at which the periodic solution y/s still exists, corresponds to the solution which is shifted by n/2 with respect to the forcing, i.e., for y/s sin(gx). 

Fig. 21 Above the solid line the spatially periodic solution is unstable. The dashed line marks the existence boundary above which the spatially periodic solution does not exist due to the criterion given by (64). The dot-dashed curve marks the existence boundary obtained numerically from (63). The parameters are e = 1 and q = 0.5...

We have found that in the 2D case, similar to ID, there exists a critical driving amplitude ac above which the spinodal decomposition ends up in the stationary periodic solution a striped structure with the period of the forcing. The critical amplitude turned out to be about three to five times larger than in the ID case. In particular, for q = 6n/Lx with Lx = 256 one has in 2D ac = 0.014 whereas for ID ac = 0.0045. Thus, for 2D the upper curve in Fig. 20 moves slightly upward (the linear stability curve as remains unchanged). 

Tn the cationic polymerization and copolymerization of trioxane in the - melt or in solution, an induction period usually exists, during which no solid polymer is formed and the reaction medium remains clear. Nevertheless, reactions are known to occur during this period. By using BF3 or an ether ate as catalyst, in homopolymerization, Kern and Jaacks (I) reported the formation of formaldehyde via depolymerization of polyoxymethylene cations. 

The theory of nonlinear oscillations can describe the periodic solution that appears beyond the instability of the steady state. Stable states exist before the instability. The perturbations correspond to complex values of the normal mode frequencies and spiral toward the steady state to a focus. As soon as the steady state becomes unstable, a stable periodic... 

Systematic studies of the influence of border pressure on the kinetics of foam column destruction and foam lifetime have been performed in [18,24,41,64-71], Foams were produced from solution of various surfactants, including proteins, to which electrolytes were added (NaCI and KC1). The latter provide the formation of foams with different types of foam films (thin, common black and Newton black). The apparatus and measuring cells used are given in Fig. 1.4. The rates of foam column destruction as a function of pressure drop are plotted in Fig. 6.11 [68]. Increased pressure drop accelerates the rate of foam destruction and considerably shortens its lifetime. Furthermore, the increase in Ap boosts the tendency to avalanche-like destruction of the foam column as a whole and the process itself begins at higher values of foam dispersity. This means that at high pressure drops the foam lifetime is determined mainly by its induction period of existence, i.e. the time interval before the onset of its avalanche-like destruction. This time proves to be an appropriate and precise characteristic of foam column destruction. 

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