Periodic solutions uniqueness

FIGURE 29 The development of the stroboscopic phase-plane. Segments (a) and (d) show the trajectory settling down to a limit-cycle through a sequence of points at times that are multiples of t. This is drawn out in the time dimension in (b) and shown in its regularity in (c). If there is a unique periodic solution, the stroboscopic plane will show a sequence of states converging on (e). 

For a chemical reaction system, the characteristics of the periodic solutions are uniquely determined by the kinetic constants as well as by the concentrations of the reactants and final products. Starting from the neighborhood of steady state as an initial condition, the system asymptotically attains a closed orbit or limit cycle. Therefore, for long times, the concentrations sustain periodic undamped oscillations. The characteristics of these oscillations are independent of the initial conditions, and the system always approaches the same asymptotic trajectory. Generally, the further a system is in the unstable region, the faster it approaches the limit cycle. 

The principal open questions that arise in the treatment of the chemo-stat with periodic washout rate, discussed in Chapter 7, are analogous to those mentioned in connection with Chapter 6. Namely, can sufficient conditions be given for the uniqueness of the positive periodic solution (fixed point of the Poincare map) that represents the coexistence of the two populations What can be said of the case of more than two competitors ... 

We conclude that any solution w t, s) coincides with the unique periodic solution of (3.67), as soon as so t) > s, independently of the initial condition. This is a very strong stability statement of finite time convergence, uniformly on bounded subsets s. For further details see [34]. 

Therefore, 9(0 is the periodic solution of the equation (7.13) (along with x Jit, xq)). The relation (7.30) follows thus firom the uniqueness of this solution. 

Section 12 Bilateral Approximation to Periodic Solutions of Systems with Lag 65 this solution is unique and, consequendy, that the following equality holds... 

By virtue of the principle of contracting mappings the equation (3.12) has a unique periodic solution z-zjf) satisfying the inequality... 

Then the system (1.1) possesses the periodic solution x = x (t) which is unique in a certain vicinity of x (i). This solution belongs to the region D and satisfies the inequality... 

Finally, the uniqueness of this periodic solution in a 5-neighborhood of x (t) follows from the existence of Green s function G (t, x). 

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 process called the limit cycle occurs. This behavior is independent of the initial conditions, and the system approaches in time the same periodic solution determined by the nonlinear differential equations. The periodic solution is characterized by its period and amplitude. The limit cycle is unique and stable with respect to small fluctuations. 

I have only been able to establish uniqueness and stability of the periodic solution for Eq. (27) for the case where n is infinite. For this case, the... 

Since 5>0, d[fip),f(q)]contraction mapping (see Appendix) and there must be a unique stable limit cycle in the four regions. In Fig. 14, we give this construction for the parameters used to compute Fig. 12. Although there is good agreement between the dynamics in the piecewise linear and the continuous equations, no proof of stable limit cycle oscillations has been found for Eq. (48) or (50). However, there has been a recent proof for the existence of nonlocal periodic solutions of Eq. (45) using fixed-point methods. ... 

Traditional reaction kinetics has dealt with the large class of chemical reactions that are characterised by having a unique and stable stationary point (i.e. all reactions tend to the equilibrium ). The complementary class of reactions is characterised either by the existence of more than one stationary point, or by an unstable stationary point (which could possibly bifurcate to periodic solutions). Other extraordinarities such as chaotic solutions are also contained in the second class. The term exotic kinetics refers to different types of qualitative behaviour (in terms of deterministic models) to sustained oscillation, multistationarity and chaotic effects. Other irregular effects, e.g. hyperchaos (Rossler, 1979) can be expected in higher dimensions. 

The simple enzyme kinetic system does not exhibit oscillatory behaviour. The existence, uniqueness and global asymptotic stability of a periodic solution of a Michaelis-Menten mechanism was proved (Dai, 1979) for the case when the reaction occurred in a volume bounded by a membrane. The permeability of the membrane to a given species was specified as the function of another species. The form of the model is... 

Dai, L. S. (1979). On the existence, uniqueness and global asymptotic stability of the periodic solution of the modified Michaelis-Menten mechanism. J. Diff. Eqs., 31, 392-417. 

Hastings, S. P. (1977). On the uniqueness and global asymptotic stability of periodic solutions for a third order system. Rocky Mountain J., 7, 513-... 

Equation for the Period. The solution of eq. (4.2) is not unique, because an arbitrary phase shift of the periodic solution is possible. So the additional equation, which is required for the computation of the period, has to be an anchor equation that fixes the phase. Depending on the considered system, different anchor conditions may be appropriate ... 

Condition V from Section B guarantees the existence of a unique 277-periodic solution A (0). 

Interdiffusion of bilayered thin films also can be measured with XRD. The diffraction pattern initially consists of two peaks from the pure layers and after annealing, the diffracted intensity between these peaks grows because of interdiffusion of the layers. An analysis of this intensity yields the concentration profile, which enables a calculation of diffusion coefficients, and diffusion coefficients cm /s are readily measured. With the use of multilayered specimens, extremely small diffusion coefficients (-10 cm /s) can be measured with XRD. Alternative methods of measuring concentration profiles and diffusion coefficients include depth profiling (which suffers from artifacts), RBS (which can not resolve adjacent elements in the periodic table), and radiotracer methods (which are difficult). For XRD (except for multilayered specimens), there must be a unique relationship between composition and the d-spacings in the initial films and any solid solutions or compounds that form this permits calculation of the compo-... 

A unique determination of a solution u ip,t ) necessitates imposing the condition of periodicity... 

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