Existence of Periodic Solutions

First let us show that solutions of (1) are bounded (Murray, 1974a). 

Obviously ( q, 7]q, Pq) G B. What does the vector field (1) look like 

Thus solutions of (1) enter B through this face as well, except possibly at the point (l/l 2hq/(l+q),l). However, at this point 

Furthermore, any solution which starts in the positive octant ( 0,  

Similarly, if 6 1/q, then 0 and 6 must decrease until 6 l/q. If 0 p l, then pp 6 1 ud thus eventually we must have pp 0 because eventually 6 1 Thus eventually p must increase until p 1. Continuing this argument proves the assertion. 

Proceeding to the limit in (2.35) as o and taking condition (ii) into account, we 

It follows from Theorem 1.1 that the process of finding a periodic solution of the system (1.10) can be reduced to the computation of the functions x (f, Xq) provided that this solution exists and the point Xq through which it passes at r = 0, is known. 

If x (t, Xq) are known, the problem of the existence of periodic solutions can be solved as follows. 

the problem of the existence of periodic solutions for (1.1) is connected with the problem of the existence of zeros of the function T(Xq). The points jCo for which Tix ) = 0 are singular points of the mapping 

We can find the mapping (3.4) only approximately, e.g., by calculating the functions 

---

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. 

J. Serrin, A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal., 3 (1959) 120-122. 

There are several other classical results about the existence of periodic solutions for Lienard s equation and its relatives. See Stoker (1950), Minorsky (1962), Andronov et al, (1973), and Jordan and Smith (1987). 

Takoudis et al. (1981) proposed a model for a bimolecular Langmuir-Hinshelwood surface reaction with two empty sites in its reaction step. The two chemisorbed species were assumed to adsorb competitively on the surface. The two dimensional model with reaction rates as parameters were shown to exhibit oscillations. Bifurcation of this model was also discussed. Takoudis et al. (1982) described a procedure for obtaining necessary and sufficient conditions for the existence of periodic solutions in surface reactions with constant temperature. An analytic method for the analysis of bifurcation to periodic solutions was developed. 

Message, P.J. and Taylor, D.B. (1978) On asymmetric periodic solutions of the plane restricted problem of three bodies, and bifurcations of families. In Szebehely, V., editor, Dynamics of planets and satellites and theories of their motion (Proceedings of Symposium No. fl of the International Astronomical Union), pages 319-323. Message, P.J. (1980) On the existence of periodic solutions of Poincare s second sort in the general problem of three bodies moving in a plane. Celestial Mechanics, vol. 21, pages 55-61. 

Message, P.J. (1982c) On the existence of periodic solutions of Poincare s third sort in the general problem of three bodies in three dimensions. Celestial Mechanics, vol. 28, pages 107-118. 

As follows from the above discussion, the problem of existence of periodic solutions for the system (3.12) is solved by the following statement. 

Since the equation (7.13) turns into the equation (7.1) for T(xo) = 0, the investigation of the problem of existence of periodic solutions is reduced to the investigation of the problem of existence of zeros of the function T(Xq). This means that a single co-periodic solution of the equation (7.1) corresponds to each zero of the function T(xq) and that the number of periodic solutions of eqn.(7.1) is equal to the number of zeros of T(pcq). Since... 

Let us now investigate the problem of existence of periodic solutions for the system (11.1). If it is known that the periodic solution of the system (11.1) exists and passes through the point Xq which is also known, then, according to Theorem 1.18, the problem of determination of this solution is reduced to the calculation of the function x Hxq). Let us denote... 

Thus, the problem of existence of periodic solutions to the system of difference equations (11.1) is reduced to the problem of existence of zeros of the function S(xo). In the general case, it is impossible to rind the limiting function x ( q) of the sequence (11.7), and consequently, it is impossible to determine S(Xq). Therefore, by using the approximate solutions x xq), we calculate the function... 

Existence of Periodic Solutions for Systems of Differential Equations with Lag... 

Let us clarify under what conditions the existence of Bubnov-Galerkin s approximations jc (t) of an arbitrary order m mo involves the existence of periodic solutions to the system of differential equations (1.1). 

The existence of periodic solutions of the system of equations (3.11) is established by the following statement. 

Alternative principle for the existence of periodic solutions of differential... 

Hastings, S., Tyson, J. Webster, P. (1977). Existence of periodic solutions for negative feedback control systems, J. Diff. Eqs., 25, 39-64. 

The general definitions may be applied with slight modifications./or appropriate generalization of the definitions/ in the multidimensional case. Hamiltonian and CRE systems can only be defined for an even number of variables. Some other notions can also be Introduced, the most Important among these is that div f=0, a property equivalent to being Hamiltonian in the two-dimensional case. This property and the existence of /global, time independent/ first integrals are closely connected with the existence of periodic solutions /Toth, 1987/ Several conjectures will be formulated here. 

The eigenvalue analysis does not reveal much information regarding the behavior of the nonlinear system once instability occurs. The existence of periodic solutions (limit cycles), region of attraction of the stable trivial solution, and the effects of system parameters on these features as well as the size of the limit cycles (amplitude of steady-state vibrations) are important problems that cannot be solved using the linearized system s equations. 

---