Asymptotically autonomous system

In many of the arguments involving chemostats it was shown that the omega limit set had to lie in a restricted set, and the equations were analyzed on that set one simply could choose initial conditions in the restricted set at time zero. The equation defining the restricted set - in effect, a conservation principle - allowed one variable to be eliminated from the system. We want to abstract this idea and make it rigorous. The omega limit set lies in a lower-dimensional set, and the trajectories in that set satisfy a smaller system of differential equations. However, it is not clear (and, indeed, not true [T3]) that the asymptotic behavior of the two systems is necessarily the same. (A very nice paper of Thieme [Tl] gives examples and helpful theorems for asymptotically autonomous systems. A classical result in this direction is a paper of Markus [M].) In this appendix, a theorem is presented which justifies the procedure on the basis of stability. 

Arcsine distribution, 105, 111 Assumption of molecular chaos, 17 Asymptotic theory, 384 of relaxation oscillations, 388 Asynchronous excitation, 373 Asynchronous quenching, 373 Autocorrelation function, 146,174 Autocovariance function, 174 Autonomous problems, 340 nonresonance oscillations, 350 resonance oscillations, 350 Autonomous systems, 356 problems of, 323 Autoperiodic oscillation, 372 Averages, 100... 

Proof. The quantity under the integral sign in the definition of A in (5.2) is the trace of the Jacobian matrix for the system (5.1) evaluated along the periodic orbit. Theorem 4.2 then applies. A periodic orbit for an autonomous system has one Floquet multiplier equal to 1. Since there are only two multipliers and one of them is 1, is the remaining one. The periodic orbit is asymptotically orbitally stable because, in view of Lemma 5.1, A<0. ... 

A major difference between competitive and cooperative systems is that cycles may occur as attractors in competitive systems. However, three-dimensional systems behave like two-dimensional general autonomous equations in that the possible omega limit sets are similarly restricted. Two important results are given next. These allow the Poincare-Bendix-son conclusions to be used in determining asymptotic behavior of three-dimensional competitive systems in the same manner used previously for two-dimensional autonomous systems. The following theorem of Hirsch is our Theorem C.7 (see Appendix C, where it is stated for cooperative systems). 

M] L. Markus (1953), Asymptotically autonomous differential systems, in Contributions to the Theory of Nonlinear Oscillation, vol. 3. Princeton, NJ Princeton University Press, pp. 17-29. 

Asymptotic integration of quasi-linear autonomous systems with lag. Uktrain. Mat. Zhum., 18(3), (1966), 117-119. 

The asymptotic behavior of a two-dimensional autonomous dynamical system is, in general, known to have fixed points or lines where f = s = 0, or to have limit cycles where f, i 0. Since the system is irreversible as the concentrations of chiral products always increase at the cost of the substrate A ... 

The main result which we establish here is that the evolution of the x-variable in this system at /i > 0 is well described by an autonomous equation (i.e. independent on the angular variable p). An immediate advantage of this is that such equations are easily integrated (since x is one-dimensional) which allows for obtaining long-time asymptotics for the local dynamics near a saddle-node. 

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