Autonomous equation

Dividing this by the differential equation for 17, gives us the pair of autonomous equations... 

The section addresses the problem of specifying constraints (10), (16), (27), (32), and (40) on macroscopic kinetics as applied to various problems. Formalization of these constraints as well as constructions of MEIS are on the whole based on the Boltzmann assumption on the equilibrium of "kinetic" trajectories of motion toward point xeq and the possibility to describe them by autonomous equations of the form x - f(x). 

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). 

An autonomous equation is an equation in which the independent variable does not appear explicitly. 

The Tikhonov theorem has an important generalization, called the centre manifold theorem, which will be discussed in Sections 5.4.5-5.4.7. In classification of catastrophes occurring in dynamical systems and represented by systems of autonomous equations, the centre manifold theorem plays the role of the splitting lemma (see Section 2.3.4). 

It should be stressed that the Poincare-Bendixon theorem given above does not hold for systems of three or more autonomous equations hence, in such systems non-periodicity of a nonstationary trajectory remaining within a confined region is possible. 

Therefore, (6.66) is now an autonomous equation (Polyanin and Zaitsev 2003) with one of the roots of the particular solution taking the following form... 

This is a set of autonomous nonlinear differential equations. Note that the above set of substitutions converted the nonautonomous Eq. (ti) to a set of autonomous equations. 

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. 

Now,wecansubstitutethisexpressionfor Cb intermsof Ca into the differential equation describing the change in Ca, make it autonomous, and derive an expression for the time dependenceofCa ... 

Its temporal evolution is specified by an autonomous system of N, possibly coupled, ordinary first-order differential equations ... 

A) Definition of Stability According to Liapounov.—Given a system of differential equations of an autonomous system... 

Autonomous (A) Versus Nonautonomous (NA) Problems. Practically all nonlinear problems of the theory of oscillations reduce to the differential equation of the form... 

It is recalled that a differential equation is called autonomous if the independent variable t (time) does not enter explicitly otherwise it is called nonautonomous. 

Mathematically, these are trajectories connecting equilibrium points of a system of autonomous ordinary differential equations. 

Equation (13) appears to be a good approximation for describing isothermal chemiluminescence kinetics for homogeneous systems where oxidation takes place uniformly. However, as has been shown by several authors [53-58], the different sections of a polymer sample may oxidize with its autonomous kinetics determined by different rates of primary initiation. A chemiluminescence imaging technique revealed that the light emission may be spread from some sites of the polymer film and the isothermal chemiluminescence vs. time runs are then modified, particularly in the stage of an advanced oxidation reaction [59]. 

The new generalized Domar equation (8.16) can be derived by following the steps from (5.7) to (5.12), with (5.7) replaced at the outset by (8.13) - assuming zero autonomous capitalist consumption (B0 = 0). See Trigg (2002b) for the complete derivation. 

It is readily seen that the set of equations (76) consists of three equations of motion in the real variables ReIm c, w. If, (x) = constant, chaos in the system does not appear since the set (76) becomes a two-dimensional autonomous system. The maximal Lyapunov exponents for the systems (75) and (72)-(74) plotted versus the pulse duration T are presented in Fig. 36. We note that within the classical system (75) by fluently varying the length of the pulse T, we turn order into chaos and chaos into order. For 0 < T < 0.84 and 1.08 < 7) < 7.5, the maximal Lyapunov exponents Li are negative or equal to zero and, consequently, lead to limit cycles and quasiperiodic orbits. In the points where L] = 0, the system switches its periodicity. The situation changes dramatically if,... 

In the present section we are concerned with genuine internal noise. We consider a closed, isolated many-body system, whose evolution is given by a Schrodinger equation. Remember that in the classical case in III.2 we gave a macroscopic description in terms of a reduced set of macroscopic variables, which obey an autonomous set of differential equations. These equations are approximate and deviations appear in the form of fluctuations, which are a vestige of the large number of eliminated microscopic variables. Our task is to carry out this program in the framework of quantum mechanics. 

The idea of the phase plane3 is to let the time be a parametric variable along the curve (u(t), v(r)) in the u, v plane. The equations are often autonomous (i.e., the right-hand sides are not functions of t) but, when they are not, they can be made so by adding w(t) = t. Then the third differential equation is w = 1. If necessary, a nonvanishing function of F and G, for example K(u, v), can be divided into each equation. This distorts the time that must be recovered from a third equation. A particularly useful transformation of this sort is given by K2 = F2 + G2, for then the independent variable is the path length, s, and... 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

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