Autonomous ordinary differential

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

Models of a diffusionless chemical reaction are described by systems of autonomous ordinary differential equations of first order ... 

Deterministic dynamic systems generated by autonomous ordinary differential equations. 

The heart of the approach can be conveyed through a simple example. Consider a single, autonomous ordinary differential equation,... 

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). 

ORDINARY DIFFERENTIAL EQUATIONS AND STABILITY THEORY An Introduction, David A. SAnchez. Brief, modem treatment. Linear equation, stability theory for autonomous and nonautonomous systems, etc. 164pp. 5H 8tt. 

In this section, we compile some results from nonlinear analysis that are used in the text. The implicit function theorem and Sard s theorem are stated. A brief overview of degree theory is given and applied to prove some results stated in Chapters 5 and 6. The section ends with an outline of the construction of a Poincare map for a periodic solution of an autonomous system of ordinary differential equations and the calculation of its Jacobian (Lemma 6.2 of Chapter 3 is proved). 

The system of equations (4.20) falls into a class of systems of ordinary differential equations it is an autonomous dynamical system, see equation (1.6) in Section 1.2. Furthermore, system (4.20) is linear — all unknowns are in the first power. As will be shown later, the systems of equations of this type may be readily solved. Linearity of the system of kinetic equations is a consequence of unimolecularity of elementary reactions in the mechanism (4.19). 

The new set now has N+l coupled ordinary differential equations (Eqs. 7.5 and 7.4). Thus, the standard form of Eq. 7.1 is recovered, and we are not constrained by the time appearing explicitly or implicitly. In this way, numerical algorithms are developed only to deal with autonomous systems. 

For each choice of T°, this is an autonomous second-order ordinary differential equation for X. as a function X of z ... 

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

The most frequently encountered numerical problem in nonlinear chemical dynamics is that of solving a set of ordinary, nonlinear, first-order, coupled, autonomous differential equations, such as those describing the BZ reaction. We hope you understand by now what nonlinear means, but let us comment on the other modifiers. The equations are ordinary because they do not contain partial derivatives (we consider partial differential equations in the next section), first order because the highest derivative is the first derivative, and coupled because the time derivative of one species depends on the concentrations of other species. In the absence of time-dependent external forcing, rate equations are autonomous, meaning that time does not appear explicitly on the right-hand side. 

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