Differential equations autonomous

An autonomous diflferential equation does not carry the independent variable explicitly, e.g., 

On the other hand, the following is a non-autonomous differential equation  

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The capabilities of MEIS and the models of kinetics and nonequilibrium thermodynamics were compared based on the theoretical analysis and concrete examples. The main MEIS advantage was shown to consist in simplicity of initial assumptions on the equilibrium of modeled processes, their possible description by using the autonomous differential equations and the monotonicity of characteristic thermodynamic functions. Simplicity of the assumptions and universality of the applied principles of equilibrium and extremality lead to the lack of need in special formalized descriptions that automatically satisfy the Gibbs phase rule, the Prigogine theorem, the Curie principle, and some other factors comparative simplicity of the applied mathematical apparatus (differential equations are replaced by algebraic and transcendent ones) and easiness of initial information preparation possibility of sufficiently complete consideration of specific features of the modeled phenomena. 

Tl] H. R. Thieme (1992), Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, Journal of Mathematical Biology 30 755-63. 

T3] H. R. Thieme (1994), Asymptotically autonomous differential equations in the plane, Rocky Mountain Journal of Mathematics 24 351-80. 

Any non-autonomous differential equation can be transformed into a set of autonomous differential equations by introducing additional state variables. For example, with the introduction of the new state variable... 

DCD models have achieved a particular significance in the last decade in connection with chaotic phenomena. There are at least two distinct methods of relating DCD models to CCD models. The easier, but less rigorous, way is by the discretisation of time. An autonomous differential equation... 

The identification of relations between statics and dynamics became a constituting part in the explanation of unity of the laws of mechanics in (Lagrange, 1788). Deriving the equations of trajectories from the equation of state (1) turned out to be possible owing to the assumptions made about observance of the relativity principle of Galileo and the third law of Newton and, hence, about representability of any trajectory in the form of a continuous sequence of equilibrium states. From the representability, in turn, follow the most important properties of the Lagrange motion curves existence of the functions of states (independent of attainability path) at each point possibility to describe the curves by autonomous differential equations that have the form x = f x) dependence of the optimal configuration of any part of the curve upon its initial point only. These properties correspond to the extreme principles of the optimal control theory. 

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. 

A non autonomous differential equation x = f t,x) can be written in autonomous form where the right hand side of the differential equation is not explicitly depending on time, by augmenting the system by the trivial equation t = 1 ... 

We will consider only methods fulfilling this condition and assuming for the rest of this chapter autonomous differential equations for ease of notation. 

Experiments in open stirred reactors have become common place these days for the study of time-dependent phenomena. By dint of the usual hypothesis of ideality, i.e. that the medium is instantaneously of homogeneous composition ( ), the concentrations are space-independent functions, whence a considerable simplification of the equations describing the behaviour. The time variations of the concentrations within an ideal reactor (as in the scheme of fig. 1), can thus be expressed simply with the set of autonomous differential equations (8) ... 

In deterministic framework time homogenity is associated to autonomous differential equations ... 

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 ... 

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. 

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. 

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

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). 

Choice of the mathematical apparatus of macroscopic equilibrium descriptions. Problems in modeling the nonholonomic, nonscleronomous, and nonconservative systems. Possibilities for using differential equations (autonomous and nonautonomous) and MP. 

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. 

Let 7r(Ar, t) denote the dynamical system generated by the autonomous system of differential equations... 

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). 

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. 

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