State equations autonomous

Equation (3.5) is called the state equation since it describes the state of the system through the state variable y as a function of the independent variable t. It is assumed that F and g have continuous partial derivatives with respect to y and u. Note that this problem is autonomous in the sense that the independent variable does not appear explicitly in F or y. When it does, the problem is easily convertible to the autonomous form (see Exercise 3.4). 

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... 

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. 

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). 

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). 

Let us now turn to a brief discussion of the examination of systems (1.6) by catastrophe theory techniques. To begin with, in systems (1.6) such static catastrophes may occur for which the condition (1.9) is satisfied. It follows from equations (1.6), (1.9) that in the stationary state of an autonomous system the condition... 

That is, the expected multiplication rate depends explicitly only on the expected state, and not on time. In other words, Eq. (18) is the differential equation of an autonomous system. Most of the models that have been proposed have the property of being autonomous. 

Explicit introduction of time (t or to) in the function F creates a much more difficult problem than that expressed by Eqs. (145)-(147), since the equations are no longer autonomous. Hence, in what follows, we ignore the dependence of F on t assume that F has the same form in the unsteady state as in the steady state, and work with the approximate equations already given. This procedure should be adequate, provided that environmental conditions are not changing too fast that is, not changing much in the mean lifetime of a cell. Of course, if we had an adequate measure of cell structure, we should not have to resort to this kind of subterfuge. 

This is a system of nonlinear autonomous delay-differential equations. Linearization around the steady-state motion (the constant deflection of the tool, Xq and jo) gives... 

In mechanics, the state of rest is called the state of static equilibrium and Equation E2.6 is viewed as the law of equilibrium. In fact, it is a mounting rule (similar to the Kirchhoff rule in electricity for components in series) based on a law of conservation and not on an autonomous principle This equation results merely from hypotheses made on the system [indeformability (E2.3), isolation, and the absence of exchange between varieties] and from the application of the First Principle of Thermodynamics (Equation E2.2). In the Formal Graph theory, this law saying that a null sum of forces expresses a static equilibrium is considered as unsuitable because the notion of equilibrium is much more general than this mechanical conception. 

The trajectory of a general autonomous system of differential equations can wander anywhere in the state-space. What kind of restrictions are obtained if one considers the trajectories of a kinetic differential equation It was mentioned earlier (Subsection 4.1.2) that the solutions of a kinetic differential equation remain in the first orthant if they started there. More refined statements regarding positivity and nonnegativity have also been stated as Problem 6 of Subsection 4.1.4. Now let us try to delineate an as narrow as possible set in the state-space for the trajectories. As a next step towards this goal let us write the kinetic differential equation (4.6) in the form... 

These equations were solved numerically and in Fig. 15.5 we show the ratio of the efficiency as defined in (15.29) for an oscillatory influx of reactants to that efficiency for a steady (constant) influx vs. w, the frequency of the oscillatory influx, (15.25). The autonomous system, with a steady influx, is in a stable focus, that is the autonomous system on being perturbed briefly returns to the stable state with an oscillatory component. In order to emphasize the effects of an oscillatory influx, conditions were chosen for the steady influx such that only 10% of the heat input is converted to work. 

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