Nonautonomous equation

You might worry that (2) is not general enough because it doesn t include any explicit time dependence. How do we deal with time-dependent or nonautonomous equations like the forced harmonic oscillator mx + bx + kx = F cos t In this case too there s an easy trick that allows us to rewrite the system in the form (2). We let x, = x and Xj = X as before but now we introduce x, = t. Then x, = 1 and so the equivalent system is... 

We do not allow f to depend explicitly on time. Time-dependent or nonautonomous equations of the form x = f(x.t) are more complicated, because one needs two pieces of information, x and r, to predict the future state of the system. Thus x = f x,t should really be regarded as a two-dimensional or second-order system, and will therefore be discussed later in the book. 

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. 

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. 

So far, all of our examples of strange attractors have come from autonomous systems, in which the governing equations have no explicit time-dependence. As soon as we consider forced oscillators and other nonautonomous systems, strange attractors start turning up everywhere. That is why we have ignored driven systems until now—we simply didn t have the tools to deal with them. 

It is important to recognize that the criteria of instability for Gk> 0 and stability for Gk - Ois strictly valid only for Gk independent of /. Indeed, if Gk depends on t, the equation (4 312) is nonautonomous, and it is well known that stability can generally be established only by exact integration of the equation. If Gk is slowly varying, a local analysis based on an instantaneous value of Gk will be qualitatively correct for a finite time interval, but for more general time-dependent forms for Gk, we should not be surprised if the situation turns out to be more complex. To see an example of this, we can consider the special case in which the bubble volume changes periodically with time about some mean value, as may occur in response to an oscillatory variation in p at a frequency different from the resonant frequency co0 (given just prior to Eq. (4-229)). Thus, we let... 

The Lie transforms for which the Hamiltonian W explicitly depends on time preserve the formal properties of the autonomous fT(p, q). Hamilton s equations of motion do not change form for nonautonomous systems ... 

As is well known, any nth order homogeneous system of nonautonomous linear ordinary differential equations... 

The deterministic model is the (nonautonomous, nonpolynomial) induced kinetic differential equation of the reactions in Fig. 7.13. This model was described in detail by Herodek et al. (1982). Now we give a formal description of a small part of the model. As an example let us consider the time evolution of summer phytoplankton. Our assumption is that it takes part in the elementary reactions No. 2, 6, 13, 26, 34, therefore the equation for is ... 

Our goal is to investigate contrast structures for more general equations than (5.1), nonautonomous, in particular. 

We consider dynamical systems governed by nonautonomous differential equations in... 

In theory, if a laboratory experiment is repeated say one hour later than the first execution, then the same concentration-time curves should be obtained (ignoring experimental error for now). Accordingly, the time in the kinetic system of differential equations is not the wall-clock time, but a relative time from the beginning of the experiment. Such a differential equation system is called an autonomous system of ODEs. In other cases, such as in atmospheric chemical or biological circadian rhythm models, the actual physical time is important, because a part of the parameters (the rate coefficients belonging to the photochemical reactions) depend on the strength of sunshine, which is a function of the absolute time. In this case, the kinetic system of ODEs is nonautonomous. 

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. 

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