Nonautonomous system

We note in passing that this simplification would not be possible in the case of a nonautonomous system... 

Figure 7. Computed curves of dimensionless heat generation fg (0) and heat removal r ( ) functions showing stable behavior of dimensionless temperature for nonautonomous systems (3)...

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. 

Note that Figure 12.5.5 is not a true phase portrait, because the system is nonau-tonomous. As we mentioned in Section 1.2, the state of the system is given by (x,y,z), not (x,y) alone, since all three variables are needed to compute the system s subsequent evolution. Figure 12.5.5 should be regarded as a two-dimensional projection of a three-dimensional trajectory. The tangled appearance of the projection is typical for nonautonomous systems. 

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 ... 

For nonautonomous systems, an additional term involving the time derivatives of W(p, q s) must be included in Eq. (A.66) [45, 46, 53]. In this Appendix, we have described how Lie transforms provide us with an important breakthrough in the CPT free from any cumbersome mixed-variable generating function as one encounters in the traditional Poincare-Von Zeipel approach. After the breakthrough in CPT by the introduction of the Lie transforms, a few modifications have been established in the late 1970s by Dewar [56] and Drag and Finn [47]. Dewar established the general formulation of Lie canonical perturbation theories for systems in which the transformation is not... 

Nonautonomous systems can always be included in the formalism of the autonomous system by defining time to be a further state variable and including it as an element of the state vector. 

Since a batch process is a nonautonomous system, with a solution motion, which evolves over a finite-time period, the standard definitions of asymptotic stability of critical points, which are appropriate for continuous processes, cannot be applied. The same is true for the definitions of nonlinear detectability [15, 17, 27, 30] and stabilizability. Moreover, while in a continuous process those definitions apply to a critical point, in a semibatch process the definitions apply to one particular motion or operation policy [20], depending on the kind of load as well as on the material dosage and heat exchange policies. The batch motion deviations, which are caused by initial state and exogenous input disturbances, exhibit accumulative or irreversible deviations. If the deviations are acceptably small, the motion is... 

It is noted that the system is now autonomous, although the initial system is nonautonomous. 

Thus ( ) = c 2 is a classical intensity. The system (75) is nonautonomous if the function 3F is explicitly time-dependent. The autonomized version of Eq.(75)... 

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). 

This phenomenon of increased conversion, yield and productivity through deliberate unsteady-state operation of a fermentor has been known for some time. Deliberate unsteady-state operation is associated with nonautonomous or externally forced systems. The unsteady-state operation of the system (periodic operation) is an intrinsic characteristic of this system in certain regions of the parameters. Moreover, this system shows not only periodic attractors but also chaotic attractors. This static and dynamic bifurcation and chaotic behavior is due to the nonlinear coupling of the system which causes all of these phenomena. And this in turn gives us the ability to achieve higher conversion, yield and productivity rates. 

Phase plane plots are not the best means of investigating these complex dynamics, for in such cases (which are at least 3-dimensional) the three (or more) dimensional phase planes can be quite complex as shown in Figures 16 and 17 (A-2) for two of the best known attractors, the Lorenz strange attractor [80] and the Rossler strange attractor [82, 83]. Instead stroboscopic maps for forced systems (nonautonomous) and Poincare maps for autonomous systems are better suited for investigating these types of complex dynamic behavior. 

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. 

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. 

Vol. 1907 M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems (2007)... 

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

Claims similar in the form are also valid for nonautonomous Hamiltonian systems. For details see, for instance, [61]. 

In order to apply the GMM to the dynamical system in (3.26), canonical transformations are required first to simplify terms in Hq, second to simplify terms in He, and third to suspend nonautonomous terms in The GHA will be applied to two of the following three canonical transformations, because the rotation of axes transformation is well known so that the GHA will not be applied for that transformation even though it is still applicable. 

The extension of the Herglotz algorithm to nonautonomous dynamical systems (GHA) significantly reduces the effort required to suspend the nonautonomous Hamiltonian component in (3.33d). 

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. 

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