Autonomous Nonlinear Equation

An autonomous equation is an equation in which the independent variable does not appear explicitly. 

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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. 

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. 

Autonomous (A) Versus Nonautonomous (NA) Problems. Practically all nonlinear problems of the theory of oscillations reduce to the differential equation of the form... 

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). 

Although these arguments have been presented for reaction systems whose rates are forced by an external oscillator, they remain true for autonomous biochemical oscillations where ot and are nonlinear functions of metabolite concentrations. That is, the rate of removal of a labeled compound through a reaction step whose rate is oscillating due to nonlinear kinetics will be enhanced over an equivalent system that maintains the same mean chemical flux and mean concentrations of metabolites but does not oscillate. This has been demonstrated numerically ( 6) on the reaction system (1) from the previous section using the full kinetic 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). 

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... 

For the sake of simplicity, let us now set up the case of a second-order bandpass filter and a comparator with saturation levels This closed-loop system verifies the required premises the system is autonomous, the nonlinearity is both separable and frequency-independent, and the linear transfer function contains enough low-pass filtering to neglect the higher harmonics at the comparator output. Choosing adequately the band-pass filter, it can be forced that the first-order characteristic equation for the closed-loop system of Fig. 4 has an oscillation solution being and the oscillation frequency and... 

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