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Topological Analysis

In: Nonlinear Physics with Mathematica for Scientists and Engineers

Author

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  • Richard H. Enns

    (Simon Fraser University, Department of Physics)

  • George C. McGuire

    (University College of the Fraser Valley, Department of Physics)

Abstract

Most nonlinear systems cannot be solved exactly so we must resort to a variety of approaches in order to obtain an approximate solution. Where applicable, the phase plane portrait can serve as a valuable tool for qualitatively determining the types of possible solutions before resorting to numerical or (usually approximate) analytical methods for specific initial conditions. In this chapter, the concept of phase plane analysis will be examined in some depth, not for a specific problem, but for a wide class of physical problems described by the following system of first order equations: 4.1 $$\begin{array}{*{20}{c}} {\frac{{dx}}{{dt}} = P(x,y),} & {\frac{{dy}}{{dt}} = Q(x,y),} \\ \end{array}$$ where P, Q are, in general, nonlinear functions of x and y and the independent variable has been taken here to be time t. In the laser competition equations (2.32), t would, of course, be replaced with z, the spatial coordinate The mathematician would refer to this set of equations as being autonomous, meaning that P and Q do not explicitly depend on t. Why it is desirable to restrict the discussion for the moment to autonomous equations will become readily apparent.1

Suggested Citation

  • Richard H. Enns & George C. McGuire, 2004. "Topological Analysis," Springer Books, in: Nonlinear Physics with Mathematica for Scientists and Engineers, chapter 0, pages 125-166, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4612-0211-0_4
    DOI: 10.1007/978-1-4612-0211-0_4
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