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The Qualitative Theory and the Phase Plane

In: Fundamentals of Ordinary Differential Equations

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  • Uri Elias

    (Technion – Israel Institute of Technology, Department of Mathematics)

Abstract

In this chapter, we draw qualitative conclusions about the solutions of autonomous second-order system of differential equations by utilizing geometric considerations, without solving the equations explicitly. Even for linear systems with constant coefficients, it is useful to look at a geometric picture rather than at an explicit but complicated formula. We define trajectories, phase plane, and critical points and make a clear distinction between solutions and their trajectories. Second-order linear, homogeneous systems with constant coefficients are classified according to their eigenvalues, and their phase planes are drawn. Stability of the critical point is determined by the eigenvalues. We make some informal notes about the phase plane of nonlinear autonomous systems and linearization near critical points. The phase plane of the nonlinear pendulum equation is fully analyzed.

Suggested Citation

  • Uri Elias, 2025. "The Qualitative Theory and the Phase Plane," Springer Books, in: Fundamentals of Ordinary Differential Equations, chapter 0, pages 199-231, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-86532-9_6
    DOI: 10.1007/978-3-031-86532-9_6
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