The Qualitative Theory and the Phase Plane

Uri Elias
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Uri Elias: Technion – Israel Institute of Technology, Department of Mathematics

Chapter Chapter 6 in Fundamentals of Ordinary Differential Equations, 2025, pp 199-231 from Springer

Abstract: 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.

Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-86532-9_6

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DOI: 10.1007/978-3-031-86532-9_6

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