Qualitative Methods and Stability of ODE Solutions

Victor Henner, Alexander Nepomnyashchy, Tatyana Belozerova and Mikhail Khenner
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Victor Henner: Perm State University, Department of Physics
Alexander Nepomnyashchy: Israel Institute of Technology, Department of Mathematics
Tatyana Belozerova: Perm State University, Department of Mathematics
Mikhail Khenner: Western Kentucky University, Department of Mathematics

Chapter Chapter 5 in Ordinary Differential Equations, 2023, pp 173-217 from Springer

Abstract: Abstract The phase plane method is a useful qualitative approach to study some important properties of solutions to systems of two autonomous ordinary differential equations. (Autonomous means that the right-hand sides of the system’s equations do not depend explicitly on the independent variable.) This method has all advantages (and disadvantages) over the numerical and analytical solution methods for systems that the phase line method presented in Sect. 2.7 has over the corresponding methods for a single first-order equation. A phase plane method can be also applied to a second-order autonomous equation, because such equation can be reduced to a system of two autonomous first-order equations. That is, with a designation x1(t) = y(t) and x2(t) = y′(t), a second-order autonomous equation

Date: 2023
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DOI: 10.1007/978-3-031-25130-6_5

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