Abstract
This chapter is devoted to an overview of dynamical systems that play a fundamental role in building mathematical models of reality. After a brief introduction of modeling, we present some theorems of existence and uniqueness as well as the definitions of first integral and phase portrait. Then, we define Liapunov’s stability for autonomous systems together with some theorems of stability and instability of equilibrium. Poincare’s perturbation method is described with some applications. Finally, Weierstass’s qualitative analysis is presented.
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Notes
- 1.
The topics contained in this chapter can also be found in [5, 6, 9, 25, 28, 29, 35, 40, 55].
- 2.
This solution is obtainable at once by the method of variable separation. However, the reader can easily verify that it is really a solution for any C.
- 3.
For the application of differential equations to economy, see, for instance, [55].
- 4.
For a proof, see, for instance, [40].
- 5.
Readers will find in [35] many programs, written using Mathematica, that allow for the analysis of many stability problems.
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Romano, A., Marasco, A. (2018). An Overview of Dynamical Systems. In: Classical Mechanics with Mathematica®. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-77595-1_10
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DOI: https://doi.org/10.1007/978-3-319-77595-1_10
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