Abstract
In this chapter, we introduce and discuss the following fundamental notions that will be used throughout the book: topological equivalence of dynamical systems and their classification, bifurcations and bifurcation diagrams, and topological normal forms for bifurcations. The last section is devoted to the more abstract notion of structural stability. In this chapter we will be dealing only with dynamical systems in the state space \(X={\mathbb R}^n\). We would like to study general (qualitative) features of the behavior of dynamical systems, in particular, to classify possible types of their behavior and compare the behavior of different dynamical systems.
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Notes
- 1.
Recall that a relation between two objects (\(a\sim b\)) is called equivalence if it is reflexive (\(a \sim a\)), symmetric (\(a \sim b\) implies \(b\sim a\)), and transitive (\(a \sim b\) and \(b \sim c\) imply \(a \sim c\)).
- 2.
It is clear that \( \overline{\bigcup _{k \in \mathbb {Z}} D_k}=[x_j,x_{j+1}]~~~\textrm{and}~~~ \overline{\bigcup _{k \in \mathbb {Z}} D'_k}=[y_j,y_{j+1}]. \)
- 3.
If necessary, one may consider the phase portrait in a parameter-dependent region \(U_{\alpha } \subset {\mathbb R}^n\).
- 4.
Recall that some time-related information on the behavior of the system is lost due to topological equivalence.
- 5.
It is possible to construct a kind of topological normal form for certain global bifurcations involving homoclinic orbits.
- 6.
The eigenvalues vary smoothly as long as they remain simple.
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Kuznetsov, Y.A. (2023). Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems. In: Elements of Applied Bifurcation Theory. Applied Mathematical Sciences, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-031-22007-4_2
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DOI: https://doi.org/10.1007/978-3-031-22007-4_2
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