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Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps. (English) Zbl 0784.34030
Summary: We present a numerical technique for the analysis of local bifurcations which is based on the continuation of structurally unstable invariant sets in a suitable phase-parameter space. The invariant sets involved in our study are equilibrium points and limit cycles of autonomous ODEs, periodic solutions of time-periodic nonautonomous ODEs, fixed points and periodic orbits of iterated maps. The more general concept of a continuation strategy is also discussed. It allows the analysis of various singularities of generic systems and of their mutual relationships. The approach is extended to codimension three singularities. We introduce several bifurcation functions and show how to use them to construct well-posed continuation problems. The described continuation technique is supported by an interactive graphical program called LOCBIF. We discuss briefly the concepts of the LOCBIF interface and give some examples of typical applications.

MSC:
34C23 Bifurcation theory for ordinary differential equations
65J99 Numerical analysis in abstract spaces
37G99 Local and nonlocal bifurcation theory for dynamical systems
65D30 Numerical integration
65L99 Numerical methods for ordinary differential equations
68N99 Theory of software
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