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Rand, R. H.; Keith, W. L.

Determinacy of degenerate equilibria with linear part \(x'=y\), \(y'=0\) using MACSYMA. (English) Zbl 0624.65068

Appl. Math. Comput. 21, 1-19 (1987).
Authors’ summary: A MACSYMA program is presented which implements the method of F. Takens [Inst. Haut. Etud. Sci., Publ. Math. 43(1973), 47-100 (1974; Zbl 0279.58009)] for proving determinacy of a flow in the neighborhood of an equilibrium point by successive blowup transformations. The program is applied to nonlinear systems with linear part \(x'=y\), \(y'=0\), and an examination of the pattern of the results reveals an infinite sequence of determinacy theorems. The results are applied to the behavior of van der Pol’s equation at infinity.
Reviewer: I.N.Katz

Cited in 4 Documents

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
65H10 Numerical computation of solutions to systems of equations
37C10 Dynamics induced by flows and semiflows
34A34 Nonlinear ordinary differential equations and systems
34-04 Software, source code, etc. for problems pertaining to ordinary differential equations

Keywords:

degenerate equilibria; MACSYMA program; successive blowup transformations; determinacy theorems; van der Pol’s equation

Citations:

Zbl 0279.58009

Software:

MACSYMA
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Full Text: DOI

References:

[1] Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983), Springer · Zbl 0515.34001
[2] W.L. Keith and R.H. Rand, Dynamics of a system exhibiting the global bifurcation of a limit cycle at infinity, Internat. J. Non-Linear Mech.; W.L. Keith and R.H. Rand, Dynamics of a system exhibiting the global bifurcation of a limit cycle at infinity, Internat. J. Non-Linear Mech. · Zbl 0647.34033
[3] Minorsky, N., Nonlinear Oscillations (1962), Van Nostrand · Zbl 0123.06101
[4] Rand, R. H., Computer Algebra in Applied Mathematics: An Introduction to MACSYMA (1984), Pitman · Zbl 0583.68012
[5] R.H. Rand, and W.L. Keith, Normal form and center manifold calculations on MACSYMAApplications of Computer Algebra; R.H. Rand, and W.L. Keith, Normal form and center manifold calculations on MACSYMAApplications of Computer Algebra
[6] Takens, F., Singularities of vector fields, Publ. Math. Inst. Hautes Etudes Sci., 43, 47-100 (1974) · Zbl 0279.58009
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