Computation of universal unfolding of the double zero bifurcation in the \(Z_{2}\)-symmetric system. (English) Zbl 1405.34033
Summary: We present explicit formulae of universal unfolding of the double zero bifurcation for any autonomous ordinary differential equation systems (ODEs) with \(\mathbf{Z}_{2}\)-symmetry. These formulae only need to use coefficients of the Taylor expansion of the vector field at its equilibria and are equally suitable for both numerical and symbolic computations, and they construct the relationship between original parameters and topological structures for the double zero bifurcation analysis in any autonomous ODEs. They can be directly used to check whether there exist the original parameters such that a system can exhibit the versal double zero bifurcation. Moreover, we can use them to compute the corresponding bifurcation curves with high precision. We take Chua’s circuit as an example to demonstrate the advantages of these formulae.
MSC:
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 37G05 | Normal forms for dynamical systems |
| 37G10 | Bifurcations of singular points in dynamical systems |
| 37G40 | Dynamical aspects of symmetries, equivariant bifurcation theory |
Keywords:
\(\mathbf{Z}_{2}\)-symmetry; double zero bifurcation; normal form; universal unfolding; centre manifoldReferences:
| [1] | Algaba A., IEICE Trans. Fundam 82 pp 1722– (1999) |
| [2] | Beyn W. J., Handbook of Dynamical Systems 2 pp 149– (2002) |
| [3] | DOI: 10.1007/978-1-4612-5929-9 · doi:10.1007/978-1-4612-5929-9 |
| [4] | DOI: 10.1137/0518045 · Zbl 0624.34026 · doi:10.1137/0518045 |
| [5] | DOI: 10.1142/S0218127410026277 · Zbl 1193.34079 · doi:10.1142/S0218127410026277 |
| [6] | DOI: 10.1017/CBO9780511665639 · doi:10.1017/CBO9780511665639 |
| [7] | Gamero E., Int. Ser. Num. Math 97 pp 123– (1991) |
| [8] | DOI: 10.1007/978-1-4615-7904-5 · doi:10.1007/978-1-4615-7904-5 |
| [9] | DOI: 10.1007/978-1-4612-1140-2 · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2 |
| [10] | DOI: 10.1016/0020-7462(80)90031-1 · Zbl 0453.70015 · doi:10.1016/0020-7462(80)90031-1 |
| [11] | DOI: 10.1016/0022-0396(67)90016-2 · Zbl 0173.11001 · doi:10.1016/0022-0396(67)90016-2 |
| [12] | Khorozov E. I., Trans. Petrovski. Semin 5 pp 163– (1979) |
| [13] | DOI: 10.1016/0375-9601(86)90464-0 · doi:10.1016/0375-9601(86)90464-0 |
| [14] | DOI: 10.1017/S0022112081002139 · Zbl 0478.76050 · doi:10.1017/S0022112081002139 |
| [15] | DOI: 10.1016/j.chaos.2004.02.014 · Zbl 1063.35136 · doi:10.1016/j.chaos.2004.02.014 |
| [16] | Lu Q. S., Bifurcation and Singularity, 2. ed. (1997) |
| [17] | DOI: 10.1007/s11071-010-9914-0 · Zbl 1286.34059 · doi:10.1007/s11071-010-9914-0 |
| [18] | Poénaru V., Singularities \(\mathbb{C}\) en présence de symétrie (1976) · Zbl 0325.57008 · doi:10.1007/BFb0079196 |
| [19] | DOI: 10.1007/BF02684366 · Zbl 0279.58009 · doi:10.1007/BF02684366 |
| [20] | Takens F., Comm. Math. Inst 3 pp 1– (1974) |
| [21] | Tang Y., Foundations of Symmetry Bifurcation Theory (1998) |
| [22] | DOI: 10.1007/BF03167860 · Zbl 0578.58029 · doi:10.1007/BF03167860 |
| [23] | Wiggins S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2. ed. (2003) · Zbl 1027.37002 |
| [24] | DOI: 10.1109/81.340866 · doi:10.1109/81.340866 |
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