Poincaré maps of the double scroll. (English) Zbl 0616.58037
Dynamical systems and nonlinear oscillations, Proc. Symp., Kyoto/Jap. 1985, RIMS Kokyuroku 574, 126-134 (1985).
[For the entire collection see Zbl 0605.00012.]
The author constructs an analytic diffeomorphism of the open n-cube which is expansive and whose inverse is also expansive. The map constructed may also be embedded in a smooth flow. The double-scroll system is a piecewise linear vector field on \(R^ 3\) which is defined in three pieces. The authors discuss the bifurcations in specific families of such systems which lead to creation and disappearance of double-scroll attractors. This is related to work of C. T. Sparrow [J. Math. Anal. Appl. 83, 275-291 (1981; Zbl 0518.34037)], P. Arneodo and P. Coullet [Commun. Math. Phys. 79, 573-579 (1981; Zbl 0485.58013)] and R. W. Brockett [Decision and control, Proc. 20th IEEE Conf., incl. Symp. Adaptive processes, San Diego/Calif. 1981, Vol. 1, 76-79 (1981; Zbl 0515.93050)].
The author constructs an analytic diffeomorphism of the open n-cube which is expansive and whose inverse is also expansive. The map constructed may also be embedded in a smooth flow. The double-scroll system is a piecewise linear vector field on \(R^ 3\) which is defined in three pieces. The authors discuss the bifurcations in specific families of such systems which lead to creation and disappearance of double-scroll attractors. This is related to work of C. T. Sparrow [J. Math. Anal. Appl. 83, 275-291 (1981; Zbl 0518.34037)], P. Arneodo and P. Coullet [Commun. Math. Phys. 79, 573-579 (1981; Zbl 0485.58013)] and R. W. Brockett [Decision and control, Proc. 20th IEEE Conf., incl. Symp. Adaptive processes, San Diego/Calif. 1981, Vol. 1, 76-79 (1981; Zbl 0515.93050)].
Reviewer: R.Devaney
MSC:
| 37G99 | Local and nonlocal bifurcation theory for dynamical systems |
| 58C25 | Differentiable maps on manifolds |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
