Knotted periodic orbits in suspensions of Smale’s horseshoe: torus knots and bifurcation sequences. (English) Zbl 0593.58027

Holmes, Philip; Williams, R. F.

Arch. Ration. Mech. Anal. 90, 115-194 (1985).

Consider a quadratic mapping due to Hénon (1976), \[ (*)\quad G_{\mu,\epsilon}(x,y)=(y,-\epsilon x+\mu -y^ 2),\quad x,y\in {\mathbb{R}}. \] For suitable \((\mu,\epsilon)\in {\mathbb{R}}^ 2\), \(G_{\mu,\epsilon}\) has a hyperbolic horseshoe. In earlier work [the first author and D. C. Whitley, Philos. Trans. R. Soc. Lond., A 311, 43-102 (1984; Zbl 0556.58023)], it was found that the bifurcation sequences of maps like (*) occurring as \(\mu\) increases varied drastically as \(\epsilon\) changed from 0 to 1. But the authors say that the results of the above paper were only partially satisfactory. This paper improves some of them by considering the knot type of periodic orbits.
For \(\epsilon =1\) and \(\mu\in (-1,3)\), \(G_{\mu,1}\) has an elliptic fixed point b, where the eigenvalues \(\lambda_ i\) of \(DG_{\mu,1}(b)\) satisfy \(\lambda_ 1\lambda_ 2=1\). As \(\mu\in (-1,3)\) passes a countable set of values \(\mu\) (p,q), the \(\lambda_ i=e^{\pm i\theta}\) pass each root of unity \((\theta =2\pi p/q)\), at which pairs of periodic orbits of period q bifurcate from b. One orbit is a saddle and the other a sink for \(\epsilon <1\), a source for \(\epsilon >1\) and elliptic for \(\epsilon =1\). The orbits vanish by coalescing pairwise in saddle-node bifurcations. In the \(\epsilon =1\) case, these orbits are called resonant Hamiltonian bifurcations.
Consider the suspension of the Hénon map \(G_{\mu,\epsilon}\). The two closed orbits obtained by suspension from the two periodic orbits created in a saddle-node bifurcation above (resonant bifurcation) have the same knot type. Moreover, the knot type cannot change as the parameters \(\epsilon\), \(\mu\) vary. It is shown that the orbits created in the resonant Hamiltonian bifurcation at \(\mu =\mu (p,q)\) are (p,q) torus knots of period q (Proposition 5.1).
In relation to these resonant bifurcations, this paper studies the torus knots and the bifurcations in the suspension of horseshoe using the method of knot holders (Birman-Williams (1973)), Braid theory and kneading theory. The following are the main theorems.
Theorem 6.1.2a. Among the (p,q)-torus knots in the Smale horseshoe there are (i) two and only two of period q for \(q>2p\), and infinitely many if the period is allowed to be arbitrary; (ii) none of period q if \(q<2p\); (iii) none at all if \(q<3p/2.\)
Theorem 6.1.2b. the number \(N_{p,q}\) of (p,q) torus knots of period \(p+q\) in the Lorenz knot holder satisfies \(2[q/p]\leq N_{p,q}\leq q-p+1\). There are infinitely many if the period is allowed to be arbitrary. Theorem 7.1.1. Let \((p_ i,q_ i)\), \(i=1,2\) be two pairs of relatively prime integers with \(p_ i<q_ i/2\), \(q_ i\geq 5\) and \(p_ 1/q_ 1>p_ 2/q_ 2\). Let \(\zeta_ i(\epsilon)\), \(\epsilon\in [0,1)\) denote the \(\mu\) value for which the unique pair of \((p_ i,q_ i)\) torus horseshoe knots of period q appear in a saddle-node bifurcation for the natural suspension of the Hénon map (*). Then \(\lim_{\epsilon \to 1}\zeta_ 1(\epsilon)>\lim_{\epsilon \to 1}\zeta_ 2(\epsilon)\) and \(\zeta_ 1(0)<\zeta_ 2(0).\)
Corollary 7.1.2. Infinitely many saddle-node bifurcation curves cross each other on the (\(\mu\),\(\epsilon)\) parameter plane between \(\epsilon =0\) and \(\epsilon =1\). In particular, each resonant torus bifurcation sequence of period q for the area preserving suspension \((\epsilon =1)\) is precisely reversed for the one-dimensional suspension \((\epsilon =0)\).

Reviewer: G.Ikegami

MSC:

37G99

Local and nonlocal bifurcation theory for dynamical systems

Keywords:

horseshoe diffeomorphism; torus knot; Hénon map; braid theory; kneading theory; saddle-node bifurcations; resonant Hamiltonian bifurcations; resonant bifurcation

Citations:

Zbl 0556.58023

References:

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