Author:
Li Dongchen,Turaev Dmitry
Abstract
AbstractA heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least $C^{2}$
C
2
, we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family create robust heterodimensional dynamics, i.e., a pair of non-trivial hyperbolic basic sets with different numbers of positive Lyapunov exponents, such that the unstable manifold of each of the sets intersects the stable manifold of the second set and these intersections persist for an open set of parameter values. We also give a solution to the so-called local stabilization problem of coindex-1 heterodimensional cycles in any regularity class $r=2,\ldots ,\infty ,\omega $
r
=
2
,
…
,
∞
,
ω
. The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders.
Publisher
Springer Science and Business Media LLC
Reference69 articles.
1. Abraham, R., Smale, S.: Nongenericity of $\Omega $-stability, global analysis I. Proc. Symp. Pure Math. 14, 5–8 (1970)
2. Afraimovich, V.S., Shilnikov, L.P.: On singular sets of Morse-Smale systems. Tr. Mosk. Mat. 28, 181–214 (1973)
3. Afraimovich, V.S., Bykov, V.V., Shilnikov, L.P.: On the structurally unstable attracting limit sets of Lorenz attractor type. Trans. Mosc. Math. Soc. 2, 153–215 (1983)
4. Asaoka, M.: Hyperbolic sets exhibiting $C^{1}$-persistent homoclinic tangency for higher dimensions. Proc. Am. Math. Soc. 136(2), 677–686 (2008)
5. Asaoka, M., Shinohara, K., Turaev, D.V.: Degenerate behavior in non-hyperbolic semigroup actions on the interval: fast growth of periodic points and universal dynamics. Math. Ann. 368(3–4), 1277–1309 (2017)
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献
1. Cascades of heterodimensional cycles via period doubling;Communications in Nonlinear Science and Numerical Simulation;2024-09