Bubbling, riddling, blowout and critical curves

Abstract

This paper provides a survey of some recent results and examples concerning the use of the method of critical curves in the study of chaos synchronization in discrete dynamical systems with an invariant one-dimensional submanifold. Some examples of two-dimensional discrete dynamical systems, which exhibit synchronization of chaoti1c trajectories with the related phenomena of bubbling, on–off intermittency, blowout and riddles basins, are examined by the usual local analysis in terms of transverse Lyapunov exponents, whereas segments of critical curves are used to obtain the boundary of a two-dimensional compact trapping region containing the one-dimensional Milnor chaotic attractor on which synchronized dynamics occur. Thanks to the folding action of critical curves, the existence of such a compact region may strongly influence the effects of bubbling and blowout bifurcations, as it acts like a ‘trapping vessel’ inside which bubbling and blowout phenomena are bounded by the global dynamical forces of the dynamical system.

Keywords:

• Synchronization
• coupling
• nonlinear dynamics
• critical curves
• noninvertible maps

AMS Subject Classifications:

• 37G35
• 37C70
• 58J70

Acknowledgements

We thank Laura Gardini for inviting us to submit this paper and for her useful suggestions. The usual disclaimers apply. This work has been performed within the framework of COST Action IS1104 ‘The EU in the new economic complex geography: models, tools and policy evaluation’ and under the auspices of GNFM, Gruppo Nazionale di Fisica Matematica (Italy).

Notes

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

1 For a more complete treatment see e.g. [14].

2 A very long sequence of such bursts, which can be observed when ${\mathrm{\Lambda }}_{\perp }^{\text{nat}}$ is close to zero, has been called on–off intermittency in [30], even if this term is more suitable after the blowout bifurcation, when ${\mathrm{\Lambda }}_{\perp }^{\text{nat}}$ becomes positive.

3 This terminology, and notation, originates from the notion of critical points as it is used in the classical works of Julia and Fatou.

4 See e.g. [28] or [27] for a description of the geometric properties of a noninvertible map related to the folding (or foliation) of its phase space.