Chaos **13** (2003) 973-981.
doi:10.1063/1.1586531
## Phase resetting effects for robust cycles between chaotic sets

Peter Ashwin(1)
Michael Field(2)
Alastair M. Rucklidge(3)
Rob Sturman(3).

(1) School of Mathematical Sciences, Laver Building,

University of Exeter, Exeter EX4 4QE, UK

(2) Department of Mathematics, University of Houston,

Houston, TX 77204-3008, USA

(3) Department of Applied Mathematics,

University of Leeds, Leeds, LS2 9JT, UK

**Abstract.**
In the presence of symmetries or invariant subspaces, attractors in dynamical
systems can become very complicated owing to the interaction with the invariant
subspaces. This gives rise to a number of new phenomena including that of
robust attractors showing chaotic itinerancy. At the simplest level this is an
attracting heteroclinic cycle between equilibria, but cycles between more
general invariant sets are also possible.
This paper introduces and discusses an instructive example of an ODE where one
can observe and analyse robust cycling behaviour. By design, we can show that
there is a robust cycle between invariant sets that may be chaotic saddles
(whose internal dynamics correspond to a Rossler system), and/or saddle
equilibria.

For this model, we distinguish between cycling that include *phase
resetting* connections (where there is only one connecting trajectory) and
more general *non-phase resetting* cases where there may be an infinite
number (even a continuum) of connections. In the non-phase resetting case
there is a question of *connection selection*: which connections are
observed for typical attracted trajectories? We discuss the instability of this
cycling to resonances of Lyapunov exponents and relate this to a conjecture
that phase resetting cycles typically lead to stable periodic orbits at
instability whereas more general cases may give rise to `stuck on' cycling.

Finally, we discuss how the presence of positive Lyapunov exponents of the
chaotic saddle mean that we need to be very careful in interpreting numerical
simulations where the return times become long; this can critically influence
the simulation of phase-resetting and connection selection.

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