SIAM J. Applied Dynamical Systems **11** (2012) 1360-1401.
doi:10.1137/120864684
## Resonance bifurcations of robust heteroclinic networks

V. Kirk(1),
Claire M. Postlethwaite(1),
and
A.M. Rucklidge(2)

(1) Department of Mathematics,

University of Auckland, Private Bag 92019,

Auckland, New Zealand

(2) Department of Applied Mathematics,

University of Leeds, Leeds, LS2 9JT, UK

**Abstract.**
Robust heteroclinic cycles are known to change stability in resonance
bifurcations, which occur when an algebraic condition on the eigenvalues of the
system is satisfied and which typically result in the creation or destruction
of a long-period periodic orbit. Resonance bifurcations for heteroclinic
networks are potentially more complicated because different subcycles in the
network can undergo resonance at different parameter values, but have, until
now, not been systematically studied. In this article we present the first
investigation of resonance bifurcations in heteroclinic networks. Specifically,
we study two heteroclinic networks in $\R^4$ and consider the dynamics that
occurs as various subcycles in each network change stability. The two cases are
distinguished by whether or not one of the equilibria in the network has real
or complex contracting eigenvalues. We construct two-dimensional Poincar\'e
return maps and use these to investigate the dynamics of trajectories near the
network; a complicating feature of the analysis is that at least one
equilibrium solution in each network has a two-dimensional unstable manifold.
We use the technique developed in [18] to keep track of all trajectories within
these two-dimensional unstable manifolds. In the case with real eigenvalues, we
show that the asymptotically stable network loses stability first when one of
two distinguished cycles in the network goes through resonance and two or six
periodic orbits appear. In some circumstances, asymptotically stable periodic
orbits can bifurcate from the network even though the subcycle from which they
bifurcate is never asymptotically stable. In the complex case, we show that an
infinite number of stable and unstable periodic orbits are created at
resonance, and these may coexist with a chaotic attractor. In both cases, we
show that near to the parameter values where individual cycles go through
resonance, the periodic orbits created in the different resonances do not
interact, i.e., the periodic orbits created in the resonance of one cycle are
not involved in the resonance of the other cycle. However, there is a further
resonance, for which the eigenvalue combination is a property of the entire
network, after which the periodic orbits which originated from the individual
resonances may interact. We illustrate some of our results with a numerical
example.

Preprint version of this paper
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### Movies relevant to this paper: real case

The red solid curves are the
small nullclines in region~$1$, the green solid curves are the
small nullclines in region~$2$, and the blue solid curves are the
$\theta_3$ nullclines. The red and green dashed curves are the
approximate small nullclines. Cf figs 5.5-5.7.

### Movies relevant to this paper: complex case

The red solid curves are the
small nullclines in region~$1$, the green solid curves are the
small nullclines in region~$2$, and the blue solid curves are the
$\theta_3$ nullclines. Cf figs 5.10-5.13.