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Abstract
Global bifurcations involving saddle periodic orbits have recentlybeen recognized as being involved in various new types of organizingcenters for complicated dynamics. The main emphasis has been onheteroclinic connections between saddle equilibria and saddleperiodic orbits, called EtoP orbits for short, which can be found invector fields in $\mathbb{R}^3$. Thanks to the development of dedicatednumerical techniques, EtoP orbits have been found in a number ofthree-dimensional model vector fields arising in applications.   We are concerned here with the case of heteroclinic connectionsbetween two saddle periodic orbits, called PtoP orbits for short. Ahomoclinic orbit from a periodic orbit to itself is an example of aPtoP connection, but is generically structurally stable in a phasespace of any dimension. The issue that we address here is that, untilnow, no example of a concrete vector field with a non-structurallystable PtoP connection was known. We present an example of a PtoPheteroclinic cycle of codimension one between two different saddleperiodic orbits in a four-dimensional vector field model ofintracellular calcium dynamics. We first show that this model is agood candidate system for the existence of such a PtoP cycle and thendemonstrate how a PtoP cycle can be detected and continued in systemparameters using a numerical setup that is based on Lin's method.
Highlights
In numerous fields of application one finds mathematical models with continuous time that take the general form of a vector field x = f (x, λ), (1)where f : Rn × Rm → Rn is sufficiently smooth, say, twice differentiable for the purpose of this paper
To understand the dynamics of (1) one needs to study how the phase space is organised by invariant objects, including equilibria and periodic orbits, and, when the equilibria or periodic orbits are of saddle type, their global stable and unstable manifolds
We have presented the first example of a concrete vector field in which a nonstructurally stable PtoP heteroclinic cycle connecting two saddle periodic orbits has been located numerically
Summary
Doedel et al [23] define projection boundary conditions via the adjoint variational equation along a periodic orbit, and continue codimension-one EtoP connections in the Lorenz system and in three-dimensional models of an electronic circuit and of a food-chain; in [24] these authors compute a codimension-zero PtoP homoclinic orbit of the food-chain model. In contrast to the above methods, Krauskopf and Rieß [42] represent an EtoP orbit of codimension d in any phase space dimension by two separate orbit segments Their numerical setup is an implementation of Lin’s method [45], which is a theoretical tool for the analysis of recurrent dynamics, in particular near homoclinic orbits and heteroclinic cycles; see, for example, [36, 49, 51, 52, 54].
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