Journal
CHAOS
Volume 30, Issue 8, Pages -Publisher
AMER INST PHYSICS
DOI: 10.1063/5.0010149
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Funding
- Spanish MINECO-FEDER [PGC2018-098676-B-100]
- Catalan Grant [2017SGR1049]
- Catalan Institution for research and advanced studies via an ICREA academia price 2018
- FPI Grant from project MINECO-FEDER-UE [MTM2012-31714]
- Institute of Computer Science of the Czech Academy of Sciences [RVO:67985807]
- RyC project [RYC-2014-15866]
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In this paper, we use the parameterization method to provide a complete description of the dynamics of an n-dimensional oscillator beyond the classical phase reduction. The parameterization method allows us, via efficient algorithms, to obtain a parameterization of the attracting invariant manifold of the limit cycle in terms of the phase-amplitude variables. The method has several advantages. It provides analytically a Fourier-Taylor expansion of the parameterization up to any order, as well as a simplification of the dynamics that allows for a numerical globalization of the manifolds. Thus, one can obtain the local and global isochrons and isostables, including the slow attracting manifold, up to high accuracy, which offer a geometrical portrait of the oscillatory dynamics. Furthermore, it provides straightforwardly the infinitesimal phase and amplitude response functions, that is, the extended infinitesimal phase and amplitude response curves, which monitor the phase and amplitude shifts beyond the asymptotic state. Thus, the methodology presented yields an accurate description of the phase dynamics for perturbations not restricted to the limit cycle but to its attracting invariant manifold. Finally, we explore some strategies to reduce the dimension of the dynamics, including the reduction of the dynamics to the slow stable submanifold. We illustrate our methods by applying them to different three-dimensional single neuron and neural population models in neuroscience.
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