Numerical periodic normalization for codim 2 bifurcations of limit cycles : computational formulas, numerical implementation, and examples

Explicit computational formulas for the coefficients of the periodic normal forms for codimension 2 (codim 2) bifurcations of limit cycles in generic autonomous ODEs are derived. All cases (except the weak resonances) with no more than three Floquet multipliers on the unit circle are covered. The resulting formulas are independent of the dimension of the phase space and involve solutions of certain boundary-value problems on the interval [0, T], where T is the period of the critical cycle, as well as multilinear functions from the Taylor expansion of the ODE right-hand side near the cycle. The formulas allow one to distinguish between various bifurcation scenarios near codim 2 bifurcations of limit cycles. Our formulation makes it possible to use robust numerical boundary-value algorithms based on orthogonal collocation, rather than shooting techniques, which greatly expands its applicability. The implementation is described in detail with numerical examples, where numerous codim 2 bifurcations of limit cycles are analyzed for the first time

Numerical computation of normal form coefficients of ODEs in MATLAB

Normal form coefficients of codim-1 and codim-2 bifurcations of equilibria of ODEs are important since their sign and size determine the bifurcation scenario near the bifurcation points. Multilinear forms with derivatives up to the fifth order are needed in these coefficients. So far, in the Matlab bifurcation software MatCont for ODEs, these derivatives are computed either by finite differences or by symbolic differentiation. However, both approaches have disadvantages. Finite differences do not usually have the required accuracy and for symbolic differentiation the Matlab Symbolic Toolbox is needed. Automatic differentiation is an alternative since this technique is as accurate as symbolic derivatives and no extra software is needed. In this paper, we discuss the pros and cons of these three kinds of differentiation in a specific context by the use of several examples

Numerical implementation of a homotopy method for computing homoclinic and heteroclinic orbits

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Using MatCont in a two-parameter bifurcation study of models for cell cycle controls

A recent application field of bifurcation theory is in modelling the cell cycle. We refer in particular to the work of J.J. Tyson and B. Novak where the fundamental idea is that the cell cycle is an alternation between two stable steady states of a system of kinetic equations. We study and extend the basic model of Tyson and Novak using the Matlab numerical bifurcation software MatCont, in a two-parameter setting and highlight several new features. We show that the limit point curves in the two-variable model behave in an ungeneric way under variation of the natural parameters and that the hysteresis loop in the model is not the usual loop caused by the existence of a codimension-2 cusp point. We continue orbits homoclinic-to-saddle-node (HSN) in the three-variable model and find that these orbits die in a non-central orbit homoclinic-to-saddle-node under a natural parameter variation. As an extension we introduce a model in which cell division appears as a continuous-in-time limit cycle. We perform a continuation of this limit cycle under a natural parameter variation and show that it loses stability in a limit point of cycles bifurcation. Alternatively, we study the cell cycle as a boundary value problem as proposed by Tyson and Novak. This leads to an interpretation of the cell as a slow-fast system and we derive several conclusions on the relation between the growth rate of the cells and the cell size at division, and on the controllability of the process

Interactive initialization and continuation of homoclinic and heteroclinic orbis in MATLAB

MATCONT is a MATLAB continuation package for the interactive numerical study of a range of parameterized nonlinear dynamical systems, in particular ODEs, that allows to compute curves of equilibria, limit points, Hopf points, limit cycles, flip, fold and torus bifurcation points of limit cycles. It is now possible to continue homoclinic-to-hyperbolic-saddle and homoclinic-to-saddle-node orbits in MATCONT. The implementation is done using the continuation of invariant subspaces, with the Riccati equations included in the defining system. A key feature is the possibility to initiate both types of homoclinic orbits interactively, starting from an equilibrium point and using a homotopy method. All known codimension-two homoclinic bifurcations are tested for during continuation. The test functions for inclination-flip bifurcations are implemented in a new and more efficient way. Heteroclinic orbits can now also be continued and an analogous homotopy method can be used for the initialization