Analysis of bifurcations of limit cycles with Lyapunov exponents and numerical normal forms

Abstract

In this paper we focus on the combination of normal form and Lyapunov exponent computations in the numerical study of the three codim 2 bifurcations of limit cycles with dimension of the center manifold equal to 4 or to 5 in generic autonomous ODEs. The normal form formulas are independent of the dimension of the phase space and involve solutions of certain linear boundary-value problems. The formulas allow one to distinguish between the complicated bifurcation scenarios which can happen near these codim 2 bifurcations, where 3-tori and 4-tori can be present. We apply our techniques to the study of a known laser model, a novel model from population biology, and a model of mechanical vibrations. These models exhibit Limit Point-Neimark-Sacker, Period-Doubling-Neimark-Sacker, and double Neimark-Sacker bifurcations. Lyapunov exponents are computed to numerically confirm the results of the normal form analysis, in particular with respect to the existence of stable invariant tori of various dimensions. Conversely, the normal forms are essential to understand the significance of the Lyapunov exponents.

A Degenerate 2:3 Resonant Hopf--Hopf Bifurcation as Organizing Center of the Dynamics: Numerical Semiglobal Results

“…These equations (with unfolding parameters β 1,2 ) approximate the second iterate of the Poincaré map associated to the critical limit cycle (see [6,8]). …”

“…Fold-Neimark-Sacker bifurcation (section 2.2.1 of [8]): ξ 1 = β 1 + ξ 2 1 + s |ξ 2 | 2 , ξ 2 = (β 2 + iω 1 ) ξ 2 + (θ + iυ) ξ 1 ξ 2 + ξ 2 1 ξ 2 . …”

“…A second addition to the bifurcation structure is the fold-Neimark-Sacker bifurcation point F NS b . The normal form coefficients, computed according to the procedure described in [7,8], reveal that the unfolding corresponds to the one shown in Figure A1(c) of [8] for s = 1, θ < 0, and E > 0 and contemplates the existence of an unstable 3D torus. This torus is destroyed via a complicated global phenomena (homoclinic tangle) occurring in an exponentially small wedge in the parameter space (see [18,32,8]).…”

Dynamical Systems with a Codimension-One Invariant Manifold: The Unfoldings and Its Bifurcations

“…This type of dynamical systems occurs frequently in applications especially in the mathematical models for population biology [Jansen, 2001;Saputra et al, 2010a;De Witte et al, 2014] and for the spread of diseases [Chitnis et al, 2006;Llensa et al, 2014]. It is because in population models, if a species dies out it cannot be regenerated therefore it is always natural to have coordinate axes as the invariant manifold.…”

Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

“…Compare computed and predicted periods figure(10) ; clf ; hold on ; % Plot computed periods on nsbr (1) and nsbr (2) omegas1 = arrayfun ( @ ( p ) p . period , nsbr .…”