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

Abstract

In this paper a degenerate case of a 2:3 resonant Hopf-Hopf bifurcation is studied. This codimensionfour bifurcation occurs when the frequencies of both Hopf bifurcation branches have the relation 2/3, and one of them presents the vanishing of the first Lyapunov coefficient. The bifurcation is analyzed by means of numerical two-and three-parameter bifurcation diagrams. The two-parameter bifurcation diagrams reveal the interaction of cyclic-fold, period-doubling (or flip), and NeimarkSacker bifurcations. A nontrivial bifurcation structure is detected in the main three-parameter space. It is characterized by a fold-flip (F F ) bubble interacting with curves of fold-Neimark-Sacker (F NS), generalized period-doubling (GP D), 1:2 strong resonances (R1:2), 1:1 strong resonances of periodtwo cycles (R (2) 1:1 ), and Chenciner bifurcations (CH). Two codimension-three points with nontrivial Floquet multipliers (1, −1, −1), where the bifurcation curves F F , F NS, R1:2, and CH interact, are detected. A second pair of codimension-three points appears when F F interacts with GP D and R(2) 1:1 (and CH in one of the points). Finally, it is shown that this degenerate 2:3 resonant Hopf-Hopf bifurcation acts as an organizing center of the dynamics, since the structure of bifurcation curves and its singular points are unfolded by this singularity.

Normal-normal resonances in a double Hopf bifurcation

“…The strong 1:1 resonance, to which we do not contribute, has been studied in [23,36,41]. Of the weak resonances, the 1:2 resonance has gotten the most attention [28,29,32,34,38,44]; next to it, the 1:3 resonance [34] and the 2:3 resonance [37,39] have been investigated as well.…”

“…Volkov [44] takes n ≥ 2 and studies the persistence of n-dimensional quasi-periodic tori at a 1:2 double Hopf normal resonance. Finally, Revel et al [37][38][39] study 1:2 and 2:3 resonant double Hopf bifurcations numerically.…”

Double Hopf bifurcation analysis in the memory-based diffusion system

“…The complex dynamics arising form double Hopf bifurcation has been recently studied by many authors for various dynamical systems, refering to [20,22,33,44,49] for ordinary differential equations, to [2,3,6,7,14,18,27,31,45,46] for delay differential equations. More recently, based on the theory of normal forms for partial functional differential equations developed by Faria [12], the double Hopf bifurcation in the reaction-diffusion system with delay has attracted the attention of the researchers [4,8,9,24].…”

Another Case of Degenerated Discrete Chenciner Dynamic System and Economics

“…The non-degenerate Chenciner bifurcation was firstly studied in the papers [6,13,14]. More recently this bifurcation appears in many papers from different areas of research, in "biology, physics, economy, informatics" [15] as well as multidisciplinary and applied sciences [12,[16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. For example, in [31], the Chenciner bifurcation was observed when a potential mechanism from bifurcation analyses was used for studying the occurrence of modulated oscillations in synchronous machine nonlinear dynamics, being reported for the first time in power engineering for this bifurcation.…”