Strong and Weak Resonances in Delayed Differential Systems

Abstract

In this paper, a general and systematic scheme is provided to research strong and weak resonances derived from delay-induced various double Hopf bifurcations in delayed differential systems. The method of multiple scales is extended to obtain a common complex amplitude equation when the double Hopf bifurcation with frequency ratio k 1 :k 2 occurs in the systems under consideration. By analyzing the complex amplitude equation, we give the conditions of the strong and weak resonances respectively in the analytical expressions. The weak resonances correspond to the codimension-two double Hopf bifurcations since the amplitudes and the phases may be decoupled, but the strong resonances to the codimension-three double Hopf bifurcations in the system. It is seen that the weak resonances happen in the system even for a lower-order ratio, i.e. k 1 + k 2 ≤ 4. As applications, two examples are displayed. Three cases of the delay-induced resonance with 1:2, 1:3, 1:5 and 1: √ 2 are discussed in detail and the corresponding normals are represented. Thus, the relative dynamical behaviors can be easily classified in the physical parameter space in terms of nonlinear dynamics. The results show the provided conditions may be used to determine that a resonance is strong or weak.

On the analysis of the double Hopf bifurcation in machining processes via centre manifold reduction

“…The formula of a 22 can be obtained from that of a 11 by replacing The above formulae are valid for non-resonant double Hopf bifurcation, where the angular frequencies ω 1 and ω 2 are not related by small integer numbers. Note that for specific values of the damping ratio ζ , it might be possible to obtain weak resonant double Hopf bifurcations [44], but their analysis is out of scope of this paper. The formula (4.29) of a 11 is the same as that obtained for the Poincaré-Lyapunov constant via the analysis of the single Hopf bifurcation in [11].…”

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].…”

Normal forms of non-resonance and weak resonance double Hopf bifurcation in the retarded functional differential equations and applications

“…Therefore, the related research subjects of dynamical behavior are developed very rapidly. Many scholars have extended and improved the method of the dynamical behavior of ODE [8,19,28,29,41] to study the dynamical behavior of DDE, and they have obtained a lot of very good results, such as stability and Hopf bifurcation [1,5,14,15,26,30], synchronization [24], Bogdanov-Takens bifurcation [6,17], zero-Hopf bifurcation [11,25,33,36,37], triple zero bifurcation [10], and double Hopf bifurcation [2][3][4]12,13,18,[21][22][23]34,35,38,40,42,45]. Some researchers also have studied the direction of numerical Hopf bifurcation and stability of bifurcating invariant curve for the delay differential equations by using multistep method and Runge-Kutta method [20,31,32].…”

“…Recently, there have been a lot of methods and techniques used to study the double Hopf bifurcation of delay differential equations, such as the center manifold type specification method [2][3][4]18,21,22], multi-scale method [4,23,34,35,41,42], perturbation incremental scheme [38][39][40], Liappunov-Schmidt reduction [7,44]. The center manifold reduction and multiple time scales are two useful techniques for computing the normal form of differential equations.…”