Stability and multiple bifurcations of a damped harmonic oscillator with delayed feedback near zero eigenvalue singularity

Abstract

We investigate the dynamics of a damped harmonic oscillator with delayed feedback near zero eigenvalue singularity. We perform a linearized stability analysis and multiple bifurcations of the zero solution of the system near zero eigenvalue singularity. Taking the time delay as the bifurcation parameter, the presence of steady-state bifurcation, Bogdanov-Takens bifurcation, triple zero, and Hopf-zero singularities is demonstrated. In the case when the system has a simple zero eigenvalue, center manifold reduction and normal form theory are used to investigate the stability and the types of steady-state bifurcation. The stability of the zero solution of the system near the simple zero eigenvalue singularity is completely solved.

Delay-induced Turing instability in reaction-diffusion equations

“…As λ, the root of (5), especially the real part of such a root, is dependent on τ [6,10], we know the stability of E * changes only when the real part of λ changes its sign, which happens when τ = τ c at which λ = ±iω k , ω k > 0 [7,8,10]. Next we find such a τ c .…”

“…However, in practice, the delay may be large [7,8,18,19] or, as will be seen in this paper below, the above proposed technique is no longer valid. Reference [9] also made an attempt to find a critical delay and they proposed ω c τ c = arccos(…”

“…Assume system (3) has an equilibrium E * = (u * i ), at which we define a ij = ∂f i ∂u j and b ij = ∂f i ∂u jτ . Then the linearization of (3) at E * has an associated characteristic equation [7,8,10],…”

“…However, in practice, time delays in biological systems may be large-see Refs. [7,8,18,19], for example. For a model described by reaction-diffusion equations, could a large time delay induce a Turing instability?…”

“…In ecology, time delay is commonly used to measure a maturation, gestation period, or reaction time of a predator population [1,4,5], and mathematically it may result in a much richer dynamics for a system, such as inducing the instability of an equilibrium so that Hopf bifurcation occurs [6][7][8][9][10][11][12]. Therefore, systems with time delays have received wide attention over the past years.…”