Abstract
Distributed delays modeled by 'weak generic kernels' are introduced in the well-known coupled Landau-Stuart system, as well as a chaotic van der Pol-Rayleigh system with parametric forcing. The systems are close via the 'linear chain trick'. Linear stability analysis of the systems and conditions for Hopf bifurcation which initiates oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. The value of the delay parameter $a=a_{Hopf}$ at Hopf bifurcation picks out the onset of Amplitude Death(AD) in all three systems, with oscillations at larger values (corresponding to weaker delay). In the Landau-Stuart system, the Hopf-generated limit cycles for $a>a_{Hopf}$ turn out to be remarkably stable under very large variations of all other system parameters beyond the Hopf bifurcation point, and do not undergo further symmetry breaking, cyclic-fold, flip, transcritical or Neimark-Sacker bifurcations. This is to be expected as the corresponding undelayed systems are robust oscillators over wide ranges of their respective parameters. Numerical simulations reveal strong distortion and rotation of the limit cycles in phase space as the parameters are pushed far into the post-Hopf regime, and reveal other features, such as how the oscillation amplitudes and time periods of the physical variables on the limit cycle attractor change as the delay and other parameters are varied. For the chaotic system, very strong delays may still lead to the cessation of oscillations and the onset of AD (even for relatively large values of the system forcing which tends to oppose this phenomenon). Varying of the other important system parameter, the parametric excitation, leads to a rich sequence of dynamical behaviors, with the bifurcations leading from one regime (or type of attractor) into the next being carefully tracked.
Highlights
As is well-known, nonlinear dynamical systems, especially coupled ones, are of wide interest in many areas of science and technology
Since we have chosen the delay a as bifurcation parameter, and larger delays or lower a values have a stabilizing effect, we know that for our delayed Landau-Stuart system, the post-Hopf regime is for a values larger than the aHopf value found using the second root of the polynomial in the last equation of Section 3.1
For a < aHopf, the strong delay stabilizes the oscillations and yields a stable fixed point. This is the regime of Amplitude Death(AD) for the system caused by the delay
Summary
As is well-known, nonlinear dynamical systems, especially coupled ones, are of wide interest in many areas of science and technology. The first is suppression of oscillation to a single or homogeneous steady state (nowadays referred to as AD), versus the second or Oscillation Death (OD)[Koseska et al, 2013] where the oscillators asymptotically populate different fixed points or ’inhomogeneous steady states’, some of which may not have been stable, or perhaps not even present, for the uncoupled oscillators. Both AD and OD are known to occur in various settings.
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