Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to time-periodic galloping solutions
نویسنده
چکیده
We show that transition to longitudinal instability of strong detonation solutions of reactive compressible Navier–Stokes equations is generically associated with Hopf bifurcation to nearby time-periodic “galloping”, or “pulsating”, solutions, in agreement with physical and numerical observation. The analysis is by pointwise semigroup techniques introduced by the authors and collaborators in previous works.
منابع مشابه
Galloping instability of viscous shock waves
Motivated by physical and numerical observations of time oscillatory “galloping”, “spinning”, and “cellular” instabilities of detonation waves, we study Poincaré–Hopf bifurcation of traveling-wave solutions of viscous conservation laws. The main difficulty is the absence of a spectral gap between oscillatory modes and essential spectrum, preventing standard reduction to a finite-dimensional cen...
متن کاملThe refined inviscid stability condition and cellular instability of viscous shock waves
Combining work of Serre and Zumbrun, Benzoni-Gavage, Serre, and Zumbrun, and Texier and Zumbrun, we propose as a mechanism for the onset of cellular instability of viscous shock and detonation waves in a finite-cross-section duct the violation of the refined planar stability condition of Zumbrun–Serre, a viscous correction of the inviscid planar stability condition of Majda. More precisely, we ...
متن کاملHOPF BIFURCATION CONTROL WITH PD CONTROLLER
In this paper, we investigate the problem of bifurcation control for a delayed logistic growth model. By choosing the timedelay as the bifurcation parameter, we present a Proportional - Derivative (PD) Controller to control Hopf bifurcation. We show that the onset of Hopf bifurcation can be delayed or advanced via a PD Controller by setting proper controlling parameter. Under consideration mode...
متن کاملStability of detonation profiles in the ZND limit
Confirming a conjecture of Lyng–Raoofi–Texier–Zumbrun, we show that stability of strong detonation waves in the ZND, or small-viscosity, limit is equivalent to stability of the limiting ZND detonation together with stability of the viscous profile associated with the component Neumann shock. More, on bounded frequencies the nonstable eigenvalues of the viscous detonation wave converge to those ...
متن کاملTransition from Rotating Waves to Modulated Rotating Waves on the Sphere
In this article, we study parameter-dependent systems of reaction-diffusion equations on the sphere, which are equivariant under the group SO(3) of all rigid rotations on a sphere. Two main types of spatial-temporal patterns that can appear in such systems are rotating waves (equilibrium in a co-rotating frame) and modulated rotating waves (periodic solution in a corotating frame). The transiti...
متن کامل