Related papers: Separatrix configurations in holomorphic flows
In this follow-up paper, we investigate the global geometry and topology of dynamical systems $\dot{x} = F(x)$ with entire vector field $F$, building on and constructively extending the local structure of simple and higher-order equilibria.…
Separatrices divide the phase space of some holomorphic dynamical systems into separate basins of attraction or 'stability regions' for distinct fixed points. 'Bundling' (high density) and mutual 'repulsion' of trajectories are often…
Multiple time scales in dynamical systems lead to a bundling of trajectories onto slow invariant manifolds (SIMs). Although they are absent in two-dimensional holomorphic dynamical systems, a bundling of orbits is often observed as well.…
An approach is presented for identifying separatrices in phase space generated from noisy time series data sets representative of measured experimental data. These separatrices are identified as ridges in the phase space distribution of…
The structure of the basin of attraction of a stable equilibrium point is investigated for a dynamical system (W97) often used to model transition to turbulence in shear flows. The basin boundary contains not only an equilibrium point Xlb…
In shear flows like pipe flow and plane Couette flow there is an extended range of parameters where linearly stable laminar flow coexists with a transient turbulent dynamics. When increasing the amplitude of a perturbation on top of the…
We consider the semilinear wave equation with subconformal power nonlinearity in two space dimensions. We construct a finite-time blow-up solution with an isolated characteristic blow-up point at the origin, and a blow-up surface which is…
We study numerically transitional coherent structures in a boundary-layer flow with homogeneous suction at the wall (the so-called asymptotic suction boundary layer ASBL). The dynamics restricted to the laminar-turbulent separatrix is…
Reminiscent of physical phase transitions separatrices divide the phase space of dynamical systems with multiple equilibria into regions of distinct flow behavior and asymptotics. We introduce complex time in order to study corresponding…
The laminar-turbulent boundary S is the set separating initial conditions which relaminarise uneventfully from those which become turbulent. Phase space trajectories on this hypersurface in cylindrical pipe flow look to be chaotic and show…
In this paper, we study existence of boundary blow-up solutions for elliptic equations involving regional fractional Laplacian. We also discuss the optimality of our results.
We establish the first complete classification of finite-time blow-up scenarios for strong solutions to the three-dimensional incompressible Euler equations with surface tension in a bounded domain possessing a closed, moving free boundary.…
In a previous article, \cite{Zill3}, we have established linear inviscid damping for a large class of monotone shear flows in a finite periodic channel and have further shown that boundary effects asymptotically lead to the formation of…
We consider non-local in time semilinear subdiffusion equations on a bounded domain, where the kernel in the integro-differential operator belongs to a large class, which covers many relevant cases from physics applications, in particular…
We have studied a chaotic transport in a two-dimensional periodic vortical flow under a time-dependent perturbation with period T where the global diffusion occurs along the stochastic web. By using the Melnikov method we construct the…
We consider a 2 d.o.f. Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. The ratio of the time derivatives of slow and fast variables is of order $0<\eps \ll 1$. At frozen…
Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou \cite{HouLuo14} proposed a new finite time blow up scenario based on extensive numerical…
We conduct an analysis of the blow up at the triple junction of three fluids in equilibrium. Energy minimizers have been shown to exist in the class of functions of bounded variations, and the classical theory implies that an interface…
In this paper, we consider the finite-time blowup of hollow vortices. These are solutions of the two-dimensional Euler equations for which the fluid domain is the complement of finitely many Jordan curves $\Gamma_1, \ldots, \Gamma_M$, and…
The existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast-slow type having Bogdanov-Takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation…