Multiscale systems

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Multiscale systems

Collaborator: M. Radziunas, D. Rachinskii, K.R. Schneider, M. Wolfrum, S. Yanchuk

Cooperation with: V.F. Butuzov, N.N. Nefedov, A.B. Vasil'eva (Moscow State University, Russia)

Supported by: DFG: Cooperation Project ``Singularly perturbed systems and exchange of stability'' of German and Russian scientists in the framework of the Memorandum of Understanding between DFG and RFFI

Description:

Delayed loss of stability for singularly perturbed systems.

We have studied the phenomenon of delayed loss of stability for singularly perturbed systems of ordinary differential equations in case that the associated autonomous system undergoes a Hopf bifurcation at the zero equilibrium as some parameter changes. In that case, the system ``notices'' that the equilibrium has lost its stability only after some delay. In contrast to the well-known standard situation, we were interested in a class of systems where the linearization of the associated system is independent of the slowly changing parameter. For this class, superlinear positive homogeneous terms in the expansion of the right-hand parts at zero define the bifurcation points and the stability of the equilibrium and the cycles. We have derived simple formulas to estimate the asymptotic delay for the delayed loss of stability phenomenon. More precisely, we found conditions which ensure that the zeros of a simple function $\psi$ defined by the positive homogeneous nonlinear terms are the Hopf bifurcation points of the associated system, the sign of $\psi$ at other points determines the stability of the zero equilibrium, and the asymptotic delay equals the distance between a bifurcation point and a zero of a primitive of $\psi$ .

Alternating contrast structures.

We consider the scalar singularly perturbed parabolic differential equation

$\displaystyle \varepsilon^{2}_{}$ $\displaystyle \left(\vphantom{ \frac{\partial^2 u}{\partial x^2}-\frac{\partial u}{\partial t}}\right.$ $\displaystyle {\frac{{\partial^2 u}}{{\partial x^2}}}$ - $\displaystyle {\frac{{\partial u}}{{\partial t}}}$ $\displaystyle \left.\vphantom{ \frac{\partial^2 u}{\partial x^2}-\frac{\partial u}{\partial t}}\right)$

f (u, x, t), x $\displaystyle \in$ (0, 1), t > 0,

(1)

where $\varepsilon$ is a small parameter, with the initial condition

u(x, 0, $\displaystyle \epsilon$ )

u⁰(x, $\displaystyle \varepsilon$ ) for 0 $\displaystyle \le$ x $\displaystyle \le$ 1,

(2)

and the boundary conditions of Neumann type

$\displaystyle {\frac{{\partial u}}{{dx}}}$ (0, t, $\displaystyle \epsilon$ )

$\displaystyle {\frac{{\partial u}}{{\partial x}}}$ (1, t, $\displaystyle \epsilon$ ) = 0 for t > 0.

(3)

It is well known that the boundary value problem (1), (3) in general has solutions exhibiting for small $\epsilon$ boundary layers (i.e., there are small regions near the boundaries x = 0 and x = 1, where the solutions rapidly change) and/or interior layers (i.e., there are small regions in the interval 0 < x < 1, where the solutions rapidly change). We call solutions of (1), (3), which have only boundary layers pure boundary layer solutions, solutions possessing an interior layer are called contrast structures.

The case that the type of the solution changes with increasing time is called alternating contrast structures, [3].

The focus of this paper is on the analytical investigation of the initial-boundary value problem (1)-(3) with periodic right-hand side in the case that a solution with a step-type interior layer exists, which moves to the boundary x = 1 or x = 0 and changes its type to a pure boundary layer solution when the interior layer arrives at the boundary. The analytical investigations are based on the method of lower and upper solutions. We distinguish three transition cases: (i) Slow transition, (ii) Fast transition, (iii) Fast-slow transition. All cases have also been investigated numerically. The case of slow transition is represented in Figure 1 for a special equation. The solution u(x, t, $\epsilon$ ) is represented for fixed t by a boldface curve.

**Fig. 1:** Slow passage
$\ProjektEPSbildNocap{0.7\textwidth}{fig2_ks1.eps}$

References:

D.I. RACHINSKII, K.R. SCHNEIDER, Dynamic Hopf bifurcations generated by nonlinear terms, to appear in: J. Differential Equations.
N.N. NEFEDOV, M. RADZIUNAS, K.R. SCHNEIDER, A.B. VASIL'EVA, Change of the type of contrast structures in parabolic Neumann problems, WIAS Preprint no. 984, 2004, to appear in: Comput. Math. Math. Phys., 45 (1) (2005).
A.B. VASIL'EVA, A.P. PETROV, A.A. PLOTNIKOV, On the theory of alternating contrast structures, Comput. Math. Math. Phys., 38 (1998), pp. 1534-1543.
M. WOLFRUM, J. HÄRTERICH, Describing a class of global attractors via symbol sequences, Discrete Contin. Dyn. Syst., 12 (2005), pp. 531-554.