Methods of Nonlinear Analysis 412-1

Winter 2002

Hermann Riecke

Homework Assignment 2

due Friday, February 22, 2002

1. Hopf Bifurcation via Multiple Scales

   Consider the Brusselator model for the reactions
   

   A ® U, (1) U ® P, (2) U ® V, (3) 2U+V ® 3U, (4)

   where A represents a substance that is abundant and therefore not a limiting factor in the reaction and P is a product that is removed instantly. This results in the reaction-diffusion system
   

   ¶tu = Ñ2 u + A - (B+1) u + u2v, (5) ¶tv = D Ñ2 v + B u - u2v (6)

   for the concentrations u and v [,] below). To simplify the calculation take A = 1. In this problem we ignore any spatial dependence.

   1. Perform a linear stability analysis of the Brusselator. Under which conditions is the destabilizing mode of the homogeneous steady state oscillatory?
   2. Use multiple time scales to derive the amplitude equation for the Hopf bifurcation.

2. Normal Form for Takens-Bogdanov Bifurcation

   Consider a dynamical system of the form
   

   ¶tx = y + F(x,y), (7) ¶ty = G(x,y), (8)

   where the functions F(x,y) and G(x,y) are strictly nonlinear, i.e. at least quadratic in x and y, but otherwise arbitrary. Thus, the linear part L consists of a Jordan block (

   0	1
   0	0

   ) (cf. Guckenheimer & Holmes Ch. 7.2, Wiggins Example 2.2.2).

   1. Perform a linear stability analysis of (7,8) and determine all eigenvectors of the linear operator.
   2. Envision a near-identify transformation that simplifies the nonlinear terms as much as possible. The nonlinear terms that cannot be removed by such transformations form a subspace and are invariant under the transformation exp(L^ts), i.e. each of the vectors V º (u,v)^t in this subspace satisfies
      

      e-Lts  V(eLts(x,y)t) = V((x,y)t). (9)

      Use this normal-form symmetry to identify to all orders in x and y which nonlinearities cannot be removed. First determine e^-Lts explicitly. To make use of the invariance condition (9) it is then useful to take the derivative of the invariance condition with respect to s.

   3. Determine explicitly the range of L in
      

      H(2) º { (u(x,y),v(x,y))t | (u(ax,ay),v(ax,ay))t = a2(u(x,y),v(x,y))t}. (10)

      Note that the results of different near-identify transformations differ by multiples of vectors from the range of L. Determine the range of L and use this to show that at the bifurcation point (codimension-2 point) the normal form for the Takens-Bogdanov bifurcation in the absence of any symmetries of the original equations (7,8) can be written to second order as
      

      ¶tx = y, (11) ¶ty = c1 xy + c2 x2. (12)

3. Squares and Stripes in a SH-Model

   Consider the modified Swift-Hohenberg model
   

   ¶t y = Ry- (Ñ2+1)2 y- a 3 y3 + b Ñ·( Ñy (Ñy)2). (13)

   It is a modification of a model derived by Gertsberg and Sivashinsky [] for the description of convection with poorly conducting top and bottom boundaries. With such boundary conditions the critical wavenumber for the onset of convection becomes very small and a systematic expansion in the small wavenumber can be performed. A detailed analysis of the possible planforms in an equation of this type has been performed in [].

   1. Use the method of your choice to derive coupled amplitude equations for the amplitudes of rolls in the x- and in the y-direction that are valid close to threshold, i.e. for R-R_c = O(e²), e << 1. Allow the wavevector (q,p) of the modes to differ slightly from the critical wavevector, (q,p) = (1+eQ,0) and (q,p) = (0,1+eP), respectively.
   2. Calculate all fixed points of the resulting amplitude equations. Determine their stability. What kind of patterns do the fixed points represent?
   3. As a function of a and b, determine all possible phase diagrams, i.e. delineate the regions of existence and of stability of all fixed points in the R-Q-plane with P = 1 fixed.
   4. Sketch the flow in the phase plane in each of the regions identified in *).
   5. Can any persistent dynamics arise in eq.(13)?