3.II.14E

Dynamical Systems | Part II, 2007

The Lorenz equations are

x˙=σ(yx)y˙=rxyxzz˙=xybz\begin{aligned} &\dot{x}=\sigma(y-x) \\ &\dot{y}=r x-y-x z \\ &\dot{z}=x y-b z \end{aligned}

where r,σr, \sigma and bb are positive constants and (x,y,z)R3(x, y, z) \in \mathbb{R}^{3}.

(i) Show that the origin is globally asymptotically stable for 0<r<10<r<1 by considering a function V(x,y,z)=12(x2+Ay2+Bz2)V(x, y, z)=\frac{1}{2}\left(x^{2}+A y^{2}+B z^{2}\right) with a suitable choice of constants AA and BB

(ii) State, without proof, the Centre Manifold Theorem.

Show that the fixed point at the origin is nonhyperbolic at r=1r=1. What are the dimensions of the linear stable and (non-extended) centre subspaces at this point?

(iii) Let σ=1\sigma=1 from now on. Make the substitutions u=x+y,v=xyu=x+y, v=x-y and μ=r1\mu=r-1 and derive the resulting equations for u˙,v˙\dot{u}, \dot{v} and z˙\dot{z}.

The extended centre manifold is given by

v=V(u,μ),z=Z(u,μ)v=V(u, \mu), \quad z=Z(u, \mu)

where VV and ZZ can be expanded as power series about u=μ=0u=\mu=0. What is known about VV and ZZ from the Centre Manifold Theorem? Assuming that μ=O(u2)\mu=O\left(u^{2}\right), determine ZZ correct to O(u2)O\left(u^{2}\right) and VV to O(u3)O\left(u^{3}\right). Hence obtain the evolution equation on the extended centre manifold correct to O(u3)O\left(u^{3}\right), and identify the type of bifurcation.

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