Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2012 / 7 / a

A critical point of a dynamical system or equilibrium is a point $(a,b)$ with $f(a,b)=g(a,b)=0$, so the constant function $(x,y)=(a,b)$ is a solution. Put $u=x-a$, $v=y-b$. For a continuously differentiable vector field, its linearization iswith remainder $o(\sqrt{u^2+v^2})$ in the original nonlinear system. For distinct eigenvalues, $M$ is diagonalizable over $\mathbb{C}$, and the linear solutions are combinations of $e^{{\lambda }_{j}t}{w}_j$, where $w_j$ are corresponding eigenvectors. Real parts determine growth or decay; nonzero imaginary parts give rotation. For hyperbolic equilibria, meaning no eigenvalue has zero real part, the Hartman-Grobman theorem transfers the local stability and qualitative orbit pattern to the nonlinear system. A purely imaginary pair is not hyperbolic: it determines a linear centre, while higher-order terms decide whether the nonlinear system actually has a centre.

A stable node hasBoth exponential modes decay as $t\to \infty$, so the equilibrium is asymptotically stable. Generic linear orbits approach tangent to the eigenvector for the less negative eigenvalue; the eigenvector directions themselves are invariant straight lines. There is no rotation because both eigenvalues are real.

An unstable node hasBoth linear modes grow forward in time. Nearby nonconstant orbits move away from the equilibrium, and approach it under reversed time; their local directions come from the two real eigenvectors.

A saddle equilibrium hasThe negative eigenvalue gives a stable manifold direction and the positive eigenvalue an unstable manifold direction. Perturbations along the latter grow, so the equilibrium is unstable even though there are special trajectories approaching it.

A centre equilibrium of the linearized real planar system hasAfter a real invertible linear change of coordinates, the system is a constant angular rotation. Its nonzero orbits are ellipses of period $2\pi /|\omega |$, so it is stable but not asymptotically stable. This is a classification of the linearization: a centre in the linearization need not be a nonlinear centre. In particular, imaginary eigenvalues alone do not prove a nonlinear centre equilibrium, which requires a punctured neighbourhood filled with periodic orbits. Part (c) illustrates exactly this limitation.

A stable spiral has the complex conjugate eigenvaluesIn suitable real coordinates the radius decays like $e^{\alpha t}$ while the angle changes at rate $\beta$. The orbit therefore spirals towards the equilibrium, which is asymptotically stable. This is also called a stable focus.

An unstable spiral hasRotation is accompanied by exponentially increasing radius in suitable real coordinates. Thus nearby nonconstant orbits spiral away, and the equilibrium is unstable. This is also called an unstable focus.