Past exam of the topics in applied mathematics course of the University of Oxford / 2010 / B / 6

The Lorenz vector field has constant divergence $\nabla \cdot F=-\sigma -1-b$. Liouville formula for a fundamental matrix for the derivative of the flow therefore givesThis concerns transported three-dimensional phase volume, not the Euclidean distance between each pair of trajectories.

A quasiperiodic orbit with the usual smooth two-frequency torus closure would lie on an invariant embedded two-torus. That torus bounds a solid torus of positive finite volume in ${\mathbb{R}}^3$. Since the vector field is tangent to its boundary, its outward flux is zero, contradicting the divergence theorem, whose volume integral is $-(\sigma +1+b)V<0$. This negative-divergence obstruction to invariant tori excludes such quasiperiodic tori. The argument uses global negative divergence and the enclosed volume; saying only that an attracting torus has zero volume would not exclude it.

At an equilibrium the sum of eigenvalues is $-(\sigma +1+b)<0$, so they cannot all have positive real parts: no equilibrium is a fully repelling source. For a periodic orbit of period $T$, one Floquet multiplier equals one in the flow direction. The product of the other two is $e^{-(\sigma +1+b)T}<1$, so they cannot both have modulus greater than one: no periodic orbit is fully repelling either. Saddle equilibria and saddle periodic orbits remain possible; contraction of volume does not imply contraction in every direction.

The first equilibrium equation gives $y=x$, and the third gives $z=x^2/b$. The second then becomes $x(r-1-x^2/b)=0$. Hence the equilibria areAt the origin, one eigenvalue is $-b$ and the remaining two solve ${\lambda }^2+(\sigma +1)\lambda +\sigma (1-r)=0$. Their discriminant is $(\sigma -1{)}^2+4\sigma r>0$. For $r<1$ both are negative, so the origin is linearly stable, with all eigenvalues real and negative. For $r>1$ one is positive and one negative, so it is a saddle equilibrium with one unstable and two stable directions. At $r=1$ its eigenvalues are $0,-(\sigma +1),-b$: linearization has a neutral direction and cannot alone prove nonlinear stability. Given the source's global attraction for $r<1$, there are no nontrivial limit cycles or persistent chaotic invariant dynamics in that regime; all initial states tend to the origin.

To establish the local pitchfork, one needs the simple eigenvalue to cross zero, the two symmetry-related branches to exist and acquire the expected local stability, and nondegeneracy of the nonlinear coefficient. The derivative of the crossing eigenvalue is $\sigma /(\sigma +1)>0$. At $C_{\pm }$, the characteristic polynomial isFor $r>1$ sufficiently close to one, all coefficients are positive and $(\sigma +b+1)b(\sigma +r)>2\sigma b(r-1)$, so the cubic Routh-Hurwitz criterion makes both new equilibria stable. Together with the symmetry $(x,y,z)\mapsto (-x,-y,z)$ and the square-root branches, this yields the supercritical pitchfork. A local centre-manifold coordinate proportional to $x$ has leading normal form $\dot{q}=\frac{\sigma }{\sigma +1}(r-1)q-\frac{\sigma }{b(\sigma +1)}{q}^3+\cdots$, explicitly displaying the nonzero stabilizing cubic coefficient. At $r=1$ that cubic also gives weak local nonlinear attraction along the centre direction.

A useful operational definition of chaos for a deterministic flow is bounded, nonperiodic long-time dynamics on an invariant set with sensitive dependence on initial conditions: arbitrarily close starting states can separate by a fixed macroscopic amount. Boundedness distinguishes recurrent irregular dynamics from simple exponential escape; aperiodicity excludes a limit cycle; sensitivity distinguishes chaotic unpredictability from a regular quasiperiodic signal. These are properties of the asymptotic dynamics, not a short irregular transient. One precise topological convention, Devaney chaos on a compact invariant set, instead specifies topological transitivity, dense periodic orbits, and sensitive dependence. Transitivity expresses indecomposability of the recurrent motion, while dense periodic orbits mean every region contains unstable recurrent structure. The convention should be stated because these formulations are not identical in complete generality.

An attractor is an invariant set approached by trajectories from a neighbourhood or positive-measure basin, according to the adopted attractor convention. Calling it strange refers to irregular, typically fractal geometry rather than a smooth point, cycle or torus. Strangeness alone does not force chaos. Strange nonchaotic attractors provide counterexamples: geometrically irregular attracting sets can lack positive exponential separation. Primary examples appear in ^[ref]. This geometric convention does not assert that every such example occurs in the autonomous three-variable Lorenz family.

Two complementary numerical diagnostics are to estimate the largest Lyapunov exponent and to construct a return map. For the first, integrate the variational equation $\dot{\eta }=DF(x(t))\eta$ alongside the Lorenz trajectory, repeatedly normalize $\eta$, and average the logarithms of the accumulated stretch factors after discarding a transient. A positive converged exponent suggests sensitive exponential separation; the full spectrum should include a zero flow exponent and have sum $-(\sigma +1+b)$, providing checks. For the second, record transverse crossings of a Poincaré section or successive maxima of a state variable, then look for the stretching and folding of an aperiodic return set rather than one point or a finite cycle. Nearby-orbit separation or a broadband power spectrum can add evidence, but plots alone do not prove chaos. Repeat with smaller integration step, longer runs and several initial conditions to distinguish numerical error, long transients and coexistence of attractors from persistent dynamics.