In a dissipative chaotic system, the Lorenz attractor has Lyapunov exponents (+0.9, 0, -14.6). In a Hamiltonian system, the exponents must come in pairs ±λ. For a two-degree-of-freedom Hamiltonian system restricted to an energy surface, the exponents are (+λ, 0, 0, -λ). Why are there two zero exponents?
AOne zero exponent is from the flow direction (as in any continuous system), and the other is from energy conservation — perturbations along the energy surface neither grow nor shrink because the system can't leave the surface
BBoth zero exponents indicate the system is not truly chaotic
CThe two zeros are an artifact of the Hamiltonian formulation and have no physical meaning
DTwo zero exponents mean the system is quasiperiodic, not chaotic
Every continuous-time system has one zero Lyapunov exponent (perturbations along the flow direction). Hamiltonian systems have an additional zero from each conserved quantity — energy conservation confines motion to a codimension-1 surface, so perturbations normal to this surface don't grow or shrink (they're simply not dynamically accessible). The pairing λ, -λ for the remaining exponents reflects the symplectic structure: expansion in one direction is exactly compensated by contraction in the conjugate direction, preserving phase space volume.
Question 2 Multiple Choice
A Poincare section of a Hamiltonian system shows islands of closed curves surrounded by a sea of scattered dots. The closed curves represent KAM tori, and the scattered dots represent:
ANumerical errors in the simulation
BChaotic orbits that wander ergodically through the region between surviving KAM tori
CUnstable fixed points of the Poincare map
DTransient orbits that will eventually settle onto a KAM torus
In a Hamiltonian system, there are no attractors — orbits don't 'settle' anywhere. The scattered dots are genuine chaotic orbits: a single initial condition, iterated thousands of times on the Poincare section, produces dots that fill the chaotic sea between KAM tori. These orbits have positive Lyapunov exponents and sensitive dependence. The structure is self-similar: zooming into the boundary between the chaotic sea and an island reveals smaller islands, thinner chaotic layers, and further structure at every scale — an infinitely complex fractal boundary.
Question 3 True / False
Arnold diffusion is impossible in Hamiltonian systems with two degrees of freedom but possible in systems with three or more.
TTrue
FFalse
Answer: True
In two degrees of freedom, the energy surface is 3-dimensional and KAM tori are 2-dimensional — they have codimension 1 and divide the energy surface into disconnected regions. Chaotic orbits are trapped between adjacent KAM tori. In three or more degrees of freedom, the energy surface is 5-dimensional (or higher) and KAM tori are n-dimensional — they no longer divide the energy surface. Chaotic orbits can thread through the gaps between tori, slowly drifting across phase space. This Arnold diffusion is extremely slow but fundamentally changes the long-term stability picture.
Question 4 Short Answer
Explain why Hamiltonian chaos doesn't produce strange attractors, and what replaces them.
Think about your answer, then reveal below.
Model answer: Strange attractors require dissipation — volume contraction collapses trajectories onto a fractal set of zero volume. Hamiltonian systems conserve volume (Liouville's theorem), so trajectories can't converge onto a lower-dimensional set. Instead, chaotic orbits fill finite-volume regions of the energy surface ergodically. The chaotic sea has the full dimension of the energy surface (minus the KAM torus barriers). What replaces the attractor is the concept of a chaotic component — a connected region of the energy surface where a single chaotic orbit is dense. The invariant measure is Liouville measure (uniform on the energy surface), not a fractal measure.
This distinction has practical consequences. In dissipative chaos, you can reconstruct the attractor from a single long trajectory (it densely covers the fractal). In Hamiltonian chaos, a single trajectory fills a region but the dimension of that region is the full energy surface dimension, not a fractal. The diagnostics differ: instead of fractal dimension, you characterize Hamiltonian chaos by the fraction of phase space that is chaotic versus regular (the stochasticity parameter).