A three-dimensional dissipative system has Lyapunov exponents (+0.9, 0, -14.6). What does each exponent tell you about the dynamics?
AAll three indicate different rates of attraction to the attractor
BThe positive exponent (+0.9) indicates chaos — exponential divergence of nearby trajectories along one direction. The zero exponent indicates the direction along the flow — neither expanding nor contracting, since nearby points on the same trajectory maintain their separation. The large negative exponent (-14.6) indicates strong contraction, collapsing volumes rapidly.
CThe positive exponent means the system is unstable and trajectories escape to infinity
DThese exponents are inconsistent — a dissipative system cannot have a positive Lyapunov exponent
These are approximately the Lyapunov exponents of the Lorenz system at standard parameters. The positive exponent produces chaos (exponential divergence → sensitive dependence). The zero exponent always exists in continuous flows — it corresponds to perturbations along the direction of motion, which neither grow nor shrink. The strongly negative exponent provides the contraction that keeps the attractor thin (low fractal dimension). The sum λ₁ + λ₂ + λ₃ ≈ -13.7 < 0 confirms dissipation: phase space volumes contract exponentially.
Question 2 Multiple Choice
A researcher computes the largest Lyapunov exponent of a system and finds λ₁ = 0. Does this rule out chaos?
AYes — chaos requires λ₁ > 0 by definition
BNo — λ₁ = 0 is consistent with quasiperiodic behavior on a torus, which some consider chaotic
Cλ₁ = 0 indicates a limit cycle, which is periodic and thus not chaotic. It rules out chaos.
DA and C are both correct descriptions: λ₁ = 0 implies periodic or quasiperiodic behavior, neither of which is chaotic
A largest Lyapunov exponent of exactly zero means nearby trajectories neither diverge nor converge on average. For a continuous flow, this indicates a stable periodic orbit (one zero exponent from the flow direction, all others negative) or quasiperiodic motion on a torus (two zero exponents). Neither is chaotic — both are predictable indefinitely. Chaos requires λ₁ > 0. However, at the boundary (λ₁ → 0⁺), one finds intermittency and the transition to chaos, which is an active research topic.
Question 3 True / False
Doubling the precision of initial condition measurements extends the prediction horizon of a chaotic system by a fixed additive amount, not by doubling the horizon.
TTrue
FFalse
Answer: True
If the initial error is δ₀ and the largest Lyapunov exponent is λ₁, the error grows as δ(t) ≈ δ₀e^{λ₁t}. Prediction fails when δ(t) reaches a threshold Δ, giving horizon t* = (1/λ₁)ln(Δ/δ₀). Halving δ₀ changes the horizon by (1/λ₁)ln(2) — a fixed additive amount, regardless of how precise you already were. Going from 3 to 6 decimal places adds the same amount of prediction time as going from 6 to 9 decimal places. This logarithmic dependence on precision is why chaos imposes a fundamental prediction limit.
Question 4 Short Answer
How do the Lyapunov exponents relate to the rate of information loss in a chaotic system?
Think about your answer, then reveal below.
Model answer: The positive Lyapunov exponents determine the rate of information loss. The sum of all positive exponents equals the Kolmogorov-Sinai entropy, which measures the rate (in bits per unit time) at which the system generates new information (equivalently, destroys information about initial conditions). A system with a larger positive Lyapunov exponent loses predictability faster. This connects dynamical systems theory to information theory: chaos is, quantitatively, the exponential generation of information about which trajectory the system is on.
The Pesin identity (for certain well-behaved systems) states h_KS = Σ(λᵢ > 0) λᵢ. For the Lorenz system with λ₁ ≈ 0.9 bits/time, the system generates about 0.9 bits of new information per unit time. This means that to maintain a prediction of fixed accuracy, you must continuously supply 0.9 bits/time of new measurement data. If you stop measuring, your prediction degrades exponentially.