The Lorenz system has three fixed points for r > 1: the origin and two symmetric points C± = (±√(b(r-1)), ±√(b(r-1)), r-1). At r = 28 with σ = 10 and b = 8/3, all three are unstable. Where do trajectories go if all fixed points are unstable?
ATrajectories escape to infinity — with no stable fixed point, nothing can confine them
BTrajectories settle onto a strange attractor — a bounded set with fractal structure that is neither a fixed point nor a periodic orbit
CTrajectories settle into a stable limit cycle that doesn't encircle any of the fixed points
DThe system enters a quasiperiodic state on a torus
Despite all fixed points being unstable, the Lorenz system is dissipative — volumes in phase space contract (the divergence of the flow is -(σ + 1 + b) < 0). This means trajectories are confined to a bounded region even though no individual fixed point attracts them. The trajectories settle onto the Lorenz attractor: a fractal set of measure zero where they loop around C+ and C- in an unpredictable pattern. This is the defining feature of a strange attractor — it attracts trajectories while having zero volume and infinite complexity.
Question 2 Multiple Choice
Lorenz discovered chaos while studying weather. He found that rounding his initial conditions from six decimal places to three produced a completely different trajectory after a short time. This illustrates:
AA bug in his computer code
BThe system being non-deterministic — different runs give different results
CSensitive dependence on initial conditions — the hallmark of chaos, where exponential divergence means even tiny differences in initial conditions lead to completely different outcomes after sufficient time
DNumerical instability in his integration scheme, not a property of the underlying equations
Lorenz's 1963 discovery is the founding moment of chaos theory. His system was deterministic — the same initial conditions always produce the same trajectory. But his rounding (a change of about 0.01%) grew exponentially until the two trajectories were completely unrelated. This is not numerical error (it persists as the integration step shrinks to zero) but a genuine property of the equations: the largest Lyapunov exponent is positive (≈ 0.9), meaning perturbations grow by a factor of e ≈ 2.7 per unit time. After about 30 time units, a difference of 10⁻³ has grown to order 1.
Question 3 True / False
The Lorenz system is symmetric under (x, y, z) → (-x, -y, z). This means that if (x(t), y(t), z(t)) is a solution, then (-x(t), -y(t), z(t)) is also a solution.
TTrue
FFalse
Answer: True
Substituting -x for x and -y for y in the equations: d(-x)/dt = σ(-y - (-x)) = -σ(y - x) = -ẋ ✓; d(-y)/dt = r(-x) - (-y) - (-x)z = -(rx - y - xz) = -ẏ ✓; dz/dt = (-x)(-y) - bz = xy - bz = ż ✓. The equations are invariant. This symmetry explains why the two lobes C+ and C- of the attractor are mirror images. The attractor itself respects this symmetry even though individual trajectories break it — a trajectory might spend more time near C+ than C- at any given moment.
Question 4 Short Answer
Describe the bifurcation sequence of the Lorenz system as r increases from 0, with σ = 10 and b = 8/3.
Think about your answer, then reveal below.
Model answer: At r = 0, the origin is the only fixed point (globally stable). At r = 1, a pitchfork bifurcation creates C+ and C- while the origin becomes unstable. For 1 < r < r_H ≈ 24.74, C± are stable spirals. At r_H, a subcritical Hopf bifurcation makes C± unstable — but the unstable limit cycles created exist for r < r_H, not r > r_H. For r slightly above r_H, the system exhibits transient chaos before settling to C±. At r ≈ 24.06, a homoclinic bifurcation creates the strange attractor, which coexists with stable C± until r_H. For r > r_H ≈ 24.74, the strange attractor is the only attractor. The classic chaotic regime at r = 28 is well beyond this transition.
The Lorenz system's route to chaos is complex and involves several bifurcation types interacting. The subcritical Hopf bifurcation at C± is crucial — it means the transition to chaos is sudden (subcritical), not gradual. The coexistence of the strange attractor with stable fixed points between r ≈ 24.06 and r_H ≈ 24.74 creates a hysteretic region where the system's long-term behavior depends on initial conditions.