Question bank — Chaotic systems — sensitivity to initial conditions, Lyapunov exponents

False — the rules are 100% deterministic; identical starts give identical futures. The apparent randomness comes only from our imperfect knowledge of the start being amplified exponentially.

False — $e^{\lambda t}$ describes the local gap between two nearby points, not the trajectory itself. The path stays on a bounded attractor by repeatedly folding back on itself.

False — pure blow-up like $\dot{x}=x$ is sensitive but unbounded and non-chaotic. Chaos needs SDIC plus boundedness plus mixing.

False — $\lambda <0$ means nearby trajectories converge, so errors die out. That is calm, predictable behaviour: a stable fixed point or limit cycle.

False — since $t_{\text{pred}}\approx \frac{1}{\lambda }\ln (\Delta /{\delta }_0)$, accuracy enters only through a logarithm, so a $1000\times$ better ${\delta }_0$ adds a fixed constant $\frac{1}{\lambda }\ln 1000$ to the horizon, not a factor of 1000.

False — identical starts follow identical deterministic rules and stay locked together forever. Divergence requires a nonzero initial gap ${\delta }_0$.

True — a dissipative flow has ${\sum }_i{\lambda }_i<0$ (volume contracts, per Liouville-type reasoning) while still having ${\lambda }_1>0$ (one stretching direction). Stretch-and-fold onto a strange attractor.

True in principle but they come in $\pm$ pairs summing to zero; volume is conserved so contraction along one direction is exactly matched by expansion along its partner — no net volume loss.

False — the $t\to \infty$ limit averages the local stretching over the whole attractor, so $\lambda$ characterises the system, not any lucky initial condition.

The local rate $f^{\prime }(x)$ varies as $x$ moves along the orbit; you must average it along the trajectory, not freeze it at the starting point. Using one value ignores every later stretch and fold.

It is not cosmetic — the linearisation $f(x+\delta )-f(x)\approx f^{\prime }(x)\delta$ is only valid for infinitesimal $\delta$. Taking ${\delta }_0\to 0$ keeps that approximation honest before averaging.

No — the $2$ is the distance-stretching factor between two nearby trajectories (the map is conjugate to the tent map, which doubles gaps), not a doubling of the population value itself.

Order matters: ${\delta }_0\to 0$ first keeps the separation infinitesimal so linear theory holds; if you let $t\to \infty$ first with finite ${\delta }_0$, the gap saturates on the bounded attractor and the log ratio no longer measures the true exponential rate.

SDIC forbids stable periodic behaviour, but chaotic systems are riddled with unstable periodic orbits (dense periodic orbits) — the trajectory just never settles onto them.

$\lambda =0$ is the marginal case of regular, quasi-periodic, or periodic motion — nearby trajectories stay a bounded, non-growing distance apart. Plenty happens; it is simply predictable.

False conclusion — better models and data push ${\delta }_0$ down and refine $\lambda$, buying a few extra days. The point is only that the logarithmic payoff makes indefinite prediction impossible, not that improvement is worthless.

Determinism is a statement about the rules (they fix the future exactly); predictability is about our knowledge of the present (always uncertain). Chaos magnifies that unavoidable uncertainty exponentially.

A constant relative growth rate — the gap growing by a fixed fraction per unit time — is precisely the defining property of the exponential; taking $\ln$ then divides out to reveal that single rate $\lambda$.

Without a finite region to fold back into, exponential stretching would just send trajectories to infinity (like $\dot{x}=x$). Boundedness forces the stretch-and-fold that mixes phase space and creates genuine chaos.

Any single positive direction is enough to make nearby trajectories diverge exponentially; the largest exponent ${\lambda }_1$ dominates the growth, so ${\lambda }_1>0$ is the on/off switch.

Dissipation contracts total volume ($\sum {\lambda }_i<0$) but that contraction can be shared unevenly: the flow still stretches along one direction (${\lambda }_1>0$) while squeezing harder elsewhere, leaving room for chaos on a strange attractor.

Because $t_{\text{pred}}\approx \frac{1}{\lambda }\ln (\Delta /{\delta }_0)$: doubling the horizon demands shrinking the initial error by a factor $e^{\lambda t}$, an exponential cost that quickly outruns any conceivable sensor.

They mean the trajectory is perpetually near a periodic pattern yet can never lock onto any (they are unstable), so the motion keeps looking almost-regular while endlessly wandering — the recognisable texture of chaos.

$\lambda =0$ — nearby trajectories neither converge nor diverge exponentially; the gap stays bounded and roughly constant, marking regular (non-chaotic) motion.

There $f^{\prime }(x^{\ast })=r(1-2x^{\ast })=0$, so $\ln |f^{\prime }|\to -\infty$: separations are crushed instantly, the strongest possible convergence, and there is no chaos.

No — it is sensitive but unbounded and non-mixing; a single trajectory just flees to infinity. Missing boundedness and mixing disqualifies it.

No — with one continuous variable the trajectory can only move monotonically toward or between fixed points; there is no room to stretch and fold, so continuous 1D flows cannot be chaotic (you need at least three dimensions, or a discrete map).

It is undefined — $\ln (|\delta (t)|/|{\delta }_0|)$ divides by zero. That is why the definition takes the limit ${\delta }_0\to 0$ rather than setting it to zero, capturing infinitesimal separation without ever reaching zero.

Not for that idealized point — sitting precisely on it, it stays forever. But any infinitesimal nudge grows exponentially away, which is exactly the SDIC that seeds chaos everywhere around it.