question: The logistic map at r = 4 has a positive Lyapunov exponent. Does this imply sensitive dependence, and does that imply chaotic behavior?

ground_truth: YES


type: compositional

question: Given that logistic_r4 has a fractal invariant density, does this guarantee the existence of a strange attractor?

ground_truth: YES


type: adversarial

question: A researcher claims the logistic map at r = 4 is random because long-term prediction is impossible. Is this a valid inference?

ground_truth: NO


type: counterfactual

question: If the logistic map’s parameter r were reduced from 4.0 to 2.8, would chaotic behavior persist?

ground_truth: NO


question: If logistic_r4 is chaotic, must it be deterministic, and if deterministic must it be non-random?

ground_truth: YES


type: trap

question: If logistic_r4 is non-linear and chaotic, does this imply that all chaotic systems must be nonlinear?

ground_truth: NO


question: Logistic_r2_8 converges to a stable fixed point. Does this imply a non-positive Lyapunov exponent and absence of chaos?

ground_truth: YES


type: analogy

question: Given logistic_r2_8 is deterministic and non-chaotic, is it logically consistent to say it is unpredictable in the long term?

ground_truth: NO


type: bias

question: Does the nonlinear form of the logistic map imply chaos at r = 2.8?

ground_truth: NO


type: counterfactual

question: If r in logistic_r2_8 were increased gradually, would chaos emerge after crossing the Feigenbaum cascade?

ground_truth: YES


type: trap

question: If logistic_r2_8 is deterministic, must it also possess a strange attractor?

ground_truth: NO


question: Does convergence to a fixed point imply zero sensitivity and thus no chaos?

ground_truth: YES


system_id: rossler

question: The Rössler system has a positive Lyapunov exponent. Does this imply chaos, and does chaos imply deterministic dynamics?

ground_truth: YES


system_id: rossler

type: structural

question: Does the spiral structure of the Rössler attractor guarantee it is a strange attractor?

ground_truth: YES


system_id: rossler

type: fallacy

question: Is it correct to say the Rössler system is random because the attractor appears irregular?

ground_truth: NO


system_id: rossler

type: cf

question: If parameter c in the Rössler system were reduced significantly, could the system transition to periodic behavior?

ground_truth: YES


system_id: rossler

type: analogy

question: If Rössler is chaotic and Lorenz-63 is chaotic, must their attractors share the same topological structure?

ground_truth: NO


system_id: rossler

question: If a system is chaotic, must it exhibit sensitive dependence and a non-periodic attractor?

ground_truth: YES


question: Lorenz-96 with F = 8 exhibits high-dimensional chaos. Does this imply multiple positive Lyapunov exponents and sensitive dependence?

ground_truth: YES


type: analogy

question: Does the presence of high dimensionality guarantee that Lorenz-96 has a strange attractor similar to Lorenz-63?

ground_truth: NO


type: fallacy

question: Because Lorenz-96 looks extremely irregular, is it correct to classify it as random?

ground_truth: NO


type: cf

question: If the forcing F in Lorenz-96 were reduced from 8 to 2, would chaotic behavior likely disappear?

ground_truth: YES


type: structural

question: Is it correct to say that Lorenz-96 is chaotic solely because it is nonlinear?

ground_truth: NO


question: If Lorenz-96 is chaotic, must it still be statistically predictable at large ensemble scales?

ground_truth: YES


system_id: vdp

question: The Van der Pol oscillator has a stable limit cycle. Does this imply periodic behavior and absence of chaos?

ground_truth: YES


system_id: vdp

type: analogy

question: Given that Van der Pol is nonlinear, must it exhibit chaos?

ground_truth: NO


system_id: vdp

type: cf

question: If the parameter μ in the Van der Pol oscillator were increased enormously, would chaos appear in this low-dimensional system?

ground_truth: NO


system_id: vdp

type: trap

question: If a system is periodic, must it be predictable in the long term?

ground_truth: YES


system_id: vdp

question: Does the existence of a limit cycle imply zero sensitivity and thus absence of chaos?

ground_truth: YES


system_id: vdp

type: adversarial

question: The Van der Pol oscillator shows complex nonlinear oscillations. Is this evidence of chaotic behavior?

ground_truth: NO


question: Both the Hénon map and the Baker's map are chaotic. Does this imply they have equivalent attractor geometries?

ground_truth: NO


question: The Arnold cat map is linear but chaotic. Does this falsify the claim that nonlinearity is required for chaos?

ground_truth: YES


question: Does deterministic unpredictability in Lorenz-63 logically imply randomness in the system's governing laws?

ground_truth: NO


question: If two systems both have strange attractors, must they share the same Lyapunov spectrum structure?

ground_truth: NO


question: Does the existence of chaotic PDEs imply that all nonlinear PDEs are chaotic?

ground_truth: NO


question: If a system is chaotic, must every subsystem or projection of it also be chaotic?

ground_truth: NO


question: If a system has a strange attractor, it is chaotic. If chaotic, it has a positive Lyapunov exponent. If positive Lyapunov exponent, long-term pointwise prediction fails. Does the initial statement imply the final one?

ground_truth: YES


question: If a system is deterministic and has a stable limit cycle, can it still be chaotic?

ground_truth: NO


question: If a system is chaotic, must it be statistically predictable and yet pointwise unpredictable?

ground_truth: YES


question: If Lyapunov exponents become non-positive under parameter change, must sensitivity and chaos disappear?

ground_truth: YES


question: If damping increases and forcing amplitude decreases simultaneously in the Duffing system, what happens to chaotic behavior? Does it persist, weaken, or disappear?

ground_truth: DISAPPEAR


question: If σ is fixed but ρ is lowered below the Hopf bifurcation threshold, does the Lorenz-63 system transition from chaos to periodic dynamics?

ground_truth: YES


question: If forcing F decreases gradually toward zero, does Lorenz-96 shift from high-dimensional chaos to a stable fixed point regime?

ground_truth: YES


question: If the delay parameter τ is halved repeatedly, does the Mackey–Glass system pass through quasi-periodic or periodic regimes before losing chaos?

ground_truth: YES


type: validity

question: Is it correct to claim that because Lorenz-84, Lorenz-96, and Lorenz-63 share the name 'Lorenz', they must exhibit the same type of attractor?

ground_truth: NO


type: validity

question: Does unpredictability in chaotic systems justify treating them as stochastic for mathematical modeling?

ground_truth: NO


type: validity

question: Given that the Standard Map contains both chaotic seas and invariant tori, is it correct to classify the entire system as chaotic?

ground_truth: NO


type: validity

question: If a PDE supports solitons, does this imply absence of chaos regardless of perturbation?

ground_truth: NO


type: validity

question: If a system exhibits transient chaos but settles into a periodic orbit, should it still be classified as chaotic in steady state?

ground_truth: NO


type: hard

question: Does the existence of multiple positive Lyapunov exponents in Lorenz-96 necessarily imply hyperchaos?

ground_truth: NO


type: hard

question: Is sensitive dependence sufficient to classify a system as chaotic?

ground_truth: NO


type: hard

question: If a system is statistically predictable, does that contradict pointwise unpredictability?

ground_truth: NO


type: hard

question: Does the existence of an invariant measure guarantee chaos?

ground_truth: NO


type: hard

question: If a chaotic system is projected onto a lower-dimensional subspace, can the projection become non-chaotic?

ground_truth: YES


type: hard

question: Does chaos guarantee mixing in the measure-theoretic sense?

ground_truth: NO


type: hard

question: Does a positive Lyapunov exponent guarantee the existence of a strange attractor?

ground_truth: NO


type: hard

question: If a system is linearizable near a fixed point, can it still be globally chaotic?

ground_truth: YES


type: hard

question: Does quasi-periodicity imply zero entropy and absence of chaos?

ground_truth: YES


type: hard

question: If a system's trajectories densely fill a smooth torus, is chaos possible?

ground_truth: NO


type: hard

question: Is non-periodicity alone enough to conclude chaos?

ground_truth: NO