90 problems

Kaplan–Yorke conjecture. The quantity $d_{KY}$ coincides with the Hausdorff dimension of any attractor.

Let $S_{\alpha}(l)$ denote the tensor complexity at subsystem size $l$, measured by the $\alpha$-Rényi entropy. Consider tensorized recurrent architectures used for chaotic time-se…

Consider the Hénon map … with parameters $a,b\in\mathbb R$. Hénon's conjecture. For $b=0.3$ and a certain parameter $a$, this map has a chaotic attractor. The source says that this…

A robust chaos conjecture asserts that chaos is a robust, high-probability behavior for high-dimensional, bounded, nonlinear dynamical systems. The paper reports near-unity experim…

Let $f$ be a mapping (neural network) and let $U$ be a bifurcation chain set as in the hyperbolicity violation conjecture. The periodic-window probability conjecture asserts that,…

Let $p$ denote the quantity characterizing the phase-chaotic states, and let the SN line be the saddle-node stability boundary for modulated amplitude waves (MAWs) of spatial perio…

In a dynamical system with invariant tori, consider the regime in which some invariant tori have been destroyed. Chaotic-dynamics conjecture. It is generally conjectured that chaot…

Generic horseshoe conjecture. Generically in the $C^k$ topology, the geodesic flow of $g$ has a horseshoe.

The discussion concerns chaotic dynamics in dynamical systems, with no more specific formal definition given in the source. Smale's conjecture. Chaotic dynamics does not exist. The…

Let a dynamical system vary between the two extreme cases of an ergodic dynamical system and a completely integrable dynamical system. Chaoticity-reduction conjecture. Together wit…

Word-complexity conjecture. The word-complexity is

Word-complexity conjecture. Extrapolating from available data, one has

Consider a dynamical system with equilibria embedded in chaotic attractors, and let $d_L$ denote its Lyapunov dimension, defined by maximizing the Kaplan–Yorke expression over the…

Let a Hamiltonian dynamical system exhibit mixed chaos, with chaotic trajectories and islands of regularity in its phase space. Suppose the system is weakly coupled to a thermal ba…

Let a dynamical system undergo an abrupt transition from a fixed-point regime to a chaotic regime, and consider its invariant measure at the boundary between these regimes. Beck's…

Arnold's conjecture. Beltrami flows can exhibit Lagrangian turbulence and chaotic behavior of trajectories. Several numerical studies support this possibility. This conjecture conc…

Consider the class of two-dimensional piecewise linear discontinuous maps discussed in the paper, whose attractors include weird quasiperiodic attractors (WQAs). Under a parameter…

Fix ${\mathrm e}^+\in{\mathcal E}^+$, $\tau$, and a time sequence ${\mathcal T}^+$, and let $J_{+\infty}=J_{+\infty}^{{\mathrm e}^+}(\tau,{\mathcal T}^+)\subset J_0$ be the compact…

Let ${\mathcal E}^+$ and ${\mathcal E}^-$ be the sets of one-sided binary sequences, let $\tau$ be the phase parameter, and let ${\mathcal T}^+$ and ${\mathcal T}^-$ be the time se…

Consider a planar piecewise-smooth system in one of the four scenarios described by the source, with a non-degenerate zero of the Melnikov function and hence persistence of the hom…

Let ${\mathcal E}^+=\{0,1\}^{{\mathbb Z}^+}$ be the set of one-sided binary sequences, let $\tau\in[0,1]$, and consider the corresponding initial conditions for the planar system.…

The paper considers a piecewise-smooth planar non-autonomous system with four possible geometric scenarios for the local unstable and stable manifolds near a homoclinic trajectory.…

Let $n\geq 1$ and let $\xi$ be a parameter in $\mathcal{R}^{(4)}_n$, where $\mathcal{R}^{(4)}_n$ is the region defined in the source by the corresponding inequalities involving…

Let $\xi$ be a parameter in $\mathcal{R}^{(4)}_0$, where $\mathcal{R}^{(4)}_0$ is the region defined by $\phi_{\rm min}(\xi)>0$, $\phi_{\rm min}(g(\xi))\leq 0$, and $\alpha(\xi)<0$…

Let $\xi$ be a parameter in the first region $\mathcal{R}^{(3)}_0$ of the non-invertible parameter region $\Phi^{(3)}$, where the BCNF is the map under study. Unique-attractor conj…