20 problems

Consider realizations of collections of digraphs by the construction described in the paper, with respective substructure realizations $cmathbf{N}_j$. Conjecture on stability of ot…

Let $cmathbf{N}_k$ be a subnetwork arising in a realization constructed by the simplex method. Stability conjecture for the subnetworks. The subnetworks $cmathbf{N}_k$ usually poss…

Let a network be created by the simplex method, and suppose it contains at least one equilibrium with a higher-dimensional unstable manifold, of dimension at least $2$. The generat…

Consider the two-dimensional RPSSL network and its two-dimensional unstable manifold. Two-dimensional RPSSL visibility conjecture. The two-dimensional RPSSL network is visible, bec…

The one-dimensional RPSSL network has trajectories with irregular visits to all parts of the network, together with rare excursions away from the network. Indefinite-excursion conj…

Extension conjecture. The result proved in this work should hold for any attracting and clean heteroclinic network whose connections are one-dimensional.

Gaussian scaling-limit conjecture. The random vector $\sqrt{D_{\varepsilon}}X_{\varepsilon,t(\varepsilon)}$ converges in distribution to a centered Gaussian vector with independent…

Interesting-dynamics conjecture. Interesting dynamics may be found in the white region at the top of the stability diagram.

Stability-tongues conjecture. The remaining white space between the $A$, $AAB$, and $BB$ regions in the parameter-space figure is completely filled with overlapping stability tongu…

Infinite-sausages conjecture. The grey (uncoloured) region is actually filled with infinitely many sausages corresponding to different patterns.

Let $f_\gamma$ be the one-parameter family of periodically perturbed forced equations … with an attracting heteroclinic network $\Gamma$, and let $\mathcal{F}_\gamma$ be the return…

Consider spiral waves in the cyclic-dominance reaction–diffusion system, and let the coexistence equilibrium point be the spatially homogeneous equilibrium with…

Generalized realization conjecture. The conclusion of Theorem holds even for directed graphs $G$ that may contain $2$-cycles and $\Delta$-cliques.

Consider the parameter-dependent dynamical system and heteroclinic network described in the paper, with the assumptions of Theorem governing the stability indices of its two hetero…

Consider an embedding heteroclinic network with transverse eigenvalues associated to its heteroclinic connections. Ashwin–Post conjecture. A sufficient, but not necessary, conditio…

Consider a heteroclinic network with lift-off effects along sequences of connections, and suppose that memory effects are constrained by the contracting and expanding eigenvalues.…

Let $\mathcal{G}$ be the graph realised by a heteroclinic network $X$, let $\tilde{X}$ be a larger network containing $X$, and let $\zeta>0$ denote the noise amplitude. Cylinder-co…

Consider a heteroclinic network with contracting, expanding, transverse, and radial eigenvalues at its equilibria. Eigenvalue stability conjecture. A sufficient, but not necessary,…

Let $X$ be the constructed heteroclinic network, and let $\zeta>0$ denote the noise amplitude. Consider typical trajectories of the corresponding noisy system and the equilibria an…

Let $X$ be the constructed heteroclinic network and let $\tilde{X}$ be a larger network containing $X$. The parameter values determine the stability properties of these networks. A…