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We derive a necessary and sufficient condition for Turing instabilities to occur in two-component systems of reaction-diffusion equations with Neumann boundary conditions. We apply this condition to reaction-diffusion systems built from…
We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of…
From the bicovariant first order differential calculus on inhomogeneous Hopf algebra ${\cal B}$ we construct the set of right-invariant Maurer-Cartan one-forms considered as a right-invariant basis of a bicovariant ${\cal B}$-bimodule over…
We investigate Riemannian quantum-geometric structures in semiclassical transport features of two-dimensional multigap topological phases. In particular, we study nonlinear Hall-like bulk electric current responses and, accordingly,…
The Hodge decomposition is a fundamental result in differential geometry and algebraic topology, particularly in the study of differential forms on a Riemannian manifold. Despite extensive research in the past few decades,…
Collision refers to a striking nonlinear interaction in dissipative systems, revealing the particle-like properties of solitons. In dual-wavelength mode-locked fiber lasers, collisions are inherent and periodic. However, how collisions…
Topology of the Generic Hamiltonian Dynamical Systems on the Riemann Surfaces given by the real part of the generic holomorphic 1-forms, is studied. Our approach is based on the notion of Transversal Canonical Basis of Cycles (TCB). This…
The existing paradigm for topological insulators asserts that an energy gap separates conduction and valence bands with opposite topological invariants. Here, we propose that \textit{equal}-energy bands with opposite Chern invariants can be…
In this paper, we consider the nonlinear dynamical behaviors of some tabu leaning neuron models. We first consider a tabu learning single neuron model. By choosing the memory decay rate as a bifurcation parameter, we prove that Hopf…
We develop functoriality for Morse theory, namely, to a pair of Morse-Smale systems and a generic smooth map between the underlying manifolds we associate a chain map between the corresponding Morse complexes, which descends to the correct…
We consider the possible concentration in phase space of a sequence of eigenfunctions (or, more generally, a quasimode) of an operator whose principal symbol has completely integrable Hamilton flow. The semiclassical wavefront set $WF_h$ of…
In this article, we construct infinitely many (small Seifert fibred, hyperbolic and toroidal) rational homology $3$-spheres that admit co-orientable taut foliations, but none with vanishing Euler class. In the context of the $L$-space…
We analyze quantum-geometric bounds on optical weights in topological phases with pairs of bands hosting nontrivial Euler class, a multigap invariant characterizing non-Abelian band topology. We show how the bounds constrain the combined…
Convection in an infinite fluid layer is often modelled by considering a finite box with periodic boundary conditions in the two horizontal directions. The translational invariance of the problem implies that any solution can be translated…
The Hamiltonian Hopf bifurcation has an integrable normal form that describes the passage of the eigenvalues of an equilibrium through the 1: -1 resonance. At the bifurcation the pure imaginary eigenvalues of the elliptic equilibrium turn…
In the present paper the exhaustive topological classification of nonsingular Morse-Smale flows of $n$-manifolds with two limit cycles is presented. Hyperbolicity of periodic orbits implies that among them one is attracting and another is…
Let $F$ be a transversely oriented foliation of codimension 1 on a closed manifold $M$, and let $\phi=\{\phi^t\}$ be a foliated flow on $(M,F)$. Assume the closed orbits of $\phi$ are simple and its preserved leaves are transversely simple.…
The occurrence of a topological phase transition can be demonstrated by a direct observation of a change in the topological invariant. For holographic topological semimetals, a topological Hamiltonian method needs to be employed to…
A transition state for a Hamiltonian system is a closed, invariant, oriented, codimension-2 submanifold of an energy-level that can be spanned by two compact codimension-1 surfaces of unidirectional flux whose union, called a dividing…
The presence of slow-fast Hopf (or singular Hopf) points in slow-fast systems in the plane is often deduced from the shape of a vector field brought into normal form. It can however be quite cumbersome to put a system in normal form. In the…