On the Structure of Periodic Eigenvalues of the Vectorial pp-Laplacian

In this paper we will solve an open problem raised by Man\'asevich and Mawhin twenty years ago on the structure of the periodic eigenvalues of the vectorial $p$-Laplacian. This is an Euler-Lagrangian equation on the plane or in higher dimensional Euclidean spaces. The main result obtained is that for any exponent $p$ other than $2$, the vectorial $p$-Laplacian on the plane will admit infinitely many different sequences of periodic eigenvalues with a given period. These sequences of eigenvalues are constructed using the notion of scaling momenta we will introduce. The whole proof is based on the complete integrability of the equivalent Hamiltonian system, the tricky reduction to $2$-dimensional dynamical systems, and a number-theoretical distinguishing between different sequences of eigenvalues. Some numerical simulations to the new sequences of eigenvalues and eigenfunctions will be given. Several further conjectures towards to the panorama of the spectral sets will be imposed.Comment: 35 pages, 10 figure

Orbital stability of smooth solitary waves for the bb-family of Camassa-Holm equations

In this paper, we study the stability of smooth solitary waves for the $b$-family of Camassa-Holm equations. We verify the stability criterion analytically for the general case $b>1$ by the idea of the monotonicity of the period function for planar Hamiltonian systems and show that the smooth solitary waves are orbitally stable, which gives a positive answer to the open problem proposed by Lafortune and Pelinovsky [S. Lafortune, D. E. Pelinovsky, Stability of smooth solitary waves in the $b$-Camassa-Holm equation]

Lyapunov exponent and almost sure asymptotic stability of a stochastic SIRS model

Epidemiological models with bilinear incidence rate usually have an asymptotically stable trivial equilibrium corresponding to the disease-free state, or an asymptotically stable nontrivial equilibrium (i. e. interior equilibrium) corresponding to the endemic state. In this paper, we consider an epidemiological model, which is a SIRS (susceptible-infected-removed-susceptible) model in uenced by random perturbations. We prove that the solutions of the system are positive for all positive initial conditions and that the solutions are global, that is, there is no finite explosion time. We present necessary and suficient condition for the almost sure asymptotic stability of the steady state of the stochastic system

The cyclicity of the period annulus of a reversible quadratic system

We prove that perturbing the periodic annulus of the reversible quadratic polynomial differential system x˙ = y + ax2, y˙ = −x with a ≠ 0 inside the class of all quadratic polynomial differential systems we can obtain at most two limit cycle, including their multiplicities. Since the first integral of the unperturbed system contains an exponential function, the traditional methods can not be applied, except in [6] a computer-assisted method was used. In this paper we provide a method for studying the problem. This is also the first purely mathematical proof of the conjecture formulated by F. Dumortier and R. Roussarie in [5] for q ≤ 2. The method may be used in other problems

The number of limit cycles of Josephson equation

In this paper, the existence and number of non-contractible limit cycles of the Josephson equation $\beta \frac{d^{2}\Phi}{dt^{2}}+(1+\gamma \cos \Phi)\frac{d\Phi}{dt}+\sin \Phi=\alpha$ are studied, where $\phi\in \mathbb S^{1}$ and $(\alpha,\beta,\gamma)\in \mathbb R^{3}$. Concretely, by using some appropriate transformations, we prove that such type of limit cycles are changed to limit cycles of some Abel equation. By developing the methods on limit cycles of Abel equation, we prove that there are at most two non-contractible limit cycles, and the upper bound is sharp. At last, combining with the results of the paper (Chen and Tang, J. Differential Equations, 2020), we show that the sum of the number of contractible and non-contractible limit cycles of the Josephson equation is also at most two, and give the possible configurations of limit cycles when two limit cycles appear.Comment: 25 pages, 15 figure

Exchange graphs of cluster algebras have the non-leaving-face property

The claim in the title is proved