Dynamical systems method and surjectivity of nonlinear maps

If F : H → H is a $C_{\mathrm{loc}}^2$-map in a real Hilbert space, sup_{u ∈ B(u0,R)}∥[F′(u)]⁻¹∥ ⩽ m(R), and ${\mathrm{limsup}}_{R>0}\frac{R}{m(R)}=\infty$, then F is surjective.     

A note on stronger forms of sensitivity for dynamical systems

Let (X,d) be a compact metric space and (κ(X),d_H) be the space of all non-empty compact subsets of X equipped with the Hausdorff metric d_H. The dynamical system (X,f) induces another dynamical system $(\kappa (X)\text{,}\overline{f})$, where f:X → X is a continuous map and $\overline{f}:\kappa (X)\to \kappa (X)$ is defined by $\overline{f}(A)=\{f(a):a\in A\}$ for any A ∈ κ(X). In this paper, we introduce the notion of ergodic sensitivity which is a stronger form of sensitivity, and present some sufficient conditions for a dynamical system (X,f) to be ergodically sensitive. Also, it is shown that $\overline{f}$ is syndetically sensitive (resp. multi-sensitive) if and only if f is syndetically sensitive (resp. multi-sensitive). As applications of our results, several examples are given. In particular, it is shown that if a continuous map of a compact metric space is chaotic in the sense of Devaney, then it is ergodically sensitive. Our results improve and extend some existing ones.     

Thirteen limit cycles for a class of Hamiltonian systems under seven-order perturbed terms

In this paper we study the existence, number and distribution of limit cycles of the perturbed Hamiltonian system:$\begin{array}{cc}\hfill & x^{\prime }=4y({\mathit{abx}}^2-{\mathit{by}}^2+1)+\epsilon x\left({\mathit{ux}}^n+{\mathit{vy}}^n-b\frac{\beta +1}{\mu +1}{x}^{\mu }{y}^{\beta }-{\mathit{ux}}^2-\lambda \right)\hfill \\ \hfill & y^{\prime }=4x({\mathit{ax}}^2-{\mathit{aby}}^2-1)+\epsilon y({\mathit{ux}}^n+{\mathit{vy}}^n+{\mathit{bx}}^{\mu }{y}^{\beta }-{\mathit{vy}}^2-\lambda )\hfill \end{array}$where μ + β = n, 0 < a < b < 1, 0 < ε ≪ 1, u, v, λ are the real parameters and n = 2k, k an integer positive.Applying the Abelian integral method [Blows TR, Perko LM. Bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems. SIAM Rev 1994;36:341–76] in the case n = 6 we find that the system can have at least 13 limit cycles.Numerical explorations allow us to draw the distribution of limit cycles.