Inverse Jacobian multipliers and Hopf bifurcation on center manifolds

In this paper we consider a class of higher dimensional differential systems in $\mathbb R^n$ which have a two dimensional center manifold at the origin with a pair of pure imaginary eigenvalues. First we characterize the existence of either analytic or $C^\infty$ inverse Jacobian multipliers of the systems around the origin, which is either a center or a focus on the center manifold. Later we study the cyclicity of the system at the origin through Hopf bifurcation by using the vanishing multiplicity of the inverse Jacobian multiplier.Comment: 22. Journal of Differential Equation, 201

Analytic integrable systems: Analytic normalization and embedding flows

In this paper we mainly study the existence of analytic normalization and the normal form of finite dimensional complete analytic integrable dynamical systems. More details, we will prove that any complete analytic integrable diffeomorphism $F(x)=Bx+f(x)$ in $(\mathbb C^n,0)$ with $B$ having eigenvalues not modulus $1$ and $f(x)=O(|x|^2)$ is locally analytically conjugate to its normal form. Meanwhile, we also prove that any complete analytic integrable differential system $\dot x=Ax+f(x)$ in $(\mathbb C^n,0)$ with $A$ having nonzero eigenvalues and $f(x)=O(|x|^2)$ is locally analytically conjugate to its normal form. Furthermore we will prove that any complete analytic integrable diffeomorphism defined on an analytic manifold can be embedded in a complete analytic integrable flow. We note that parts of our results are the improvement of Moser's one in {\it Comm. Pure Appl. Math.} 9$($1956$)$, 673--692 and of Poincar\'e's one in {\it Rendiconti del circolo matematico di Palermo} 5$($1897$)$, 193--239. These results also improve the ones in {\it J. Diff. Eqns.} 244$($2008$)$, 1080--1092 in the sense that the linear part of the systems can be nonhyperbolic, and the one in {\it Math. Res. Lett.} 9$($2002$)$, 217--228 in the way that our paper presents the concrete expression of the normal form in a restricted case.Comment: 32. Journal of Differential Equations, 201

The 1616th Hilbert problem on algebraic limit cycles

For real planar polynomial differential systems there appeared a simple version of the $16$th Hilbert problem on algebraic limit cycles: {\it Is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree $m$?} In [J. Differential Equations, 248(2010), 1401--1409] Llibre, Ram\'irez and Sadovskia solved the problem, providing an exact upper bound, in the case of invariant algebraic curves generic for the vector fields, and they posed the following conjecture: {\it Is $1+(m-1)(m-2)/2$ the maximal number of algebraic limit cycles that a polynomial vector field of degree $m$ can have?} In this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves. This result includes the Llibre {\it et al}\,'s as a special one. For the polynomial vector fields having only non--dicritical invariant algebraic curves we answer the simple version of the 16th Hilbert problem.Comment: 16. Journal Differential Equations, 201

Nonuniform Dichotomy Spectrum and Normal Forms for Nonautonomous Differential Systems

The aim of this paper is to study the normal forms of nonautonomous differential systems. For doing so, we first investigate the nonuniform dichotomy spectrum of the linear evolution operators that admit a nonuniform exponential dichotomy, where the linear evolution operators are defined by nonautonomous differential equations $\dot x=A(t)x$ in $\R^n$. Using the nonuniform dichotomy spectrum we obtain the normal forms of the nonautonomous linear differential equations. Finally we establish the finite jet normal forms of the nonlinear differential systems $\dot x=A(t)x+f(t,x)$ in $\R^n$, which is based on the nonuniform dichotomy spectrum and the normal forms of the nonautonomous linear systems.Comment: 28 page