59,061 research outputs found

### 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

### The $16$th 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

### 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

### 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

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