Jianguo Si

Analytic solutions of an iterative functional differential equation

Abstract This paper is concerned with an iterative functional differential equation x ′( z ) = x ( m ) ( z ), where x ( m ) ( z ) = x ( x (… x ( z ))) is the m th iterate of the function x ( z ). By constructing a convergent power series solution y ( z ) of a companion equation of the form αy ′( αz ) = y ′( z ) y ( α m z ), analytic solutions of the form y ( αy −1 ( z )) for the original differential equation are obtained.

Analytic Solutions of an Iterative Functional Differential Equation which may Violate the Diophantine Condition

Existence of analytic solutions of an iterative functional differential equation is studied by reducing locally the equation to another functional differential equation without iteration of the unknown function. As done previously we first discuss the case that the constant fu2009(0) is not on the unit circle in C and the case that the constant lies on the circle but fulfills the Diophantine condition. Furthermore, we investigate the case that the constant is a unit root, which violates the Diophantine condition.

Analytic solutions of a nonlinear iterative equation near neutral fixed points and poles

In this paper existence of analytic solutions of a nonlinear iterative equations is studied when given functions are all analytic and when given functions have poles. As well as in many previous works, we reduce this problem to finding analytic solutions of a functional equation without iteration of the unknown function f. For technical reasons, in previous works an indeterminate constant related to the eigenvalue of the linearized f at its fixed point O is required to fulfill the Diophantine condition that O is an irrationally neutral fixed point of f. In this paper the case of rationally neutral fixed points is also discussed, where the Diophantine condition is not required.

Analytic Solutions of a Second-order Functional Differential Equation with a State Dependent Delay

This paper is concerned with a second-order functional differential equation of the form x″(z) = x(az + bx(z)). By constructing a convergent power series solution of an auxiliary equation of the form α2y″ (αz) y′ (z) = αy′(z)y″(z)+(y′(z))3[y(α2z)−ay(αz)], analytic solutions of the form (y(αyt - 1(z)) − az)/b for the original differential equation are obtained.

Analytic solutions of a second-order iterative functional differential equation☆

Abstract This paper is concerned with a second-order iterative functional differential equation x ″( z ) = ( x m ( z )) 2 . Its analytic solutions are discussed by locally reducing the equation to another functional differential equation with proportional delays μ 2 y ″(μ z ) y ′( z ) = μ y ′(μ z ) y ″( z ) + [ y ′( z )] 3 [ y (μ m z )] 2 and by constructing a convergent power series solutions for the latter equation.

Analytic invariant curves for a planar map

Abstract In this paper, we study the existence of analytic invariant curves for two-dimensional maps of the form F(x,y) = (x + y, y + G(x) + H(x + y)).

Analytic solutions of a polynomial-like iterative functional equation

In this paper existence of local analytic solutions of a polynomial-like iterative func-tional equation is studied.As well as in previous work,we reduce this problem with the Schrue383der transformation to finding analytic solutions of a functional equation without iteration of the un-known function f.For technical reasons,in previous work the constant α given in the Schrue383der transformation,i.e.,the eigenvalue of the linearized f at its fixed point O,is required to fulfill that α is off the unit circle S1 or lies on the circle with the Diophantine condition.In this paper,we obtain results of analytic solutions in the case of α at resonance,i.e.,at a root of the unity and the case of α near resonance under the Brjuno condition.

Quasi-periodic solutions and stability of the equilibrium for quasi-periodically forced planar reversible and Hamiltonian systems under the Bruno condition

In this paper we investigate the existence of quasi-periodic solutions of non-autonomous two-dimensional reversible and Hamiltonian systems under the Bruno condition. As an application we study the dynamical stability of the trivial solution at the origin of a quasi-periodically forced planar system. Under a mild non-degeneracy condition we give a criterion that is necessary and sufficient for a large class of systems.

Local analytic solutions of a functional differential equation with a deviating argument depending on the state derivative near resonance

This paper is concerned with a functional differential equation x^(^n^)(z)=1x(az+bx^(z)) with a deviating argument depending on the state derivative, where a,b are two complex numbers. We first discuss the existence of analytic solutions for some special cases of the above equation. Then, by reducing the equation with the Schroder transformation to another functional differential equation with proportional delay, an existence theorem is established for analytic solutions of the original equation. For the constant @l given in the Schroder transformation, besides the case 0<|@l|<1, we focus on those @l on the unit circle S^1, i.e., |@l|=1. We discuss not only those @ls at resonance, i.e. at a root of the unity, but also those @l near resonance under the Brjuno condition.

Analytic Solutions of aq-Difference Equation and Applications to Iterative Equations*

In this paper, a nonlinear q-difference equation is investigated in the complex field C for existence of local invertible analytic solutions. We discuss not only in the case but also for Our results can be applied to obtain existence of analytic solutions for some iterative equations.