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Weinian Zhang

Publications and source records attributed to Weinian Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Admissible characterization in delay equations and robustness of exponential dichotomies against small-delay perturbations

Robustness of exponential dichotomies against small-delay perturbations presents a fundamental obstacle: the lack of an effective admissible characterization in nonautonomous delay equations. Considerable efforts have been devoted to obtaining such a characterization for differential equations in Banach spaces. However, even unlike ordinary differential equations, the variation of constants formula for delay equations requires extending the phase space to a space of discontinuous functions, and the dependence on the past states leads to another challenge in establishing exponential dichotomy via admissibility, i.e., constructing a suitable admissible pair and then estimating the growth/decay rates along the unstable/stable subspaces via the admissibility property. In this paper, we provide an admissible characterization based on two pairs of Banach spaces that yields explicit dichotomy exponents. Through this characterization and an operator perturbation method, we prove that small-delay perturbations preserve exponential dichotomies.

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arXiv2026

Differentiable normal linearization of partially hyperbolic dynamical systems

A result on $C^0$ linearization which is differentiable at the hyperbolic fixed point is known. In this paper, we further investigate a partially hyperbolic diffeomorphism $F$ to find a local $C^0$ conjugacy, which is $C^1$ on the center manifold, to linearize the hyperbolic component (normal to the center direction) and obtain its Takens' normal form. Our result is optimal, as it needs no non-resonant condition usually required for smooth conjugacy (e.g., as in the Takens' theorem) and the $C^{1,\alpha}$ $(\alpha>0)$ smoothness condition is sharp. For the proof, the center direction obstructs the decoupling of $F$ as the stable and unstable foliations do not intersect. We overcome this difficulty via a semi-decoupling method only with the unstable foliation, where a modified Lyapunov-Perron equation needs to be established along the center direction. Subsequent issues of cocycle reduction and differentiable linearization for an expansive fiber-preserving mapping are then addressed by the Whitney's extension theory and a lifting technique, respectively. In the local context, our result improves the result of $C^0$ normal linearization by [C. Pugh and M. Shub, Invent. Math., 10 (1970): 187-198] to a differentiable one.

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arXiv2026

Global center of polynomial Newton system and its non-isochronicity

Using a new compactification (toroidal compactification) and desingularization, we obtain a complete characterization of monodromy at infinity for polynomial Newton system of arbitrary degree, in which we establish an equivalence between the monodromy and the non-existence of 1/2-fractional formal invariant curves. Combining the complete characterization with either Darboux integrability or algebraic reducibility of local centers, we obtain conditions for all cases of global center. Furthermore, investigating the asymptotic behavior of the period function of orbits near infinity, we prove the non-isochronicity for the global center, which consequently solves an open problem proposed by Conti.

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arXiv2025

Sequential dichotomies and uniformities for a neutral equation

Sequential dichotomies of general delay equations are not uniform, which was proved two decades ago. This however reminds whether the countably infinite many dichotomies of a neutral equation have the sequential uniformity. In this paper, considering a scalar neutral equation, we give a negative answer and prove that the series of the projections of dichotomies is divergent.

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arXiv2025

Normal forms of piecewise-smooth systems with a monodromic singular point

Normal form theory is developed deeply for planar smooth systems but has few results for piecewise-smooth systems because difficulties arise from continuity of the near-identity transformation, which is constructed piecewise. In this paper, we overcome the difficulties to study normal forms for piecewise-smooth systems with FF, FP, or PP equilibrium and obtain explicit any-order normal forms by finding piecewise-analytic homeomorphisms and deriving a new normal form for analytic systems. Our theorems of normal forms not only generalize previous results from second-order to any-order, from FF type to all FF, FP, PP types, but also provide a new method to compute Lyapunov constants, which are applied to solve the center problem and any-order Hopf bifurcations of piecewise-smooth systems.

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arXiv2024

An equation in nonlinear combination of iterates

In this paper we deal with an equation in nonlinear combination of iterates. Although it can be reduced by the logarithm conjugacy to a form for application of Schauder's or Banach's fixed point theorems, a difficulty called Zero Problem is encountered for continuous solutions because the domain does not contain $0$. So we consider solutions with weaker regularity, using the Knaster-Tarski fixed point theorem for complete lattices to give order-preserving solutions. Then we give semi-continuous solutions and integrable solutions.

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arXiv2023

Orbits and tsectors in irregular exceptional directions of full-null degenerate singular point

Near full-null degenerate singular points of analytic vector fields, asymptotic behaviors of orbits are not given by eigenvectors but totally decided by nonlinearities. Especially, in the case of high full-null degeneracy, i.e., the lowest degree of nonlinearities is high, such a singular point may have irregular exceptional directions and the blow-up technique can be hardly applied, which leaves a problem how to determine numbers of orbits and (elliptic, hyperbolic and parabolic) tangential sectors in this case. In this paper we work on this problem. Using Newton polygons to decompose nonlinearities into principal parts and remainder parts, we convert the problem to the numbers of nonzero real roots of edge-polynomials of principal parts. Computing Newton polygons for multiplication and differentiation of analytic functions and giving Newton polygons for addition, which was not found in literatures, we determine semi-definiteness of the Lie-bracket of principal parts and therefore obtain criteria for those numbers.

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arXiv2023

Smooth invariant foliations without a bunching condition and Belitskii's $C^{1}$ linearization for random dynamical systems

Smooth linearization is one of the central themes in the study of dynamical systems. The classical Belitskii's $C^1$ linearization theorem has been widely used in the investigation of dynamical behaviors such as bifurcations, mixing, and chaotic behaviors due to its minimal requirement of partial second order non-resonances and low regularity of systems. In this article, we revisit Belitskii's $C^1$ linearization theorem by taking an approach based on smooth invariant foliations and study this problem for a larger class of dynamical systems ({\it random dynamical systems}). We assumed that the linearized system satisfies the condition of Multiplicative Ergodic Theorem and the associated Lyapunov exponents satisfy Belitskii's partial second order non-resonant conditions. We first establish the existence of $C^{1,\beta}$ stable and unstable foliations without assuming the bunching condition for Lyapunov exponents, then prove a $C^{1,\beta}$ linearization theorem of Belitskii type for random dynamical systems. As a result, we show that the classical Belitskii's $C^1$ linearization theorem for a $C^{2}$ diffeomorphism $F$ indeed holds without assuming all eigenspaces of the linear system $DF(0)$ are invariant under the nonlinear system $F$, a requirement previously imposed by Belitskii in his proof.

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arXiv2022

Description of tempered exponential dichotomies by admissibility with no Lyapunov norms

Tempered exponential dichotomy formulates the nonuniform hyperbolicity for random dynamical systems. It was described by admissibility of a pair of function classes defined with Lyapunov norms, For MET-systems (systems satisfying the assumptions of multiplicative ergodic theorem (abbreviated as MET)), it can be described by admissibility of a pair without a Lyapunov norm. However, it is not known how to choose a suitable Lyapunov norms before a tempered exponential dichotomy is given, and there are examples of random systems which are not MET-systems but have a tempered exponential dichotomy. In this paper we give a description of tempered exponential dichotomy for general random systems, which may not be MET-systems, purely by measurable admissibility of three pairs of function classes with no Lyapunov norms. Further, restricting to the MET-systems, we obtain a simpler description of only one pair with no Lyapunov norms. Finally, we use our results to prove the roughness of tempered exponential dichotomies for parametric random systems and give a H\"older continuous dependence of the associated projections on the parameter.

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arXiv2021

Iteration and iterative equation on lattices

In this paper we investigate iteration of maps on lattices and the corresponding polynomial-like iterative equation. Since a lattice need not have a metric space structure, neither the Schauder fixed point theorem nor the Banach fixed point theorem is available. Using Tarski's fixed point theorem, we prove the existence of order-preserving solutions on convex complete sublattices of Riesz spaces. Further, in $\mathbb{R}^n$ and $\mathbb{R}$, special cases of Riesz space, we discuss upper semi-continuous solutions and integrable solutions respectively. Finally, we indicate more special cases of Riesz space for discussion on the iterative equation.

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arXiv2021

Continuous solutions of an iterative equation with multiplication

Iterative equation is an equality with an unknown function and its iterates. There were not found a result on iterative equations with multiplication of iterates of the unknown function on $\mathbb{R}$. In this paper we use an exponential function to reduce the equation in conjugation to the well-known form of polynomial-like iterative equation, but we encountered two difficulties: the reduction restricts our discussion of the equation to $\mathbb{R}_+$; the reduced polynomial-like iterative equation is defined on the whole $\mathbb{R}$ but known results were given on comapct intervals. We revisit the polynomial-like iterative equation on the whole $\mathbb{R}$ and give existence, uniqueness, stability and construction of continuous solutions of our equation on $\mathbb{R}_+$. Then we technically extend our solutions from $\mathbb{R}_+$ to $\mathbb{R}_-$.

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arXiv2020

Iterative roots of exclusive multifunctions

In this paper we investigate iterative roots of strictly monotone upper semi-continuous multifunctions having finitely many jumps. Known results are concerning roots of order 2 for multifunctions of exact one jump. For the general investigation, we introduce a concept `intensity' to formulate the growth of jumps under iteration and find a class of strictly monotone and upper semi-continuous multifunctions of intensity 1, called exclusive multifunctions, each of which has an absorbing interval. Then we use the absorbing interval to construct iterative roots of order $n$ for those exclusive multifunctions.

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arXiv2019

Smooth Linearization of Nonautonomous Differential Equations with a Nonuniform Dichotomy

In this paper we give a smooth linearization theorem for nonautonomous differential equations with a nonuniform strong exponential dichotomy. In terms of discretized evolution operator with hyperbolic fixed point 0, we formulate its spectrum and then give a spectral bound condition for the linearization of such equations to be simultaneously differentiable at 0 and Hölder continuous near 0. Restricted in the autonomous case, our result is the first one that gives a rigorous proof for simultaneously differentiable and Hölder linearization of hyperbolic systems without any non-resonant conditions.

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arXiv2019

Smooth Linearization of Nonautonomous Difference Equations with a Nonuniform Dichotomy

In this paper we give a smooth linearization theorem for nonautonomous difference equations with a nonuniform strong exponential dichotomy. The linear part of such a nonautonomous difference equation is defined by a sequence of invertible linear operators on $\mathbb{R}^d$. Reducing the linear part to a bounded linear operator on a Banach space, we discuss the spectrum and its spectral gaps. Then we obtain a gap condition for $C^1$ linearization of such a nonautonomous difference equation. We finally extend the result to the infinite dimensional case. Our theorems improve known results even in the case of uniform strong exponential dichotomies.

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arXiv2018

Continuous solutions of a second order iterative equation

In this paper we study the existence of continuous solutions and their constructions for a second order iterative functional equation, which involves iterate of the unknown function and a nonlinear term. Imposing Lipschitz conditions to those given functions, we prove the existence of continuous solutions on the whole $\mathbb{R}$ by applying the contraction principle. In the case without Lipschitz conditions we hardly use the contraction principle, but we construct continuous solutions on $\mathbb{R}$ recursively with a partition of $\mathbb{R}$.

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arXiv2017

Dynamics of epidemic models with asymptomatic infection and seasonal succession

In this paper, we consider a compartmental SIRS epidemic model with asymptomatic infection and seasonal succession, which is a periodic discontinuous differential system. The basic reproduction number $\mathcal{R}_0$ is defined and valuated directly for this model, and the uniformly persistent of the disease and threshold dynamics are obtained. Specially, global dynamics of the model without seasonal force are studied. It is shown that the model has only a disease-free equilibrium which is globally stable if $\mathcal{R}_0\le 1$, and as $\mathcal{R}_0>1$ the disease-free equilibrium is unstable and the model has an endemic equilibrium, which is globally stable if the recovering rates of asymptomatic infective and symptomatic infective are close. These theoretical results provide an intuitive basis for understanding that the asymptomatic infective individuals and the disease seasonal transmission promote the evolution of epidemic, which allow us to predict the outcomes of control strategies during the course of the epidemic.

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arXiv2013

Sharp regularity of linearization for $C^{1,1}$ hyperbolic diffeomorphisms

$C^1$ linearization is of special significance because it preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions. However, $C^1$ smoothness is not enough to guarantee $C^1$ linearization. For $C^{1,1}$ hyperbolic diffeomorphisms on Banach spaces $C^1$ linearization was proved under a gap condition together with a band condition of the spectrum. In this paper, the result of $C^1$ linearization in Banach spaces is strengthened to $C^{1,β}$ linearization with a constant $β>0$ under a weaker band condition by a decomposition with invariant foliations. The weaker band condition allows the spectrum to be a union of more than two but finitely many bands but restricts those bands to be bounded by a number depending on the supremum of contractive spectrum and the infimum of expansive spectrum. Furthermore, we give an estimate for the exponent $β$ and prove that the estimate is sharp in the planar case.

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