This paper aims to prove that the collection of all non-iterates, those self-maps which do not have an iterative root of any order, is dense in the set of all self-maps. Establishing the non-existence of iterative roots of a class of noninjective maps, we prove that the collection of all non-iterates is dense in the set of all self-maps, which has a measure zero set or countable set of discontinuities.

This paper is based on the work [JDE, 2024, 396, 147-171], which gives that the general cubic polynomial differential system has an isochronous global center at the origin if and only if the system can be reduced to the form ẋ=-y+ax^2 , ẏ=x-2axy+2a^2x^3 . In this paper we discuss the monotonicity and the convexity of period function for the system ẋ=-y+ax^2 , ẏ=x-2axy+bx^3 , a more general form with a free coefficient b of the cubic term. We prove that the period function is strictly decreasing for b>2a^2 , isochronous for b=2a^2 and strictly increasing for b<2a^2 . Furthermore, we prove that the period function is convex for b 2a^2.

It is known in control theory that the problem how an interconnection of ISS (abbreviation of input-to-state stability) subsystems remains ISS leads to a cycle condition, which is a set of inequalities involving the composition of unbounded Kamke functions. In this paper we convert the problem on those inequalities to functional equations and obtain 𝒦_∞ solutions by finding fixed points in a complete metric space or a locally convex topological linear space. We also give 𝒦_∞ solutions by piecewise construction. Our results can be applied to discussing ISS of interconnection.

Since the Cantor invariant set of Smale's horse-shoe map was proved to be caused by transversally homoclinic intersection, great attention was paid to bifurcations of homoclinic or heteroclinic loops. After the Melnikov method was used effectively for planar systems, efforts were made to degenerate homoclinic or heteroclinic loops in higher dimensional spaces. Known results are concerning homoclinic solutions arising from a nondegenerate or degenerate homoclinic solution. In this paper we discuss multiple homoclinic solutions bifurcated from a degenerate heteroclinic loop in singular perturbation. Singular perturbation prevents from using the Lyapunov-Schmidt reduction because of no differentiability in the parameter. For homoclinic loops we need to apply the cut-connect technique to the unperturbed heteroclinic loops, each branch of which may have a different degenerate degree. We overcome the difficulties to obtain conditions for coexistence of specific numbers of homoclinic solutions. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The existence of increasing convex continuous solutions to polynomial-like iterative equations on compact intervals was investigated in Zhang et al. (2006) and Xu and Zhang (2007) using the Schauder fixed point theorem and the Banach contraction principle; however, all of those results were proved under the severe assumption that the given function is a Lipschitzian map on its domain. In this paper we investigate their convex solutions with a weaker regularity, avoiding the Lipschitz requirement. Using the Knaster-Tarski fixed point theorem, we provide sufficient conditions for the existence of increasing convex semi-continuous solutions to these equations with no Lipschitz assumption on the given function. In particular, this fixed point theorem allows us to deduce further structure of the sets of such solutions to these equations, showing that they are complete lattices and thus proving the existence of minimum and maximum solutions to these equations rather than just the existence or uniqueness of solutions, as proved using the above fixed point theorems.

ABSTRACT In this paper, we introduce the concept of nonuniform ‐exponential dichotomies to formulate an asymptotic behavior weaker than the known nonuniform ‐exponential dichotomy. We discuss the roughness of such dichotomies on the whole line and the half line separately without global invertibility of the evolution family. Our results extend the known results to nonuniform ‐exponential dichotomies in the case that or .

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,α (α>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.

In this paper, without the assumption of bounded growth, we prove that if an evolution family is sufficiently close to one which has an exponential dichotomy, then it also has an exponential dichotomy. It was proved by Henry (1981) in the case where the evolution family has bounded growth. The difficulty caused by the absence of bounded growth, which is an essential condition in Henry’s proof, is resolved by proving a result on system-dependence of extended Green functions. We also extend this result to exponential dichotomies on a half line ℝ+ or ℝ− as well as on any nonempty closed interval of ℝ.

In this paper we discuss the traveling waves of the resonant Schrödinger equation with weak non-locality and dual-power law nonlinearity. We focus on the 6 important types of bounded traveling waves such as solitary wave, kink wave, periodic wave, compacton, peakon and periodic peakon and use a tuple of 6 non-negative integers, called wave tuple, to present the numbers of those waves. We give conditions of parameters for various wave tuples and obtain totally 15 different wave tuples for the equation. This provides 4 coexisting modes and shows that the largest coexisting number is 5.

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.

The non-commutativity of quaternion multiplication presents a fundamental obstacle in analyzing linear quaternion-valued differential equations (QDEs). While the exponential solution for homogeneous linear QDEs by Campos and Mawhin is a cornerstone of the field, its reliance on a restrictive commutativity condition limits it to a narrow, complex-like subclass of functions. This work overcomes this limitation by introducing a novel algorithmic framework that solves the homogeneous initial value problem without any commutativity assumptions. We first demonstrate that the commutativity condition is equivalent to confining the dynamics to a complex-valued subspace. Our primary contribution is a method that systematically reduces the QDE to a solvable real nonlinear differential equation. We further derive closed-form solutions for key non-commutative cases. These results dramatically expand the solvable landscape of linear QDEs, with direct applications in control theory, quantum mechanics, and hypercomplex signal processing, where non-commutative dynamics are intrinsic. We demonstrate the power of our approach by applying it to Robinson’s Quaternion Kinematical Differential Equations and a problem in medical image communication security.

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.

In this paper, we investigate multi-periodicity in a predator–prey system with a fear effect. Overcoming the difficulties in the calculation of focal values and the irreducible decomposition of the algebraic varieties of focal values under some restrictions of biological sense by using the stratified resultant elimination, we find that the weak focus is of multiplicity at most four. Based on this, we identify conditions for the occurrence of exactly one, two, or three small cycles from Hopf bifurcations by determining the independence of focal values. Moreover, applying the Poincaré–Bendixson theorem, we also explore large cycles that are periodic orbits different from those arising from Hopf bifurcations. Further, we prove the existence of the global attractor and obtain its structure by integrating all results about the system. Our work indicates that there are several ways of coexistence for the predator and prey, characterized by monostability, bistability of node–cycle type, and bistability of cycle–cycle type.

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.

We study solutions with different periods arising from a degenerate heteroclinic loop of type (d(1), d(2)) under a single parametric perturbation. Although the existence of periodic solutions from homoclinic and heteroclinic bifurcations was well established in literature, the related multiplicity problem, i.e., determining the number of linearly independent bifurcating periodic solutions, is also interesting and needs to be investigated further. In this paper, we prove that d(1) + d(2) periodic solutions can arise via such a perturbation and give conditions for any integer n (1 < n < d(1) + d(2)) under which n periodic solutions with different periods coexist.

A Zhang-Yang's open question reads: Does a PM function F with height H(F) = 1 have an iterative root f of order n <= N(F) + 1 if the 'characteristic endpoints condition' is not satisfied? This question was answered in the case that F is strictly increasing on its characteristic interval K(F). However, a more difficult case is that F is strictly decreasing on K(F). In this paper we discuss the decreasing case, giving existence of f of order n< N(F) with H(f) = n and of order n <= N(F) + 1 with H(f) = n -1. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Known results show that, with a 9-angular switching boundary for 9 is an element of (0, pi], a planar piecewise linear differential system formed by two Hamiltonian linear subsystems has no crossing algebraic limit cycles of type I, i.e., those cycles crossing one of the two sides of the 9-angular switching boundary twice only, and at most two crossing algebraic limit cycles of type II, i.e., those cycles crossing both sides of the 9-angular switching boundary once separately. In this paper, using the Chebyshev theory and Descartes' rule to overcome difficulties in applying Gr & ouml;bner basis to solve polynomial systems, we study the number of crossing algebraic limit cycles for such a piecewise linear system having a Hamiltonian sub-system and a non-Hamiltonian sub-system. We prove that the maximum number of type I is one and the lower and upper bounds of the maximum number of type II are five and seven, respectively, and show the coexistence of type I and type II, which implies that a lower bound for the maximum number of all crossing algebraic limit cycles is six. (c) 2025 Published by Elsevier Inc.

Normal form is an important tool in the study of bifurcations but, unlike those for smooth differential systems, normal forms for piecewise-smooth systems have difficulty with the near-identity transformation, which is constructed piecewise but needs to be homeomorphic. Such a difficulty of homeomorphism was encountered when we simultaneously normalized two matrices and obtained the second-order normal form near an equilibrium of FF type for piecewise-smooth monodromic systems. In this article, we establish normal forms for piecewise-smooth monodromic systems near an equilibrium of FF, FP, or PP type. We overcome the difficulty of homeomorphism and generalize both from second-order to any given order and from FF type to all FF, FP, and PP types. This new method can be used to compute Lyapunov constants and discuss degenerate Hopf bifurcations for piecewise-smooth systems.

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.

It was found that a function with exactly one discontinuity may have a continuous iterate of second order, indicating that a discontinuity may be repaired to be a continuous one by its adjacent pair of functions of second order, called second order sui-repair. If a function has more than one discontinuities, examples show that some discontinuities may be repaired to be continuous ones by the other’s adjacent pair of functions of second order, called second order C^0 homi-repair. In this paper we investigate second order C^0 homi-repairs of removable and jumping discontinuities for functions having more than one but finitely many discontinuities. We give necessary and sufficient conditions for removable and jumping discontinuities to be C^0 repaired by the second order iteration.