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In 1973 Nussbaum proved that certain bounded solutions of autonomous delay-differential equations with analytic nonlinearities are themselves analytic. On the other hand, the two authors of this paper more recently showed that bounded solutions of certain delay-differential equations, again with analytic nonlinearities, can be $$C^\infty $$ smooth, yet not be analytic for certain ranges of the independent variable t. In this paper we extend the 1973 results to obtain analytic solutions of a broader class of delay-differential equations, including a wide variety of nonautonomous equations. Nevertheless, there are still equations with analytic nonlinearities possessing global bounded $$C^\infty $$ solutions for which analyticity is unknown. This is the case, for example, for the equation $$\begin{aligned} y'(t)=g(y(t-1))+\varepsilon \sin (t^2) \end{aligned}$$ where g is analytic and where $$y=0$$ is a hyperbolic equilibrium when $$\varepsilon =0$$ .

We consider a class of compact positive operators $$L:X\rightarrow X$$ given by $$(Lx)(t)=\int ^t_{\eta (t)}x(s)\,ds$$ , acting on the space X of continuous $$2\pi $$ -periodic functions x. Here $$\eta $$ is continuous with $$\eta (t)\le t$$ and $$\eta (t+2\pi )=\eta (t)+2\pi $$ for all $$t\in \mathbf{R}$$ . We obtain necessary and sufficient conditions for the spectral radius of L to be positive, in which case a nonnegative eigensolution to the problem $$\kappa x=Lx$$ exists for some $$\kappa >0$$ (equal to the spectral radius of L) by the Krein–Rutman theorem. If additionally $$\eta $$ is analytic, we study the set $${\mathcal {A}}\subseteq \mathbf{R}$$ of points t at which x is analytic; in general $${\mathcal {A}}$$ is a proper subset of $$\mathbf{R}$$ , although x is $$C^\infty $$ everywhere. Among other results, we obtain conditions under which the complement $${\mathcal {N}}=\mathbf{R}{\setminus }{\mathcal {A}}$$ of $${\mathcal {A}}$$ is a generalized Cantor set, namely, a nonempty closed set with empty interior and no isolated points. The proofs of this and of other such results depend strongly on the dynamical properties of the map $$t\rightarrow \eta (t)$$ .

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $\dot x(t)=-\alpha(t)x(t)-\beta(t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $\delta\beta(t)\ge 0$ with $\delta\in{-1,1}$. Positivity properties are studied, with the result that if $(-1)^k=\delta$ then the $k$-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients $\alpha(t)$ and $\beta(t)$ are periodic of the same period, and $\beta(t)$ satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of $u_0$-positivity of the exterior product is investigated when $\beta(t)$ satisfies a uniform sign condition., Comment: 84 pages

We study the singularly perturbed state-dependent delay-differential equation (⁎) e x ˙ ( t ) = − x ( t ) − k x ( t − r ) , r = r ( x ( t ) ) = 1 + x ( t ) , which is a special case of the equation e x ˙ ( t ) = g ( x ( t ) , x ( t − r ) ) , r = r ( x ( t ) ) . One knows that for every sufficiently small e > 0 , Eq. (⁎) possesses at least one so-called slowly oscillating periodic solution, and moreover, the graph of every such solution approaches a specific sawtooth-like shape as e → 0 . In this paper we obtain higher-order asymptotics of the sawtooth, including the location of the minimum and maximum of the solution with the form of the solution near these turning points, and as well an asymptotic formula for the period. Using these and other asymptotic formulas, we further show that the solution enjoys the property of superstability, namely, the nontrivial characteristic multipliers are of size O ( e ) for small e. This stability property implies that this solution is unique among all slowly oscillating periodic solutions, again for small e.

Two homogeneous measures of noncompactness β and γ on an infinite dimensional Banach space X are called “equivalent” if there exist positive constants b and c such that b β(S) ≤ γ(S) ≤ c β(S) for all bounded sets $${S\subset X}$$ . If such constants do not exist, the measures of noncompactness are “inequivalent.” We ask a foundational question which apparently has not previously been considered: For what infinite dimensional Banach spaces do there exist inequivalent measures of noncompactness on X? We provide here the first examples of inequivalent measures of noncompactness. We prove that such inequivalent measures exist if X is a Hilbert space; or if (Ω, Σ, μ) is a general measure space, 1 ≤ p ≤ ∞, and X = L p (Ω, Σ, μ); or if K is a compact Hausdorff space and X = C(K); or if K is a compact metric space, 0

If L : Y → Y is a bounded linear map on a Banach space Y, the “radius of the essential spectrum” or “essential spectral radius” ρ(L) of L is well-defined and there are well-known formulas for ρ(L) in terms of measures of noncompactness. Now let $${C \subset D}$$ be complete cones in a normed linear space (X, || · ||) and f : C → C a continuous map which is homogeneous of degree one and preserves the partial ordering induced by D. We prove (see Section 2) that various obvious analogs of the formulas for the essential spectral radius for the case f : C → C have serious defects, even when f is linear on C. We propose (see (3.5)) a definition for ρ C (f), the “cone essential spectral radius of f,” which avoids these difficulties. If $${{\tilde r}_{C}(f)}$$ denotes the (Bonsall) cone spectral radius of f, we conjecture (see Conjecture 4.1) that if $${\rho_{C}(f) < {\tilde r}_{C}(f)}$$ , then there exists $${u \in C {\backslash} \, \{0\}}$$ with f(u) = ru where r ≔ r C (f). If f satisfies certain additional conditions (for example, if f is a compact perturbation of a map which is linear on C), we obtain the conclusion of the conjecture; but in general we observe (Remark 4.7) that the conjecture is intimately related to old and difficult conjectures in asymptotic fixed point theory. In Section 5 we briefly discuss extensions of generalized max-plus operators which were our original motivation and for which Conjecture 4.1 is already nontrivial.

We study traveling waves for reaction diffusion equations on the spatially discrete domain $\Z^2$. The phenomenon of crystallographic pinning occurs when traveling waves become pinned in certain directions despite moving with non-zero wave speed in nearby directions. Mallet-Paret has shown that crystallographic pinning occurs for all rational directions, so long as the nonlinearity is close to the sawtooth. In this paper we show that crystallographic pinning holds in the horizontal and vertical directions for bistable nonlinearities which satisfy a specific computable generic condition. The proof is based on dynamical systems. In particular, it relies on an examination of the heteroclinic chains which occur as singular limits of wave profiles on the boundary of the pinning region., Comment: 55 pages

In Sections 2 and 3 of this paper we refine and generalize theorems of Nussbaum (see [42]) concerning the approximate fixed point index and the fixed point index class. In Section 4 we indicate how these results imply a wide variety of asymptotic fixed point theorems. In Section 5 we prove a generalization of the mod p theorem: if p is a prime number, f belongs to the fixed point index class and f satisfies certain natural hypothesis, then the fixed point index of f p is congruent mod p to the fixed point index of f. In Section 6 we give a counterexample to part of an asymptotic fixed point theorem of A. Tromba [55]. Sections 2, 3, and 4 comprise both new and expository material. Sections 5 and 6 comprise new results.

In this paper we examine a class of Eulerian time discretizations for a monotone cyclic feedback system with a time delay; see Mallet-Paret and Sell (1996a, 1996b) for background information. We construct an integer-valued function V for the discrete-time problem. The Main Theorem shows that V is a Lyapunov function, that is, V(xn+1)≤V(x n ) along a solution {x n }∞n=0, where the time steps can be relatively large.

We study the max-plus equation (∗)P+ψ(ξ)=maxξ⩽s⩽M(H(s)+ψ(γ(s))),ξ∈[0,M],where H:[0,M]→(−∞,∞) and γ:[0,M]→[0,M] are given functions. The function ψ:[0,M]→[−∞,∞) and the quantity P are unknown, and are, respectively, an eigenfunction and additive eigenvalue. Eigensolutions ψ are known to describe the asymptotics of certain solutions of singularly perturbed differential equations with state dependent time lags. Under general conditions we prove the existence of a finite set (a basis) of eigensolutions ϕi, for 1⩽i⩽q, with the same eigenvalue P, such that the general solution ψ to (∗) is given byψ(ξ)=(c1+ϕ1(ξ))∨(c2+ϕ2(ξ))∨⋯∨(cq+ϕq(ξ)).Here ci∈[−∞,∞) are arbitrary quantities and ∨ denotes the maximum operator. In many cases q=1 so the solution ψ is unique up to an additive constant.

We consider a class of singularly perturbed delay-differential equations of the formεẋ(t)=f(x(t),x(t−r)),where r=r(x(t)) is a state-dependent delay. We study the asymptotic shape, as ε→0, of slowly oscillating periodic solutions. In particular, we show that the limiting shape of such solutions can be explicitly described by the solution of a pair of so-called max-plus equations. We are able thereby to characterize both the regular parts of the solution graph and the internal transition layers arising from the singular perturbation structure.

We study the nonlinear eigenvalue problem $f(x) = \lambda x$ for a class of maps $f: K\to K$ which are homogeneous of degree one and order-preserving, where $K\subset X$ is a closed convex cone in a Banach space X. Solutions are obtained, in part, using a theory of the "cone spectral radius" which we develop. Principal technical tools are the generalized measure of noncompactness and related degree-theoretic techniques. We apply our results to a class of problems max $\max_{t\in J(s)} a(s, t)x(t) = \lambda x(s)$ arising from so-called "max-plus operators," where we seek a nonnegative eigenfunction $ x\in C[0, \mu]$ and eigenvalue $\lambda$. Here $J(s) = [\alpha(s), \beta(s)] \subset [0, \mu]$ for $s\in [0, \mu]$, with $a, \alpha$, and $\beta$ given functions, and the function $a$ nonnegative.

We prove a Fredholm alternative theorem for a class of asymptotically hyperbolic linear differential difference equations of mixed type. We also establish the cocycle property and the spectral flow property for such equations, providing an effective means of calculating the Fredholm index. Such systems can arise from equations which describe traveling waves in a spatial lattice.

We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely, lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c≠0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c≠0. Convergence results for solutions are obtained at the singular perturbation limit c → 0.

Neuromuscular reflexes with time-delayed negative feedback, such as the pupil light reflex, have different rates depending on the direction of movement. This asymmetry is modeled by an implicit first-order delay differential equation in which the value of the rate constant depends on the direction of movement. Stability analyses are presented for the cases when the rate is: (1) an increasing and (2) a decreasing function of the direction of movement. It is shown that the stability of equilibria in these dynamical systems depends on whether the rate constant is a decreasing or increasing function. In particular, when the asymmetry has the shape of an increasing step function, it is possible to have stability which is independent of the value of the time delay or the steepness (i.e., gain) of the negative feedback.

In this paper, we study the existence and stability of traveling waves in lattice dynamical systems, in particular, in lattice ordinary differential equations (lattice ODEs) and in coupled map lattices (CMLs). Instead of employing the moving coordinate approach as for partial differential equations, we construct a local coordinate system around a traveling wave solution of a lattice ODE, analogous to the local coordinate system around a periodic solution of an ODE. In this coordinate system the lattice ODE becomes a nonautonomous periodic differential equation, and the traveling wave corresponds to a periodic solution of this equation. We prove the asymptotic stability with asymptotic phase shift of the traveling wave solution under appropriate spectral conditions. We also show the existence of traveling waves in CML's which arise as time-discretizations of lattice ODEs. Finally, we show that these results apply to the discrete Nagumo equation.

We consider the equation $$ \dot x(t)=f(t,x(t),x(\eta(t))) $$ with a variable time-shift $\eta(t)$. Both the nonlinearity $f$ and the shift function $\eta$ are given, and are assumed to be analytic (that is, holomorphic) functions of their arguments. Typically the time-shift represents a delay, namely that $\eta(t)=t-r(t)$ with $r(t)\ge 0$. The main problem considered is to determine when solutions (generally $C^\infty$ and often periodic solutions) of the differential equation are analytic functions of $t$; and more precisely, to determine for a given solution at which values of $t$ it is analytic, and at which values it is not analytic. Both sufficient conditions for analyticity, and also for nonanalyticity, at certain values of $t$ are obtained. It is shown that for some equations there exists a solution which is $C^\infty$ everywhere, and is analytic at certain values of $t$ but is not analytic at other values of $t$. Throughout our analysis, the dynamic properties of the map $t\to \eta(t)$ play a crucial role., Comment: This is identical to the earlier version 1305.0579v1 except for author email addresses added to the cover page

In this paper recent work on the dynamics of lattice differential equations is surveyed. In particular, results on propagation failure and lattice induced anisotropy for traveling wave or plane wave solutions in higher space dimensions spatially discrete bistable reaction–diffusion systems are considered. In addition, analysis of and spatial chaos in the equilibrium states of spatially discrete reaction–diffusion systems are discussed.

For part I see ibid., vol.42, no.10, pp.746-51 (1995). We survey a class of continuous-time lattice dynamical systems, with an idealized nonlinearity. We introduce a class of equilibria called mosaic solutions, which are composed of the elements 1, -1, and 0, placed at each lattice point. A stability criterion for such solutions is given. The spatial entropy h of the set of all such stable solutions is defined, and we study how this quantity varies with parameters. Systems are qualitatively distinguished according to whether h=0 (termed pattern formation), or h>0 (termed spatial chaos). Numerical techniques for calculating h are described. >

For linear scalar parabolic equations such as $$u_t = u_{xx} + a(t,x)u_x + b(t,x)u$$ on a finite interval 0≦x≦π, with various boundary conditions, we obtain canonical Floquet solutions u n (t, x). These solutions are characterized by the property that z(u n (t, x))=n for all teℝ, where z(·) denotes the zero crossing (lap) number of Matano. The coefficients a(t, x) and b(t, x) are not assumed to be periodic in t, but if they are, the solutions u n (t, x) reduce to the standard Floquet solutions. Our results may naturally be expressed in the language of linear skew product flows. In this context, we obtain for each N≧1 an exponential dichotomy between the bundles span {u n (·,·)} =1/ and $$\overline {span} \{ u_n ( \cdot , \cdot )\} _{n = N + 1}^\infty $$ .

SynopsisThe singularly perturbed differential-delay equationis studied for a class of step-function nonlinearities f. We show that in general the discrete systemdoes not mirror the dynamics of (*), even for small ε, but that rather a different systemdoes. Here F is related to, but different from, f, and describes the evolution of transition layers. In this context, we also study the effects of smoothing out the discontinuities of f.

In this paper we begin a study of the differential-delay equation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0Jc9yq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepS0he9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabew7aLjqadI% hagaqbaiaacIcacaWG0bGaaiykaiabg2da9iabgkHiTiaadIhacaGG% OaGaamiDaiaacMcacqGHRaWkcaWGMbGaaiikaiaadIhacaGGOaGaam% iDaiabgkHiTiaadkhacaGGPaGaaiykaiaacYcacaqGGaGaaeiiaiaa% dkhacqGH9aqpcaWGYbGaaiikaiaadIhacaGGOaGaamiDaiaacMcaca% GGPaaaaa!5192! $$\varepsilon x'(t) = - x(t) + f(x(t - r)), r = r(x(t))$$ . We prove the existence of periodic solutions for 0

Numerical simulation of periodic solutions in large arrays of Josephson junctions indicates the existence of periodic solutions where each junction oscillates with the same waveform, but with equal phase lags. These solutions are called ponies on a merry-go-round or POMs for short. The authors prove the existence of POMs in the equations modelling large arrays of Josephson junctions by using global bifurcation techniques. The basic idea is to view the period of the solution and the phase lag as independent parameters and to prove, using a priori estimates, that the synchronous solution (with phase lag set to zero) can be continued to a solution with phase lag equal to (1/N)th of the period, a POM.

We consider the hypercycle system of ODEs, which models the concentration of a set of polynucleotides in a flow reactor. Under general conditions, we prove the omega-limit set of any orbit is either an equilibrium or a periodic orbit. The existence of an orbitally asymptotic stable periodic orbit is shown for a broad class of such systems.

We prove the Poincare-Bendixson theorem for monotone cyclic feedback systems; that is, systems inR n of the form $$x_i = f_i (x_i , x_{i - 1} ), i = 1, 2, ..., n (\bmod n).$$ We apply our results to a variety of models of biological systems.

We discuss recent results in the theory of lattice differential equations (LDE’s), particularly for LDE’s of diffusive type. Broadly speaking, an LDE is an infinite system of ordinary differential equations modeled on an underlying spatial lattice which reflects the underlying geometry of the lattice. Often one obtains an LDE upon discretizing a partial differential equation, however, many LDE’s occur as models in their own right and are not approximations to the continuum limit. Generally here we consider LDE’s composed of local nonlinear dynamics (often of bistable type) coupled with a discrete laplacian. Questions examined are the existence of equilibria (particularly those exhibiting regular patterns), the existence and qualitative properties of traveling waves, and the occurrence of pinning. We also describe in detail the necessary functional analysis associated with functional differential equations of mixed type, which play a central role in the proofs of these results.

In recent papers the authors have studied differential-delay equations $E_\varepsilon $ of the form $\varepsilon \dot x(t) = - x(t) + f(x(t - 1))$. For functions like $f(x) = \mu _1 + \mu _2 \sin (\mu _3 x + \mu _4 )$, such equations arise in optics, while for choices like $f(x) = \mu x^\nu e^{ - x} $ and $f(x) = \mu x^\nu (1 + x^\lambda )^{ - 1} $ and for $x \geqq 0$, the equation has been suggested in physiological models. Under varying hypotheses on f (labeled (I), (II), and (III) below), previous work has given theorems concerning existence and asymptotic properties as $\varepsilon \to 0^ + $ of periodic solutions of $E_\varepsilon $, which oscillate about a value a such that $f(\alpha ) = \alpha $. However, verifying (I), (II), or (III) for specific examples can be difficult. This paper gives general principles that help in verifying (I), (II), or (III), and then applies these results to specific classes of functions of interest.

For scalar equations $$u_t = u_{xx} + f(x, u, u_x )$$ with x e S 1 and f e C 2 we show that the classical theorem of Poincare and Bendixson holds: the ω-limit set of any bounded solution satisfies exactly one of the following alternatives: This is surprising, because the system is genuinely infinite-dimensional.

We illustrate that most existence theorems using degree theory are in principle relatively constructive. The first one presented here is the Brouwer Fixed Point Theorem. Our method is "constructive with probability one" and can be implemented by computer. Other existence theorems are also proved by the same method. The approach is based on a transversality theorem.

A set of partial differential equations are developed describing fluid flow and temperature variation in a thermosyphon with particularly simple external heating. Several exact mathematical results indicate that a Bessel-Fourier expansion should converge rapidly to a solution. Numerical solutions for the time-dependent coefficients of that expansion exhibit a transition to chaos like that shown by the Lorenz equations over a wide range of fluid material parameters.

In this paper, we study two main features of the homotopy curves which we follow when we use the homotopy method for solving the zeros of analytic maps. First, we prove that near the solution the curve behaves nicely. Secondly, we prove that the set of starting points which give smooth homotopy curves is open and dense. The second property is of particular importance in computer implementation.

Let T be a C1 map from an open subset of a separable Hilbert space into the Hilbert space, and Γ a negatively invariant compact set, that is T(Γ) ⊇ Γ. Suppose the derivative of T for x ϵ Γ is a uniform contraction on a subspace of finite codimension. Then the topological dimension of Γ is finite. This result may be used to show that for certain delay differential equations and partial differential equations, any almost periodic solution has only finitely many rationally independent frequencies, thus extending results of Cartwright for ODE's.

The fractal dimension of an attracting torusTkin×Tkis shown to be almost always equal to the Lyapunov dimension as predicted by a previous conjecture. The cases studied here can have several Lyapunov numbers greater than 1 and several less than 1

The singularly perturbed differential-delay equation $$\varepsilon \dot x(t) = - x(t) + f(x(t - 1))$$ is studied. Existence of periodic solutions is shown using a global continuation technique based on degree theory. For small ɛ these solutions are proved to have a square-wave shape, and are related to periodic points of the mapping f:R→R.When f is not monotone the convergence of x(t) to the square-wave typically is not uniform, and resembles the Gibbs phenomenon of Fourier series.

In this paper we show that the scalar reaction diffusion equation \[ u t = ν Δ u + f ( x , u ) , u ∈ R {u_t} = \nu \Delta u + f(x,u),\qquad u \in R \] with x ∈ Ω n ⊂ R n ( n = 2 , 3 ) x \in {\Omega _n} \subset {R^n}\quad (n = 2,3) and with Dirichlet, Neumann, or periodic boundary conditions, has an inertial manifold when (1) the equation is dissipative, and (2) f f is of class C 3 {C^3} and for Ω 3 = ( 0 , 2 π ) 3 {\Omega _3} = {(0,2\pi )^3} or Ω 2 = ( 0 , 2 π / a 1 ) × ( 0 , 2 π / a 2 ) {\Omega _2} = (0,2\pi /{a_1}) \times (0,2\pi /{a_2}) , where a 1 {a_1} and a 2 {a_2} are positive. The proof is based on an (abstract) Invariant Manifold Theorem for flows on a Hilbert space. It is significant that on Ω 3 {\Omega _3} the spectrum of the Laplacian Δ \Delta does not have arbitrary large gaps, as required in other theories of inertial manifolds. Our proof is based on a crucial property of the Schroedinger operator Δ + υ ( x ) \Delta + \upsilon (x) , which is valid only in space dimension n ≤ 3 n \leq 3 . This property says that Δ + υ ( x ) \Delta + \upsilon (x) can be well approximated by the constant coefficient problem Δ + υ ¯ \Delta + \bar \upsilon over large segments of the Hilbert space L 2 ( Ω ) {L^2}(\Omega ) , where υ ¯ = ( vol Ω ) − 1 ∫ Ω υ d x \bar \upsilon = {({\text {vol}}\Omega )^{ - 1}}\int _\Omega {\upsilon \;dx} is the average value of υ \upsilon . We call this property the Principle of Spatial Averaging. The proof that the Schroedinger operator satisfies the Principle of Spatial Averaging on the regions Ω 2 {\Omega _2} and Ω 3 {\Omega _3} described above follows from a gap theorem for finite families of quadratic forms, which we present in an Appendix to this paper.

Transition layers arising from square-wave-like periodic solutions of a singularly perturbed delay differential equation are studied. Such transition layers correspond to heteroclinic orbits connecting a pair of equilibria of an associated system of transition layer equations. Assuming a monotonicity condition in the nonlinearity, we prove these transition layer equations possess a unique heteroclinic orbit, and that this orbit is monotone. The proof involves a global continuation for heteroclinic orbits.