Linearized Stability for Semiflows Generated by a Class of Neutral Equations,with Applications to State-Dependent Delays

We prove a principle of linearized stability for semiflows generated by neutral functional differential equations of the form x′(t) = g(? x _t , x _t ). The state space is a closed subset in a manifold of C ²-functions. Applications include equations with state-dependent delay, as for example x′(t) = a x′(t + d(x(t))) + f (x(t + r(x(t)))) with \({a\in\mathbb{R}, d:\mathbb{R}\to(-h,0), f:\mathbb{R}\to\mathbb{R}, r:\mathbb{R}\to[-h,0]}\).     

An iterative updating method for undamped structural systems

Finite element model updating is a procedure to minimize the differences between analytical and experimental results and can be mathematically reduced to solving the following problem. Problem P: Let M _a ∈SR ^n×n and K _a ∈SR ^n×n be the analytical mass and stiffness matrices and Λ=diag{λ ₁,…,λ _p }∈R ^p×p and X=[x ₁,…,x _p ]∈R ^n×p be the measured eigenvalue and eigenvector matrices, respectively. Find \((\hat{M}, \hat{K}) \in \mathcal{S}_{MK}\) such that \(\| \hat{M}-M_{a} \|^{2}+\| \hat{K}-K_{a}\|^{2}= \min_{(M,K) \in {\mathcal{S}}_{MK}} (\| M-M_{a} \|^{2}+\|K-K_{a}\|^{2})\), where \(\mathcal{S}_{MK}=\{(M,K)| X^{T}MX=I_{p}, MX \varLambda=K X \}\) and ∥?∥ is the Frobenius norm. This paper presents an iterative method to solve Problem P. By the method, the optimal approximation solution \((\hat{M}, \hat{K})\) of Problem P can be obtained within finite iteration steps in the absence of roundoff errors by choosing a special kind of initial matrix pair. A numerical example shows that the introduced iterative algorithm is quite efficient.     

Global Attractor for a Nonlinear Oscillator Coupled to the Klein–Gordon Field

The long-time asymptotics is analyzed for all finite energy solutions to a model\(\mathbf{U}(1)\)-invariant nonlinear Klein–Gordon equation in one dimension, with the nonlinearity concentrated at a single point: each finite energy solution converges as t→ ± ∞ to the set of all “nonlinear eigenfunctions” of the form ψ(x)e^{?iω t}. The global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.We justify this mechanism by the following novel strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time spectrum in the spectral gap [ ? m,m] and satisfies the original equation. This equation implies the key spectral inclusion for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of each omega-limit trajectory to a single harmonic \(\omega\in[-m,m]\).The research is inspired by Bohr’s postulate on quantum transitions and Schrödinger’s identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled\(\mathbf{U}(1)\)-invariant Maxwell–Schrödinger and Maxwell–Dirac equations.     

Planar Vector Fields with a Given Set of Orbits

We determine all the \({\mathcal{C}^1}\) planar vector fields with a given set of orbits of the form y ? y(x) = 0 satisfying convenient assumptions. The case when these orbits are branches of an algebraic curve is also study. We show that if a quadratic vector field admits a unique irreducible invariant algebraic curve \({g(x, y) = \sum_{j=0}^S a_j(x) y^{S-j}= 0}\) with S branches with respect to the variable y, then the degree of the polynomial g is at most 4S.     

Synchronization transitions induced by partial time delay in a excitatory–inhibitory coupled neuronal network

Information transmission delays are an inherent factor of neuronal systems as a consequence of the finite propagation speeds and time lapses occurring by both dendritic and synaptic processes. In real neuronal systems, some delay between two neurons is too small and can be ignored, which results in partial time delay. In this paper, we focus on investigating influences of partial time delay on synchronization transitions in a excitatory–inhibitory (E–I) coupled neuronal networks. Here, we suppose time delay between two neurons equals to \(\tau \) with probability \(p_{\mathrm{delay}}\) and investigate effect of partial time delay on synchronization transitions of the neuronal networks by controlling \(\tau \) and \(p_{\mathrm{delay}}\) under three cases. In these three cases, excitatory synapses are always considered to delayed with probability \(p_{\mathrm{delay}}\), while inhibitory synapses are considered to be without delays (case I), delayed with probability \(p_{\mathrm{delay}}\) (case II), and always delayed (case III), respectively. It is revealed that, in the first two cases, partial time delay has little influences on synchronization of the neuronal network for small \(p_{\mathrm{delay}}\), while it could induce synchronization transitions at \(\tau \) around integer multiples of the period of individual neuron T when \(p_{\mathrm{delay}}\) is large enough, while in the case III, partial time delay could induce synchronization transitions at \(\tau \) being around odd integer multiples of T / 2 for small \(p_{\mathrm{delay}}\) and at \(\tau \) being around integer multiples of T for large \(p_{\mathrm{delay}}\). Most interesting observation is that partial time delay could induce frequent synchronization transitions at \(\tau \) being around integer multiples of T / 2 for intermediate \(p_{\mathrm{delay}}\). Moreover, effect of rewiring probability on synchronization transitions induced by partial time delay has been discussed. It is found that synchronization transitions induced by partial time delay are robust to rewiring probability for large \(p_{\mathrm{delay}}\) under the three cases.     

Regularity and Singularities of Optimal Convex Shapes in the Plane

We focus here on the analysis of the regularity or singularity of solutions Ω ₀ to shape optimization problems among convex planar sets, namely:

$J(\Omega_{0})={\rm min} \{J(\Omega), \Omega \quad {\rm convex},\Omega \in \mathcal{S}_{\rm ad}\},$

where \({\mathcal{S}_{\rm ad}}\) is a set of 2-dimensional admissible shapes and \({J:\mathcal{S}_{\rm ad}\rightarrow\mathbb{R}}\) is a shape functional. Our main goal is to obtain qualitative properties of these optimal shapes by using first and second order optimality conditions, including the infinite dimensional Lagrange multiplier due to the convexity constraint. We prove two types of results:

1. i)
   under a suitable convexity property of the functional J, we prove that Ω ₀ is a W ^2,p -set, \({p\in[1, \infty]}\). This result applies, for instance, with p = ∞ when the shape functional can be written as J(Ω) = R(Ω) + P(Ω), where R(Ω) = F(|Ω|, E _f (Ω), λ₁(Ω)) involves the area |Ω|, the Dirichlet energy E _f (Ω) or the first eigenvalue of the Laplace–Dirichlet operator λ₁(Ω), and P(Ω) is the perimeter of Ω;
    

1. ii)
   under a suitable concavity assumption on the functional J, we prove that Ω ₀ is a polygon. This result applies, for instance, when the functional is now written as J(Ω) = R(Ω) ? P(Ω), with the same notations as above.
    

     

Global Attractors of Sixth Order PDEs Describing the Faceting of Growing Surfaces

A spatially two-dimensional sixth order PDE describing the evolution of a growing crystalline surface h(x, y, t) that undergoes faceting is considered with periodic boundary conditions, as well as its reduced one-dimensional version. These equations are expressed in terms of the slopes \(u_1=h_{x}\) and \(u_2=h_y\) to establish the existence of global, connected attractors for both equations. Since unique solutions are guaranteed for initial conditions in \(\dot{H}^2_{per}\), we consider the solution operator \(S(t): \dot{H}^2_{per} \rightarrow \dot{H}^2_{per}\), to gain our results. We prove the necessary continuity, dissipation and compactness properties.     

Multipulse Phases in k-Mixtures of Bose–Einstein Condensates

For the system

$-\Delta U_i+ U_i=U_i^3-\beta U_i\sum_{j\neq i}U_j^2,\quad i=1,\dots,k,$

(with k ≧ 3), we prove the existence for β large of positive radial solutions on \({\mathbb R^N}\) . We show that as β →  + ∞, the profile of each component U _i separates, in many pulses, from the others. Moreover, we can prescribe the location of such pulses in terms of the oscillations of the changing-sign solutions of the scalar equation  ? ΔW  +  W  =  W³. Within an Hartree–Fock approximation, this provides a theoretical indication of phase separation into many nodal domains for the k-mixtures of Bose–Einstein condensates.

     

Steady Nearly Incompressible Vector Fields in Two-Dimension: Chain Rule and Renormalization

Given bounded vector field \({b : {\mathbb{R}^{d}} \to {\mathbb{R}^{d}}}\), scalar field \({u : {\mathbb{R}^{d}} \to {\mathbb{R}}}\), and a smooth function \({\beta : {\mathbb{R}} \to {\mathbb{R}}}\), we study the characterization of the distribution \({{\rm div}(\beta(u)b)}\) in terms of div b and div(ub). In the case of BV vector fields b (and under some further assumptions), such characterization was obtained by L. Ambrosio, C. De Lellis and J. Malý, up to an error term which is a measure concentrated on the so-called tangential set of b. We answer some questions posed in their paper concerning the properties of this term. In particular, we construct a nearly incompressible BV vector field b and a bounded function u for which this term is nonzero. For steady nearly incompressible vector fields b (and under some further assumptions), in the case when d = 2, we provide complete characterization of div(\({\beta(u)b}\)) in terms of div b and div(ub). Our approach relies on the structure of level sets of Lipschitz functions on \({{\mathbb{R}^{2}}}\) obtained by G. Alberti, S. Bianchini and G. Crippa. Extending our technique, we obtain new sufficient conditions when any bounded weak solution u of \({\partial_t u + b \cdot \nabla u=0}\) is renormalized, that is when it also solves \({\partial_t \beta(u) + b \cdot \nabla \beta(u)=0}\) for any smooth function \({\beta \colon{\mathbb{R}} \to {\mathbb{R}}}\). As a consequence, we obtain new a uniqueness result for this equation.     

A Note on the Convergence of Singularly Perturbed Second Order Potential-Type Equations

In this paper we study the limit as \(\varepsilon \rightarrow 0\) of the singularly perturbed second order equation \(\varepsilon ^2 \ddot{u}_\varepsilon + \nabla _{\!x} V(t,u_\varepsilon (t))=0\), where V(t, x) is a potential. We assume that \(u_0(t)\) is one of its equilibrium points such that \(\nabla _{\!x}V(t,u_0(t))=0\) and \(\nabla _{\!x}^2V(t,u_0(t))>0\). We find that, under suitable initial data, the solutions \(u_\varepsilon \) converge uniformly to \(u_0\), by imposing mild hypotheses on V. A counterexample shows that they cannot be weakened.     

Time Periodic Traveling Curved Fronts in the Periodic Lotka–Volterra Competition–Diffusion System

This paper is concerned with time periodic traveling curved fronts for periodic Lotka–Volterra competition system with diffusion in two dimensional spatial space

$$\begin{aligned} {\left\{ \begin{array}{ll} \dfrac{\partial u_{1}}{\partial t}=\Delta u_{1} +u_{1}(x,y,t)\left( r_{1}(t)-a_{1}(t)u_{1}(x,y,t)-b_{1}(t)u_{2}(x,y,t)\right) ,\\ \dfrac{\partial u_{2}}{\partial t}=d\Delta u_{2} +u_{2}(x,y,t)\left( r_{2}(t)-a_{2}(t)u_{1}(x,y,t)-b_{2}(t)u_{2}(x,y,t)\right) , \end{array}\right. } \end{aligned}$$

where \(\Delta \) denotes \(\frac{\partial ^{2}}{\partial x^{2} }+ \frac{\partial ^{2}}{\partial y^{2} }\), \(x,y\in {\mathbb {R}}\) and \(d>0\) is a constant, the functions \(r_i(t),a_i(t)\) and \(b_i(t)\) are T-periodic and Hölder continuous. Under suitable assumptions that the corresponding kinetic system admits two stable periodic solutions (p(t), 0) and (0, q(t)), the existence, uniqueness and stability of one-dimensional traveling wave solution \(\left( \Phi _{1}(x+ct,t),\Phi _{2}(x+ct,t)\right) \) connecting two periodic solutions (p(t), 0) and (0, q(t)) have been established by Bao and Wang ( J Differ Equ 255:2402–2435, 2013) recently. In this paper we continue to investigate two-dimensional traveling wave solutions of the above system under the same assumptions. First, we establish the asymptotic behaviors of one-dimensional traveling wave solutions of the system at infinity. Using these asymptotic behaviors, we then construct appropriate super- and subsolutions and prove the existence and non-existence of two-dimensional time periodic traveling curved fronts. Finally, we show that the time periodic traveling curved front is asymptotically stable.

     

Complex Dynamics in a ODE Model Related to Phase Transition

Motivated by some recent studies on the Allen–Cahn phase transition model with a periodic nonautonomous term, we prove the existence of complex dynamics for the second order equation

$$\begin{aligned} -\ddot{x} + \left( 1 + \varepsilon ^{-1} A(t)\right) G'(x) = 0, \end{aligned}$$

where A(t) is a nonnegative T-periodic function and \(\varepsilon > 0\) is sufficiently small. More precisely, we find a full symbolic dynamics made by solutions which oscillate between any two different strict local minima \(x_0\) and \(x_1\) of G(x). Such solutions stay close to \(x_0\) or \(x_1\) in some fixed intervals, according to any prescribed coin tossing sequence. For convenience in the exposition we consider (without loss of generality) the case \(x_0 =0\) and \(x_1 = 1\).

     

A Reaction–Diffusion–Advection Equation with Mixed and Free Boundary Conditions

We investigate a reaction–diffusion–advection equation of the form \(u_t-u_{xx}+\beta u_x=f(u)\) \((t>0,\,0<x<h(t))\) with mixed boundary condition at \(x=0\) and Stefan free boundary condition at \(x=h(t)\). Such a model may be applied to describe the dynamical process of a new or invasive species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundary representing the expanding front. The goal of this paper is to understand the effect of advection environment and no flux across the left boundary on the dynamics of this species. For the case \(|\beta |<c_0\), we first derive the spreading–vanishing dichotomy and sharp threshold for spreading and vanishing, and then provide a much sharper estimate for the spreading speed of h(t) and the uniform convergence of u(t, x) when spreading happens. For the case \(|\beta |\ge c_0\), some results concerning virtual spreading, vanishing and virtual vanishing are obtained. Here \(c_0\) is the minimal speed of traveling waves of the differential equation.     

New Derived from Anosov Diffeomorphisms with Pathological Center Foliation

In this paper we focused our study on derived from Anosov diffeomorphisms (DA diffeomorphisms ) of the torus \(\mathbb {T}^3,\) it is, an absolute partially hyperbolic diffeomorphism on \(\mathbb {T}^3\) homotopic to a linear Anosov automorphism of the \(\mathbb {T}^3.\) We can prove that if \(f: \mathbb {T}^3 \rightarrow \mathbb {T}^3 \) is a volume preserving DA diffeomorphism homotopic to a linear Anosov A,  such that the center Lyapunov exponent satisfies \(\lambda ^c_f(x) > \lambda ^c_A > 0,\) with x belongs to a positive volume set, then the center foliation of f is non absolutely continuous. We construct a new open class U of non Anosov and volume preserving DA diffeomorphisms, satisfying the property \(\lambda ^c_f(x) > \lambda ^c_A > 0\) for \(m-\)almost everywhere \(x \in \mathbb {T}^3.\) Particularly for every \(f \in U,\) the center foliation of f is non absolutely continuous.     

Okamoto’s Space for the First Painlevé Equation in Boutroux Coordinates

We study the completeness and connectedness of asymptotic behaviours of solutions of the first Painlevé equation d² y/dx ² = 6 y ² + x in the limit \({x\to\infty,x\in{\mathbb C}}\). This problem arises in various physical contexts including the critical behaviour near gradient catastrophe for the focusing nonlinear Schrödinger equation. We prove that the complex limit set of solutions is non-empty, compact and invariant under the flow of the limiting autonomous Hamiltonian system, that the infinity set of the vector field is a repellor for the dynamics and obtain new proofs for solutions near the equilibrium points of the autonomous flow. The results rely on a realization of Okamoto’s space, that is, the space of initial values compactified and regularized by embedding in \({{\mathbb C}{\mathbb P} 2}\) through an explicit construction of nine blowups.     

Smooth positon solutions of the focusing modified Korteweg–de Vries equation

The n-fold Darboux transformation \(T_{n}\) of the focusing real modified Korteweg–de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the n-soliton solutions of the mKdV equation are also expressed by determinants whose elements consist of the eigenvalues \(\lambda _{j}\) and the corresponding eigenfunctions of the associated Lax equation. The nonsingular n-positon solutions of the focusing mKdV equation are obtained in the special limit \(\lambda _{j}\rightarrow \lambda _{1}\), from the corresponding n-soliton solutions and by using the associated higher-order Taylor expansion. Furthermore, the decomposition method of the n-positon solution into n single-soliton solutions, the trajectories, and the corresponding “phase shifts” of the multi-positons are also investigated.     

On Properties of the Generalized Wasserstein Distance

The Wasserstein distances W_p (p \({\geqq}\) 1), defined in terms of a solution to the Monge–Kantorovich problem, are known to be a useful tool to investigate transport equations. In particular, the Benamou–Brenier formula characterizes the square of the Wasserstein distance W₂ as the infimum of the kinetic energy, or action functional, of all vector fields transporting one measure to the other. Another important property of the Wasserstein distances is the Kantorovich–Rubinstein duality, stating the equality between the distance W₁(μ, ν) of two probability measures μ, ν and the supremum of the integrals in d(μ ?ν) of Lipschitz continuous functions with Lipschitz constant bounded by one. An intrinsic limitation of Wasserstein distances is the fact that they are defined only between measures having the same mass. To overcome such a limitation, we recently introduced the generalized Wasserstein distances \({W_p^{a,b}}\), defined in terms of both the classical Wasserstein distance W_p and the total variation (or L¹) distance, see (Piccoli and Rossi in Archive for Rational Mechanics and Analysis 211(1):335–358, 2014). Here p plays the same role as for the classic Wasserstein distance, while a and b are weights for the transport and the total variation term. In this paper we prove two important properties of the generalized Wasserstein distances: (1) a generalized Benamou–Brenier formula providing the equality between \({W_2^{a,b}}\) and the supremum of an action functional, which includes a transport term (kinetic energy) and a source term; (2) a duality à la Kantorovich–Rubinstein establishing the equality between \({W_1^{1,1}}\) and the flat metric.     

Temporal Homogenization of Linear ODEs,with Applications to Parametric Super-Resonance and Energy Harvest

We consider the temporal homogenization of linear ODEs of the form \({\dot{x}=Ax+\epsilon P(t)x+f(t)}\), where P(t) is periodic and \({\epsilon}\) is small. Using a 2-scale expansion approach, we obtain the long-time approximation \({x(t)\approx {\rm exp}(At) \left( \Omega(t)+\int_0^t {\rm exp}(-A \tau) f(\tau) {\rm d}\tau \right)}\), where \({\Omega}\) solves the cell problem \({\dot{\Omega}=\epsilon B \Omega + \epsilon F(t)}\) with an effective matrix B and an explicitly-known F(t). We provide necessary and sufficient conditions for the accuracy of the approximation (over a \({{\mathcal{O}}(\epsilon^{-1})}\) time-scale), and show how B can be computed (at a cost independent of \({\epsilon}\)). As a direct application, we investigate the possibility of using RLC circuits to harvest the energy contained in small scale oscillations of ambient electromagnetic fields (such as Schumann resonances). Although a RLC circuit parametrically coupled to the field may achieve such energy extraction via parametric resonance, its resistance R needs to be smaller than a threshold \({\kappa}\) proportional to the fluctuations of the field, thereby limiting practical applications. We show that if n RLC circuits are appropriately coupled via mutual capacitances or inductances, then energy extraction can be achieved when the resistance of each circuit is smaller than \({n\kappa}\). Hence, if the resistance of each circuit has a non-zero fixed value, energy extraction can be made possible through the coupling of a sufficiently large number n of circuits (\({n\approx 1000}\) for the first mode of Schumann resonances and contemporary values of capacitances, inductances and resistances). The theory is also applied to the control of the oscillation amplitude of a (damped) oscillator.     

Convergence of Radially Symmetric Solutions for (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian Elliptic Equations with a Damping Term

This study considers the quasilinear elliptic equation with a damping term,

$$\begin{aligned} \text {div}(D(u)\nabla u) + \frac{k(|{\mathbf {x}}|)}{|{\mathbf {x}}|}\,{\mathbf {x}}\cdot (D(u)\nabla u) + \omega ^2\big (|u|^{p-2}u + |u|^{q-2}u\big ) = 0, \end{aligned}$$

where \({\mathbf {x}}\) is an N-dimensional vector in \(\big \{{\mathbf {x}} \in \mathbb {R}^N: |{\mathbf {x}}| \ge \alpha \big \}\) for some \(\alpha > 0\) and \(N \in {\mathbb {N}}\setminus \{1\}\); \(D(u) = |\nabla u|^{p-2} + |\nabla u|^{q-2}\) with \(1 < q \le p\); k is a nonnegative and locally integrable function on \([\alpha ,\infty )\); and \(\omega \) is a positive constant. A necessary and sufficient condition is given for all radially symmetric solutions to converge to zero as \(|{\mathbf {x}}|\rightarrow \infty \). Our necessary and sufficient condition is expressed by an improper integral related to the damping coefficient k. The case that k is a power function is explained in detail.