We study a nonlinear dynamical system on networks inspired by the pitchfork bifurcation normal form. The system has several interesting interpretations: as an interconnection of several pitchfork systems, a gradient dynamical system and the dominating behaviour of a general class of nonlinear dynamical systems. The equilibrium behaviour of the system exhibits a global bifurcation with respect to the

In this paper, we formulate the problem of elastodynamic transformation cloaking for Kirchoff–Love plates and elastic plates with both in-plane and out-of-plane displacements. A cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic plate (virtual problem) to that of an anisotropic and inhomogeneous elastic plate with a hole surrounded by a cloak that is to

Buoyancy-driven fluids such as many atmospheric and oceanic flows and the Rayleigh–Bénard convection are modeled by the Boussinesq systems. By rigorously estimating the large-time behavior of solutions to a special Boussinesq system, this paper reveals a fascinating phenomenon on buoyancy-driven fluids that the temperature can actually stabilize the fluids. The Boussinesq system concerned here governs

This paper considers a general stochastic SIR epidemic model driven by a multidimensional Lévy jump process with heavy-tailed increments and possible correlation between noise components. In this framework, we derive new sufficient conditions for disease extinction and persistence in the mean. Our method differs from previous approaches by the use of Kunita’s inequality instead of the Burkholder–Davis–Gundy

We develop a theory for distributed branch points and investigate their role in determining the shape and influencing the mechanics of thin hyperbolic objects. We show that branch points are the natural topological defects in hyperbolic sheets, they carry a topological index which gives them a degree of robustness, and they can influence the overall morphology of a hyperbolic surface without concentrating

In this article, we study a one-dimensional degenerate fourth-order parabolic equation (a thin-film model) with a source term. We prove existence of generalized weak solutions for the case \(n>0\) and study interface propagation properties like: finite speed propagation and waiting time phenomenon for the case \(1

We propose a convex-optimization-based framework for computation of invariant measures of polynomial dynamical systems and Markov processes, in discrete and continuous time. The set of all invariant measures is characterized as the feasible set of an infinite-dimensional linear program (LP). The objective functional of this LP is then used to single out a specific measure (or a class of measures) extremal

This paper is concerned with the conditions of existence and nonexistence of traveling wave solutions (TWS) for a class of discrete diffusive epidemic model. We find that the existence of TWS is determined by the so-called basic reproduction number and the critical wave speed: When the basic reproduction number \(\mathfrak {R}_0>1\), there exists a critical wave speed \(c^*>0\), such that for each

A nonlinear chain with sixth-order polynomial on-site potential is used to analyze the evolution of the total-to-kinetic-energy ratio during development of modulational instability of extended nonlinear vibrational modes. For the on-site potential of hard-type (soft-type) anharmonicity, the instability of \(q =\pi \) mode (\(q = 0\) mode) results in the appearance of long-living discrete breathers

We classify the extensions of n-body central configurations to \((n+1)\)-body central configurations in \(R^3\), in both the collinear case and the non-collinear case. We completely solve the two open questions posed by Hampton (Nonlinearity 18: 2299-2304, 2005). This classification is related with study on co-circular and co-spherical central configurations. We also obtain a general property of co-circular

In this paper, we study the diffusive limit of solutions to the generalized Langevin equation (GLE) in a periodic potential. Under the assumption of quasi-Markovianity, we obtain sharp longtime equilibration estimates for the GLE using techniques from the theory of hypocoercivity. We then show asymptotic results for the effective diffusion coefficient in the small correlation time regime, as well as

The multiphase Whitham modulation equations with N phases have 2N characteristics which may be of hyperbolic or elliptic type. In this paper, a nonlinear theory is developed for coalescence, where two characteristics change from hyperbolic to elliptic via collision. Firstly, a linear theory develops the structure of colliding characteristics involving the topological sign of characteristics and multiple

The main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer–Meinhardt model on an isotropically evolving domain. In the case where the inhibitor’s response to the activator’s growth is rather weak, then the shadow system of the Gierer–Meinhardt model is reduced to a single though non-local equation whose dynamics is thoroughly

We are modelling multiscale, multi-physics uncertainty in wave–current interaction (WCI). To model uncertainty in WCI, we introduce stochasticity into the wave dynamics of two classic models of WCI, namely the generalised Lagrangian mean (GLM) model and the Craik–Leibovich (CL) model. The key idea for the GLM approach is the separation of the Lagrangian (fluid) and Eulerian (wave) degrees of freedom

We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework for the computation of optimal reaction coordinates of such systems that is based on learning a parameterization of a low-dimensional transition manifold in a certain

In this paper, we propose and study a stochastic aggregation–diffusion equation of the Keller–Segel (KS) type for modeling the chemotaxis in dimensions \(d=2,3\). Unlike the classical deterministic KS system, which only allows for idiosyncratic noises, the stochastic KS equation is derived from an interacting particle system subject to both idiosyncratic and common noises. Both the unique existence

We examine micromagnetic pattern formation in chiral magnets, driven by the competition of Heisenberg exchange, Dzyaloshinskii–Moriya interaction, easy-plane anisotropy and thermodynamic Landau potentials. Based on equivariant bifurcation theory, we prove existence of lattice solutions branching off the zero magnetization state and investigate their stability. We observe in particular the stabilization

We construct a new Evans function for quasi-periodic solutions to the linearisation of the sine-Gordon equation about a periodic travelling wave. This Evans function is written in terms of fundamental solutions to a Hill’s equation. Applying the Evans-Krein function theory of Kollár and Miller (SIAM Rev 56(1):73–123, 2014) to our Evans function, we provide a new method for computing the Krein signatures

We complete the analysis initiated in Dabade et al. (J Nonlinear Sci 21:415–460, 2018) on the micromagnetics of cubic ferromagnets in which the role of magnetostriction is significant. We prove ansatz-free lower bounds for the scaling of the total micromagnetic energy including magnetostriction contribution, for a two-dimensional sample. This corresponds to the micromagnetic energy per unit length

Given two distinct subsets A, B in the state space of some dynamical system, transition path theory (TPT) was successfully used to describe the statistical behavior of transitions from A to B in the ergodic limit of the stationary system. We derive generalizations of TPT that remove the requirements of stationarity and of the ergodic limit and provide this powerful tool for the analysis of other dynamical

Specific kinds of physical and biological systems exhibit complex Mixed-Mode Oscillations mediated by folded-singularity canards in the context of slow-fast models. The present manuscript revisits these systems, specifically by analysing the dynamics near a folded singularity from the viewpoint of inflection sets of the flow. Originally, the inflection set method was developed for planar systems [Brøns

A key starting assumption in many classical interatomic potential models for materials is a site energy decomposition of the potential energy surface into contributions that only depend on a small neighbourhood. Under a natural stability condition, we construct such a spatial decomposition for self-consistent tight binding models, extending recent results for linear tight binding models to the nonlinear

Recent laboratory experiments of Bolles et al. (Phys Rev Fluids 4(1):011801, 2019) demonstrate that an abrupt change in bottom topography can trigger anomalous statistics in randomized surface waves. Motivated by these observations, Majda et al. (Proc Natl Acad Sci 116(10):3982–3987, 2019) developed a theoretical framework, based on deterministic and statistical analysis of the truncated Korteweg-de

This work proposes and investigates a new model of the rotating rigid body based on the non-twisting frame. Such a frame consists of three mutually orthogonal unit vectors whose rotation rate around one of the three axis remains zero at all times and, thus, is represented by a nonholonomic restriction. Then, the corresponding Lagrange–D’Alembert equations are formulated by employing two descriptions

We develop the contact singularity theory for singularly perturbed (or ‘slow–fast’) vector fields of the general form \(z' = H(z,\varepsilon )\), \(z\in {\mathbb {R}}^n\) and \(0 < \varepsilon \ll 1\). Our main result is the derivation of computable, coordinate-independent defining equations for contact singularities under an assumption that the leading-order term of the vector field admits a suitable

We study propagation direction of the traveling wave for the diffusive Lotka–Volterra competition system with bistable nonlinearity in a periodic habitat. By directly proving the strong stability of two semitrivial equilibria, we establish a new and sharper result on the existence of traveling wave. Using the method of upper and lower solutions, we provide two comparison theorems concerning the direction

We consider conformally invariant energies W on the group \({{\,\mathrm{GL}\,}}^{\!+}(2)\) of \(2\times 2\)-matrices with positive determinant, i.e., \(W:{{\,\mathrm{GL}\,}}^{\!+}(2)\rightarrow {\mathbb {R}}\) such that $$\begin{aligned} W(A\, F\, B) = W(F) \quad \text {for all }\; A,B\in \{a\, R\in {{\,\mathrm{GL}\,}}^{\!+}(2) \,|\,a\in (0,\infty ),\; R\in {{\,\mathrm{SO}\,}}(2)\}, \end{aligned}$$

In this paper, we report a rigorous theory of the inverse scattering transforms (ISTs) for the derivative nonlinear Schrödinger (DNLS) equation with both zero boundary conditions (ZBCs) and nonzero boundary conditions (NZBCs) at infinity and double zeros of analytical scattering coefficients. The scattering theories for both ZBCs and NZBCs are addressed. The direct scattering problem establishes the

We consider a restricted four-body problem, with a precise hierarchy between the bodies: two larger bodies and a smaller one, all three of oblate shape, and a fourth, infinitesimal body, in the neighborhood of the smaller of the three bodies. The three heavy bodies are assumed to move in a plane under their mutual gravity, and the fourth body to move in the three-dimensional space under the gravitational

General rogue waves are derived for the generalized derivative nonlinear Schrödinger (GDNLS) equations by a bilinear Kadomtsev–Petviashvili (KP) reduction method. These GDNLS equations contain the Kaup–Newell equation, the Chen–Lee–Liu equation and the Gerdjikov–Ivanov equation as special cases. In this bilinear framework, it is shown that rogue waves to all members of these equations are expressed

We answer three fundamental questions concerning monostable traveling fronts for the scalar Kolmogorov ecological equation with diffusion and spatiotemporal interaction: These are the questions about their existence, uniqueness and geometric shape. In the particular case of the food-limited model, we give a rigorous proof of the existence of a peculiar, yet substantive and nonlinearly determined class

For a large system of identical particles interacting by means of a potential, we find that a strong large scale flow velocity can induce motions in the inertial range via the potential coupling. This forcing lies in special bundles in the Fourier space, which are formed by pairs of particles. These bundles are not present in the Boltzmann, Euler and Navier–Stokes equations, because they are destroyed

Running is the basic mode of fast locomotion for legged animals. One of the most successful mathematical descriptions of this gait is the so-called spring–mass model constructed upon an inverted elastic pendulum. In the description of the grounded phase of the step, an interesting boundary value problem arises where one has to determine the leg stiffness. In this paper, we find asymptotic expansions

First-order hyperbolic partial differential equations with two internal variables have been used to model biological and epidemiological problems with two physiological structures, such as chronological age and infection age in epidemic models, age and another physiological character (maturation, size, stage) in population models, and cell-age and molecular content (cyclin content, maturity level,

The Cahn–Hilliard equation is a classic model of phase separation in binary mixtures that exhibits spontaneous coarsening of the phases. We study the Cahn–Hilliard equation with an imposed advection term in order to model the stirring and eventual mixing of the phases. The main result is that if the imposed advection is sufficiently mixing, then no phase separation occurs, and the solution instead

We discuss the Lagrangian property and the conservation of the kinetic energy for solutions of the 2D incompressible Euler equations. Existence of Lagrangian solutions is known when the initial vorticity is in \(L^p\) with \(1\le p\le \infty \), and if \(p\ge 3/2\), all weak solutions are conservative. In this work, we prove that solutions obtained via the vortex method are Lagrangian, and that they

In section 5.1 entitled “The shortest NEP from a bounded domain in prviously.

We study dynamic networks under an undirected consensus communication protocol and with one state-dependent weighted edge. We assume that the aforementioned dynamic edge can take values over the whole real numbers, and that its behaviour depends on the nodes it connects and on an extrinsic slow variable. We show that, under mild conditions on the weight, there exists a reduction such that the dynamics

In this article we investigate the phase transition phenomena that occur in a model of self-organisation through body-attitude coordination. Here, the body attitude of an agent is modelled by a rotation matrix in \({\mathbb {R}}^3\) as in Degond et al. (Math Models Methods Appl Sci 27(6):1005–1049, 2017). The starting point of this study is a BGK equation modelling the evolution of the distribution

In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible

We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection–diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or heat equation. By this coordinate transformation, the reversible effects of advection

We consider families of Hamiltonian systems in two degrees of freedom with an equilibrium in 1 : 2 resonance. Under detuning, this “Fermi resonance” typically leads to normal modes losing their stability through period-doubling bifurcations. For cubic potentials, this concerns the short axial orbits, and in galactic dynamics, the resulting stable periodic orbits are called “banana” orbits. Galactic

This paper examines a spike-adding bifurcation phenomenon whereby small-amplitude canard cycles transition into large-amplitude bursting oscillations along a single continuous branch in parameter space. We consider a class of three-dimensional singularly perturbed ODEs with two fast variables and one slow variable and singular perturbation parameter \(\varepsilon \ll 1 \) under general assumptions

The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter \(\epsilon \rightarrow 0\). Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper

In recent years, rumor propagation in social networks attracts more researchers’ attention. In this paper, we have established I2S2R rumor spreading models in both homogeneous networks and heterogeneous networks considering the effect of time delay. In the homogeneous network model, we obtain the basic reproduction number by means of the next-generation matrix. Besides, the local stability and the

Recent experimental work has revealed that interstitial fluid flow can mobilize two types of tumor cell migration mechanisms. One is a chemotactic-driven mechanism where chemokine (chemical component) bounded to the extracellular matrix (ECM) is released and skewed in the flow direction. This leads to higher chemical concentrations downstream which the tumor cells can sense and migrate toward. The

The original version of this article unfortunately contained an error in Acknowledgement section. The authors would like to correct the error with this erratum. The correct text should read as:

A normal form theory for non-quasiperiodic systems is combined with the special properties of the partially averaged Newtonian potential pointed out in Pinzari (Celest Mech Dyn Astron 131(5):22, 2019) to prove, in the averaged, planar three-body problem, the existence of a plenty of motions where, periodically, the perihelion of the inner body affords librations about one equilibrium position and its

A novel general framework for the study of \(\Gamma \)-convergence of functionals defined over pairs of measures and energy-measures is introduced. This theory allows us to identify the \(\Gamma \)-limit of these kind of functionals by knowing the \( \Gamma \)-limit of the underlying energies. In particular, the interaction between the functionals and the underlying energies results, in the case these

We prove the global existence and uniqueness of smooth solutions to the one-dimensional barotropic Navier–Stokes system with degenerate viscosity \(\mu (\rho )=\rho ^\alpha \). We establish that the smooth solutions have possibly two different far-fields, and the initial density remains positive globally in time, for the initial data satisfying the same conditions. In addition, our result works for

The stochastically forced vorticity equation associated with the two-dimensional incompressible Navier–Stokes equation on \(D_\delta :=[0,2\pi \delta ]\times [0,2\pi ]\) is considered for \(\delta \approx 1\), periodic boundary conditions, and viscosity \(0<\nu \ll 1\). An explicit family of quasi-stationary states of the deterministic vorticity equation is known to play an important role in the long-time

In this paper, we provide a novel approach to capture causal interaction in a dynamical system from time series data. In Sinha and Vaidya (in: IEEE conference on decision and control, pp 7329–7334, 2016), we have shown that the existing measures of information transfer, namely directed information, Granger causality and transfer entropy, fail to capture the causal interaction in a dynamical system

We study the integrability of an eight-parameter family of three-dimensional spherically confined steady Stokes flows introduced by Bajer and Moffatt. This volume-preserving flow was constructed to model the stretch–twist–fold mechanism of the fast dynamo magnetohydrodynamical model. In particular we obtain a complete classification of cases when the system admits an additional Darboux polynomial of

This paper presents reduced-order nonlinear filtering schemes based on a theoretical framework that combines stochastic dimensional reduction and nonlinear filtering. Here, dimensional reduction is achieved for estimating the slow-scale process in a multiscale environment by constructing a filter using stochastic averaging results. The nonlinear filter is approximated numerically using the ensemble

Predator–prey interactions are among the most complicated interactions between biological species, in which there may be both direct effect (through predation) and indirect effect (e.g., fear effect). In the literature, the indirect effect has been largely missing in predator–prey models, until some recent works. Based on the recent work (Wang et al. in J Math Biol 73:1179–1204, 2016) where a fear

In this paper, we deal with Sel’kov model with saturation law which has been applied to numerous problems in chemistry and biology. We will study the stability of the unique constant steady state, existence and nonexistence of nonconstant steady states of such models. In particular, we prove that Turing pattern may occur when the saturation coefficient is small but will not occur when the coefficient

We extend the Itô–Wentzell formula for the evolution of a time-dependent stochastic field along a semimartingale to k-form-valued stochastic processes. The result is the Kunita–Itô–Wentzell (KIW) formula for k-forms. We also establish a correspondence between the KIW formula for k-forms derived here and a certain class of stochastic fluid dynamics models which preserve the geometric structure of deterministic

In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in \(\mathbb R^3\), having the diffusion term \({\varvec{A}}_p( u)=\nabla \cdot ( |{\varvec{D}}(u)|^{p-2} {\varvec{D}}(u))\) with \( {\varvec{D}}(u) = \frac{1}{2} (\nabla u + (\nabla u)^{ \top })\), \(3/2

In this paper, we consider the stochastic dynamics of a finite system of particles in a finite volume (Kac-like particle system) which annihilate with probability \(\alpha \in (0,1)\) or collide elastically with probability \(1-\alpha \). We first establish the well-posedness of the particle system which exhibits no conserved quantities. We rigorously prove that, in some mean-field limit, a suitable

Characterising the glycemic response to a glucose stimulus is an essential tool for detecting deficiencies in humans such as diabetes. In the presence of a constant glucose infusion in healthy individuals, it is known that this control leads to slow oscillations as a result of feedback mechanisms at the organ and tissue level. In this paper, we provide a novel quantitative description of the dependence