Nonlinear Phenom Research Articles

Vector Field Models for Nematic Disclinations

In this paper, a model for defects in nematic liquid crystals that was introduced in Zhang et al. (Physica D Nonlinear Phenom 417:132828, 2021) is studied. In the literature, the setting of many models for defects is the function space SBV (special functions of bounded variation). However, the model considered herein regularizes the director field to be in a Sobolev space by introducing a second vector field tracking the defect. A relaxation result in the case of fixed parameters is proved along with some partial compactness results as the defect width vanishes.

A continuum framework for phase field with bulk-surface dynamics

This continuum mechanical theory aims at detailing the underlying rational mechanics of dynamic boundary conditions proposed by Fischer et al. (Phys Rev Lett 79:893, 1997), Goldstein et al. (Phys D Nonlinear Phenom 240:754–766, 2011), and Knopf et al. (ESAIM Math Model Numer Anal 55:229–282, 2021). As a byproduct, we generalize these theories. These types of dynamic boundary conditions are described by the coupling between the bulk and surface partial differential equations for phase fields. Our point of departure within this continuum framework is the principle of virtual powers postulated on an arbitrary part mathcal {P} where the boundary partial mathcal {P} may lose smoothness. That is, the normal field may be discontinuous along an edge partial ^2mathcal {P}. However, the edges characterizing the discontinuity of the normal field are considered smooth. Our results may be summarized as follows. We provide a generalized version of the principle of virtual powers for the bulk-surface coupling along with a generalized version of the partwise free-energy imbalance. Next, we derive the explicit form of the surface and edge microtractions along with the field equations for the bulk and surface phase fields. The final set of field equations somewhat resembles the Cahn–Hilliard equation for both the bulk and surface. Moreover, we provide a suitable set of constitutive relations and thermodynamically consistent boundary conditions. In Knopf et al. (2021), a mixed (Robin) type of boundary condition for the chemical potentials is proposed for the model in Fischer et al. (1997), Goldstein et al. (2011). In addition to this boundary condition, we also include this type of mixed boundary condition for the microstructure, that is the phase fields. Lastly, we derive the Lyapunov-decay relations for these mixed type of boundary conditions for both the microstructure and chemical potential.

Artificial Intelligence and the Common Sense of Animals.

The problem of common sense remains a major obstacle to progress in artificial intelligence. Here, we argue that common sense in humans is founded on a set of basic capacities that are possessed by many other animals, capacities pertaining to the understanding of objects, space, and causality. The field of animal cognition has developed numerous experimental protocols for studying these capacities and, thanks to progress in deep reinforcement learning (RL), it is now possible to apply these methods directly to evaluate RL agents in 3D environments. Besides evaluation, the animal cognition literature offers a rich source of behavioural data, which can serve as inspiration for RL tasks and curricula.

Ghost attractors in blinking Lorenz and Hindmarsh-Rose systems.

In this paper, we consider blinking systems, i.e., non-autonomous systems generated by randomly switching between several autonomous continuous time subsystems in each sequential fixed period of time. We study cases where a non-stationary attractor of a blinking system with fast switching unexpectedly differs from the attractors of composing subsystems. Such a non-stationary attractor is associated with an attractor of the averaged system being a ghost attractor of the blinking system [Belykh et al., Phys. D: Nonlinear Phenom. 195, 188 (2004); Hasler et al., SIAM J. Appl. Dyn. Syst. 12, 1031 (2013); Belykh et al., Eur. Phys. J. Spec. Top. 222, 2497 (2013)]. Validating the theory of stochastically blinking systems [Hasler et al., SIAM J. Appl. Dyn. Syst. 12, 1031 (2013); Hasler et al., SIAM J. Appl. Dyn. Syst. 12, 1007 (2013)], we demonstrate that fast switching between two Lorenz systems yields a ghost chaotic attractor, even though the dynamics of both systems are trivial and defined by stable equilibria. We also study a blinking Hindmarsh-Rose system obtained from the original model of neuron activity by using randomly switching sequence as an external stimulus. Despite the fact that the values of the external stimulus are selected from a set corresponding to the tonic spiking mode, the blinking model exhibits bursting activity. For both systems, we analyze changes in the dynamical behavior as the period of stochastic switching increases. Using a numerical approximation of the invariant measures of the blinking and averaged systems, we give estimates of a non-stationary and ghost attractors' proximity.

Center Manifold Theory for the Motions of Camphor Boats with Delta Function

Various collective motions of camphor boats have been studied. Camphor boats are self-driven particles that interact with each other through the change of surface tension on water surface by camphor molecules. Consequently, even in a one-dimensional circuit, a congested state or jamming can be observed (Suematsu et al. in Phys Rev E 81:056210, 2010). In this phenomenon, the unidirectional motion of each particle is considered a traveling wave, and the concentration of camphor molecules forms a pulse shape. Hence, each pair of particles interacts with each other like a pulse–pulse interaction. Thus, we expect that the center manifold theories proposed in Ei (J Dyn Differ Equ 14:85–137, 2002) and Ei et al. (Physica D 165:176–198 2002) are applicable for the analysis of the collective motion of camphor boats. However, spatial discontinuity in our model, in particular the existence of Dirac delta functions in a linearized operator, does not fulfill the requirement in the reduction process because the authors developed their theories in $$L^2$$ -framework and for smooth nonlinearity in Ei (2002) and Ei et al. (2002). In this article, we extend the results obtained in Ei (2002) and Ei et al. (2002) and establish a new center manifold approach in $$(H^1)^*$$ -framework. Finally, we succeed to rigorously reduce a mathematical model (Nagayama et al. in Physica D: Nonlinear Phenom 194:151–165, 2004) coupled with a Newtonian equation and a reaction–diffusion equation including delta functions to an ordinary differential system that represents the motions of camphor boats.

Nonnegative weak solutions of thin-film equations related to viscous flows in cylindrical geometries

Motivated by models for thin films coating cylinders in two physical cases proposed in Kerchman (J Fluid Mech 290:131–166, 1994) and Kerchman and Frenkel (Theor Comput Fluid Dyn 6:235–254, 1994), we analyze the dynamics of corresponding thin-film models. The models are governed by nonlinear, fourth-order, degenerate, parabolic PDEs. We prove, given positive and suitably regular initial data, the existence of weak solutions in all length scales of the cylinder, where all solutions are only local in time. We also prove that given a length constraint on the cylinder, long time and global in time weak solutions exist. This analytical result is motivated by numerical work on related models of Reed Ogrosky (Modeling liquid film flow inside a vertical tube, Ph.D. thesis, The University of North Carolina at Chapel Hill, 2013) in conjunction with the works (Camassa et al. in Phys Rev E 86(6):066305, 2012; Physica D Nonlinear Phenom 333:254–265, 2016; J Fluid Mech 745:682–715, 2014; J Fluid Mech 825:1056–1090, 2017).

On the Determining Wavenumber for the Nonautonomous Subcritical SQG Equation

A time-dependent determining wavenumber was introduced in Cheskidov and Dai (Phys D Nonlinear Phenom 376–377:204–215, 2018) to estimate the number of determining modes for the surface quasi-geostrophic equation. In this paper we continue this investigation focusing on the subcritical case and study trajectories inside an absorbing set bounded in $$L^\infty $$ . Utilizing this bound we find a time-independent determining wavenumber that improves the estimate obtained in Cheskidov and Dai (Phys D Nonlinear Phenom 376–377:204–215, 2018). This classical approach is more direct, but it is contingent on the existence of the $$L^\infty $$ absorbing set.

About Universality and Thermodynamics of Turbulence.

This paper investigates the universality of the Eulerian velocity structure functions using velocity fields obtained from the stereoscopic particle image velocimetry (SPIV) technique in experiments and direct numerical simulations (DNS) of the Navier-Stokes equations. It shows that the numerical and experimental velocity structure functions up to order 9 follow a log-universality (Castaing et al. Phys. D Nonlinear Phenom. 1993); this leads to a collapse on a universal curve, when units including a logarithmic dependence on the Reynolds number are used. This paper then investigates the meaning and consequences of such log-universality, and shows that it is connected with the properties of a “multifractal free energy”, based on an analogy between multifractal and thermodynamics. It shows that in such a framework, the existence of a fluctuating dissipation scale is associated with a phase transition describing the relaminarisation of rough velocity fields with different Hölder exponents. Such a phase transition has been already observed using the Lagrangian velocity structure functions, but was so far believed to be out of reach for the Eulerian data.

Complex pattern formation arising from wave instabilities in a three-agent chemical system with superdiffusion

This work investigates analytically and numerically the wave instabilities in a three-agent chemical system undergoing anomalous diffusion. Anomalous diffusion is modeled using higher-dimensional space-fractional operators. The model considers one activator and two inhibitors, and it is a superdiffusive extension of some reaction-diffusion systems in the literature (Phys D Nonlinear Phenom 199(1–2):264–277, 2004). The model under investigation is presented in generic form as a three-species reaction-diffusion system, so that it has broad applicability to a wide range of chemical and biological systems. A weakly nonlinear analysis is performed, dispersive curves of eigenvalues are plotted and their behavior is analyzed. This analysis reveals that the critical value of the wave number for wave instability increases when the superdiffusive exponent decreases. The numerical scheme is performed and some numerical simulations are conducted as evidence of the analytical predictions. We show how the evolution of spatio-temporal patterns is related to meaningful parameters. In particular, we demonstrate that the system exhibits the coexistence of regular and irregular structures, forming complex patterns that are not familiar in standard reaction-diffusion systems.

Stability of Traveling Wavefronts for a Nonlocal Dispersal System with Delay

This paper is concerned with a nonlocal epidemic model arising from the spread of an epidemic by oral-fecal transmission. Comparing with the previous works, here we extend the model in Capasso and Maddalena, Nonlinear Phenom Math Sci. 41:207–217 (1982) by including a spatial convolution term and a discrete delay term corresponding to the dispersal of bacteria in the environment and the latent period of the virus, respectively. Besides existence and asymptotic behavior, the main part of the paper is devoted to the stability of the traveling wavefronts under some monostable assumptions. By using a comparison theorem and the weighted energy method with a suitably selected weight function, we show that all the non-critical traveling waves are exponentially stable. Finally, we apply our results to a specific epidemic model and discuss the effect of time delay on the stability of wavefront.

Electrochemical Oscillation in Li-Ion Batteries

Summary Two-phase reactions are prevalent in Li-ion batteries, whereas the underlying dynamics of phase separation in a real electrode still remain elusive, since numerous electrode particles constitute a formidably complex system for existing experimental techniques. Here we present an intriguing oscillatory phenomenon in the typical phase-separating electrode material Li 4 Ti 5 O 12 . During galvanostatic processes, the voltage oscillates due to the discrete nature of multi-particle phase-separating reactions, and the subtle oscillatory signals allow us to evaluate the fraction of actively phase-separating particles in real time. Through the analysis of oscillatory phenomena, we unveil the dependence of the active fraction on the depth of charge/discharge, cycling current, and working temperature, considerably deepening our understanding of the multi-particle phase-separation reaction. Moreover, it is the first time that electrochemical oscillations have been identified in rechargeable battery systems, opening up a new frontier for both theoretical and experimental researchers.

Generating series for GUE correlators

We extend to the Toda lattice hierarchy the approach of Bertola et al. (Phys D Nonlinear Phenom 327:30–57, 2016; IMRN, 2016) to computation of logarithmic derivatives of tau-functions in terms of the so-called matrix resolvents of the corresponding difference Lax operator. As a particular application we obtain explicit generating series for connected GUE correlators. On this basis an efficient recursive procedure for computing the correlators in full genera is developed.

Chimera patterns in the Kuramoto–Battogtokh model

Kuramoto and Battogtokh (2002 Nonlinear Phenom. Complex Syst. 5 380) discovered chimera states represented by stable coexisting synchrony and asynchrony domains in a lattice of coupled oscillators. After a reformulation in terms of a local order parameter, the problem can be reduced to partial differential equations. We find uniformly rotating, spatially periodic chimera patterns as solutions of a reversible ordinary differential equation, and demonstrate a plethora of such states. In the limit of neutral coupling they reduce to analytical solutions in the form of one- and two-point chimera patterns as well as localized chimera solitons. Patterns at weakly attracting coupling are characterized by virtue of a perturbative approach. Stability analysis reveals that only the simplest chimeras with one synchronous region are stable.

A thermodynamic framework for coupled multiphase Ginzburg-Landau/Cahn-Hilliard systems for simulation of lower bainitic transformation

A thermodynamic framework for a coupled Ginzburg-Landau/Cahn-Hilliard systems is presented in this work. It is based on the concept of generalized stresses and microforce balances as introduced by Gurtin (Phys D Nonlinear Phenom 92(3):178–192, 1996). His concept is extended to compute phase changes and diffusion simultaneously on the same domain and adding coupling terms between the quantities. The resulting thermodynamic framework distinguishes between basic balance laws which are universal and constitutive equations which depend on the specific material. As an application, the transformation of austenite to lower bainite is simulated in this work. The multiphase-field model describes the evolution of bainitic ferrite and the precipitation of carbides, while the Cahn–Hilliard equation governs the carbon diffusion within the bainitic ferrite.

Chaotic asymptotic behaviour of the solutions of the Lighthill–Whitham–Richards equation

The phenomenon of chaos has been exhibited in mathematical nonlinear models that describe traffic flows, see, for instance (Li and Gao in Modern Phys Lett B 18(26–27):1395–1402, 2004; Li in Phys. D Nonlinear Phenom 207(1–2):41–51, 2005). At microscopic level, Devaney chaos and distributional chaos have been exhibited for some car-following models, such as the quick-thinking-driver model and the forward and backward control model (Barrachina et al. in 2015; Conejero et al. in Semigroup Forum, 2015). We present here the existence of chaos for the macroscopic model given by the Lighthill–Whitham–Richards equation.

Variational Dynamics of Free Triple Junctions

We propose a level set framework for representing triple junctions either with or without free endpoints. For triple junctions without free endpoints, our method uses two level set functions to represent the three segments that constitute the structure. For free triple junctions, we extend our method using the free curve work of Schaeffer and Vese (J Math Imaging Vis, 1–17, 2013), Smereka (Phys D Nonlinear Phenom 138(3–4):282–301, 2000). For curves moving under length minimizing flows, it is well known that the endpoints either intersect perpendicularly to the boundary, do not intersect the boundary of the domain or the curve itself (free endpoints), or meet at triple junctions. Although many of these cases can be formulated within the level set framework, the case of free triple junctions does not appear in the literature. Therefore, the proposed free triple junction formulation completes the important curve structure representations within the level set framework. We derive an evolution equation for the dynamics of the triple junction under length and area minimizing flow. The resulting system of partial differential equations are both coupled and highly non-linear, so the system is solved numerically using the Sobolev preconditioned descent. Qualitative numerical experiments are presented on various triple junction and free triple junction configurations, as well as an example with a quadruple junction instability. Quantitative results show convergence of the preconditioned algorithm to the correct solutions.

Knowledge Based Perceptual Anchoring

Perceptual anchoring is the process of creating and maintaining a connection between the sensor data corresponding to a physical object and its symbolic description. It is a subset of the symbol grounding problem, introduced by Harnad (Phys. D, Nonlinear Phenom. 42(1–3):335–346, 1990) and investigated over the past years in several disciplines including robotics. This PhD dissertation focuses on a method for grounding sensor data of physical objects to the corresponding semantic descriptions, in the context of cognitive robots where the challenge is to establish the connection between percepts and concepts referring to objects, their relations and properties. We examine how knowledge representation can be used together with an anchoring framework, so as to complement the meaning of percepts while supporting better linguistic interaction with the use of the corresponding concepts. The proposed method addresses the need to represent and process both perceptual and semantic knowledge, often expressed in different abstraction levels, while originating from different modalities. We then focus on the integration of anchoring with a large scale knowledge base system and with perceptual routines. This integration is applied in a number of studies, where in the context of a smart home, several evaluations spanning from spatial and commonsense reasoning to linguistic interaction and concept acquisition.

Calibration of POD reduced‐order models using Tikhonov regularization

AbstractIn this paper we compare various methods of calibration that can be used in practice to improve the accuracy of reduced‐order models based on proper orthogonal decomposition. The bench mark configuration retained corresponds to a case of relatively simple dynamics: a two‐dimensional flow around a cylinder for a Reynolds number of 200. We generalize to the first and second‐order the method of calibration based on Tikhonov regularization recently used in (Physica D: Nonlinear Phenom. 2009; 238:416–427). Finally, we show that for this flow configuration this procedure is the most effective in terms of reduction of errors. Copyright © 2009 John Wiley & Sons, Ltd.

Boundary element solution of the two-dimensional sine-Gordon equation using continuous linear elements

This article studies the boundary element solution of two-dimensional sine-Gordon (SG) equation using continuous linear elements approximation. Non-linear and in-homogenous terms are converted to the boundary by the dual reciprocity method and a predictor–corrector scheme is employed to eliminate the non-linearity. The procedure developed in this paper, is applied to various problems involving line and ring solitons where considered in references [Argyris J, Haase M, Heinrich JC. Finite element approximation to two-dimensional sine-Gordon solitons. Comput Methods Appl Mech Eng 1991;86:1–26; Bratsos AG. An explicit numerical scheme for the sine-Gordon equation in 2+1 dimensions. Appl Numer Anal Comput Math 2005;2(2):189–211, Bratsos AG. A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation. Numer Algorithms 2006;43:295–308; Bratsos AG. The solution of the two-dimensional sine-Gordon equation using the method of lines. J Comput Appl Math 2007;206:251–77; Bratsos AG. A third order numerical scheme for the two-dimensional sine-Gordon equation. Math Comput Simul 2007;76:271–8; Christiansen PL, Lomdahl PS. Numerical solutions of 2+1 dimensional sine-Gordon solitons. Physica D: Nonlinear Phenom 1981;2(3):482–94; Djidjeli K, Price WG, Twizell EH. Numerical solutions of a damped sine-Gordon equation in two space variables. J Eng Math 1995;29:347–69; Dehghan M, Mirzaei D. The dual reciprocity boundary element method (DRBEM) for two-dimensional sine-Gordon equation. Comput Methods Appl Mech Eng 2008;197:476–86]. Using continuous linear elements approximation produces more accurate results than constant ones. By using this approach all cases associated to SG equation, which exist in literature, are investigated.

Diffuse interface method for simulation of the chemical vapor deposition of pyrolytic carbon: Aspects of the mathematical formulation

AbstractThe present work is inspired by Anderson et al. (Phys. D Nonlinear Phenom. 2000; 135(1–2):175–194) and Noll (http://www.math.cmu.edu/wn0g/noll) and falls in the conceptual line of the Ginzburg–Landau class of first‐order phase‐transition models based on the concept of phase‐field parameter. Trying to keep the exposition as much general as possible, we develop below a thermodynamically consistent rationalization of the physical process of (anisotropic) deposition of pyrolytic carbon from a gas phase. The derivation line we follow is well established in the field of the modern continuum physics. From the covariance of the first principle of thermodynamics, the second Newton's law and the Liouville's theorem with respect to the one‐dimensional Lie groups of transformations, the balance laws for the temperature, linear momentum and density are formulated. This system of partial differential equations is comprehended further by the constitutive laws for the phase field, the stress, and the heat and entropy fluxes obtained in a form consistent with the Clausius–Duhem understanding of the second law. The result referred to as a local, strongly coupled initial boundary value problem of chemical vapor deposition (IBVP‐CVD) constitutes the general mathematical description of the CVD process. The weak form of isotropic IBVP‐CVD is then derived and discretized by means of the discontinuous Galerkin method. At the end of the paper, we also derive the weak formulations for the local lifting operators that provide the stabilization mechanism for the discontinuous Galerkin discretization scheme. Copyright © 2008 John Wiley & Sons, Ltd.