Narrow Results

This paper presents a novel result on the effect of coupling through both analytical and numerical investigations on linearly coupled systems including chaotic and nonchaotic systems. It is found that when a single system has potential of oscillation but is currently in a "marginal" state to produce a limit cycle via Hopf bifurcation due to the change of a parameter, an appropriate coupling strength can excite the potential limit cycle such that the coupled system oscillates synchronously. Similarly, when a stable limit cycle is at the "margin" of a chaotic attractor in a single system, a certain coupling strength can induce the potential chaotic attractor such that the coupled system has a synchronous chaotic behavior. This excitation mechanism is different from the traditional function of coupling in that the latter mainly drives the coupled system to synchronize with the ongoing dynamics of a single system but does not recover its disappearing dynamics. This newly observed synchronization is called coherent synchronization to distinguish it from various common types of synchronization. Several numerical examples are presented for quantitative description of this interesting phenomenon.

In this paper, we have studied the dynamics of two identical resistively coupled Chua's canonical circuits and have found that it is strongly affected by initial conditions, coupling strength and the presence of coexisting attractors. Depending on the coupling variable, chaotic synchronization has been observed both numerically and experimentally. Anti-phase synchronization has also been studied numerically clarifying some aspects of uncertainty in chaos synchronization.

In the study of complex systems one of the major concerns is the detection and characterization of causal interdependencies and couplings between different subsystems. The nature of such dependencies is typically not only nonlinear but also asymmetric and thus makes the use of symmetric and linear methods ineffective. Moreover, signals sampled from real world systems are noisy and short, posing additional constraints on the estimation of the underlying couplings. In this article, we compare a set of six recently introduced methods for quantifying the causal structure of bivariate time series extracted from systems with complex dynamical behavior. We discuss the usefulness of the methods for detecting asymmetric couplings and directional flow of information in the context of uni- and bidirectionally coupled deterministic chaotic systems.

This paper reveals the dynamical properties of two interacting neural networks with multiple couplings. Different time delays are introduced into the nearest-neighbor links and long-range connections in each layer and the couplings between different substructures. The delay-dependent and delay-independent stability and the oscillations bifurcated from the trivial equilibrium of the network are analyzed. The conditions of the existence of nontrivial equilibria and pitchfork bifurcation are discussed. Numerical simulations are performed to validate the theoretical results and interesting neuronal activities are observed, such as completely synchronous oscillations, three types of asynchronous oscillations, multiple switches between the rest states and different oscillations, coexistence of different oscillations, and coexistence of nontrivial equilibria and oscillations.

In this work, we propose to use the distance correlation ($DC$) and the Euclidian distance ($ED$) as powerful quantifiers to describe correlation and synchronization in spatially extended systems. As examples, we use the coupled Kuramoto oscillator model and coupled map lattices to study chimera states. Results for $DC$ and $ED$ are compared and discussed in terms of recurrence plots, local synchronization order parameter, local recurrence rate, and local $DC$ rate. The existence of correlations between desynchronized states, not visible in usual recurrence plots, becomes evident when using $DC$ and $ED$. Since correlation is a broader concept than synchronization, the proposed quantifiers amplify the characterization, description and possible application of spatially extended systems.

In refined network analysis, a compact network model is combined with distributed models for semiconductor devices in a multiphysics approach. For linear RLC networks containing diodes as distributed devices, we construct a mathematical model that combines the differential-algebraic network equations of the circuit with elliptic boundary value problems modeling the diodes. For this mixed initial-boundary value problem of partial differential-algebraic equations a first existence result is given.

In this paper, we prove the existence, uniqueness and various kinds of Ulam stability for fractional order coupled systems with fractional order boundary conditions involving Riemann–Liouville fractional derivatives. The standard fixed point theorem like Leray–Schauder alternative and Banach contraction are applied to establish the existence theory and uniqueness. Furthermore, we build sufficient conditions for the stability mentioned above by two methods. Also, an example is given to illustrate our theoretical results. The proposed problem is the generalization of third-order ordinary differential equations with classical, initial and anti-periodic boundary conditions.

We study the well-posedness for multi-dimensional and coupled systems of forward–backward SDEs when the generator can be separated into a quadratic and a subquadratic part. We obtain the existence and uniqueness of the solution on a small time interval. Moreover, the continuity and differentiability with respect to the initial value are presented.

In this paper, we mathematically study a particular process for assimilation in the brain. The research aims to establish a theoretical model at computational level of the mechanism in a cognitive process operated by the mirror-neuron system, to generate a multi-dimensional system from this model, and to analyze the fundamentals of the related cognitive process in terms of dynamical systems. Finally, to understand the interactions between two individual mirror-neuron systems, we formulate and examine coupled systems that are composed of two distinct systems. We also carry out various numerical simulations to illustrate the assimilation process.

The main purpose of this paper is to present computational methods for simulating some typical problems in the emerging computational geoscience field. Due to remarkable differences between engineering systems and Earth ones, existing computational methods, which are designed for solving engineering problems, cannot be directly used to solve geoscience problems without any modification. However, the fundamental philosophy of developing computational methods is applicable to the computational simulation of both geoscience and engineering problems. Because of their inherent approximation, computational methods must be verified before putting into application. After several computational methods and algorithms, which are developed for simulating some typical problems in the emerging computational geoscience field, are briefly introduced, a typical geoscience problem, known as the chemical dissolution-front instability problem in ore-forming systems of supercritical Zhao numbers, is selected to demonstrate how computational methods can be used to solve geoscience problems.

A number of engineering applications involve contact with bodies modeled using specialized theories of solid mechanics like beams or shells. While computational models for contact in 2D and 3D solid mechanics have been extensively developed in the literature, problems involving contact with beams or shells have received less attention. When modeling contact between a solid body represented with beam or shell theory and a domain discretized with solid finite elements, the contact model faces the typical challenges of enforcing geometric compatibility and the transfer of a complete pressure field along the contact interface, with the added complications stemming from the different underlying mathematical formulations and finite element discretizations in the connecting domains. Resultant-based beam and shell theories do not provide direct estimates of surface tractions, therefore rendering the issue of pressure transfer on beam–solid and shell–solid interfaces more problematic. In the absence of specialized contact formulations for solid–beam and solid–shell interfaces, contact models have relied almost exclusively on the Node-To-Surface (NTS) geometric compatibility approach. This formulation suffers from well-known drawbacks, including instability, surface locking and incomplete pressure fields on the interface. The NTS approach, however, remains the method most readily applicable to contact with beam or shell elements among the vast variety of available methods for computational contact modeling using finite elements. The goal of this paper is to bridge the gap in the literature on coupling domains with beam and solid finite element discretizations. We propose an interface formulation for beam–solid interfaces that ensures the transfer of a complete pressure field while enforcing geometric compatibility using standard NTS constraints. The formulation uses a stabilization approach, based on a special form of the Discontinuous Galerkin method, to enforce weak continuity between the stress fields on the solid side of the interface, and the moment and shear resultants in the contacting beam. We show that the proposed formulation is a robust approach for satisfying compatibility constraints while ensuring the transfer of a complete pressure field on beam–solid finite element interfaces that can be used with bilinear and quadratic interpolations in the solid, and Euler or Timoshenko formulations for the beam.

In this paper, a susceptible–exposed–infective–recovered–susceptible (SEIRS) epidemic model with vaccination has been formulated. We studied the global stability of the corresponding single-group model, multi-group model with strongly connected network and multi-group model without strongly connected network by means of analyzing their basic reproduction numbers and the application of Lyapunov functionals. Finally, we provide some numerical simulations to illustrate our analysis results.