One-dimensional Nonlinear Systems Research Articles

A Numerical Study for the Solution of Time Fractional Nonlinear Shallow Water Equation in Oceans

In this paper, an analytical solution for the coupled one-dimensional time fractional nonlinear shallow water system is obtained by using the homotopy perturbation method (HPM). The shallow water equations are a system of partial differential equations governing fluid flow in the oceans (sometimes), coastal regions (usually), estuaries (almost always), rivers and channels (almost always). The general characteristic of shallow water flows is that the vertical dimension is much smaller than the typical horizontal scale. This method gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. A very satisfactory approximate solution of the system with accuracy of the order 10-4 is obtained by truncating the HPM solution series at level six.

One-dimensional nonlinear chromatography system and delta-shock waves

The Riemann problem for the nonlinear chromatography system is considered. Existence and admissibility of δ-shock type solution in both variables are established for this system. By the interactions of δ-shock wave with elementary waves, the generalized Riemann problem for this system is presented, the global solutions are constructed, and the large time-asymptotic behavior of the solutions are analyzed. Moreover, by studying the limits of the solutions as perturbed parameter \({\varepsilon}\) tends to zero, one can observe that the Riemann solutions are stable for such perturbations of the initial data.

Asymptotics of the solutions of the one-dimensional nonlinear system of equations of shallow water with degenerate velocity

A system of one-dimensional nonlinear equations of shallow water with degenerate velocity is considered. The change of variables taking the given system to a nonlinear system with small nonlinearity is proposed. Formal asymptotic solutions near the point of degeneracy are obtained.

Numerical Solution of Coupled-BBM Systems of Boussinesq type by Implicit Finite Difference Method

In this paper we study a numerical solution of coupled BBM systems of Boussinesq type, which describes approximately the two ways propagation of surface wave in a uniform horizontal channel of length l filled in its undisturbed depth h. This paper is devoted to drive the matrix algebraic equation for the one-dimensional nonlinear BBM system which is obtaining from using the implicit finite difference method. The convergence analysis of the solution is proved. Numerical experiments are presented with a variety of initial conditions describing the generation and evolution of such waves, and their interactions.

Particle Model Predictive Control for Probability Density Functions

AbstractIn industrial processes such as steel making, it is often important to suppress the dispersion of end-product quality, which motivates a control problem of probability density functions (PDFs) representing the dispersion of the state of dynamical systems. Although we consider a system whose state equation is represented by a nonlinear ordinary differential equation, the time evolution of a PDF of the state is governed by a partial differential equation (PDE) with a finite dimensional control input. Model predictive control (MPC), also known as receding horizon control, is applied for the optimal control of the dispersion. An MPC method of the PDF is formulated exactly using the time evolution of the PDF, which we call exact formulation. We propose another control method for the PDF, which is formulated by Monte Carlo (MC) approximation, and call this control method particle MPC (PMPC). We apply the proposed control method to a one-dimensional nonlinear system and compare the results of the two control methods, which show that the proposed control method is valid and has a high potential for real-time implementation.

UNIFORMITY AND SPATIAL CHAOS OF SPATIAL PHYSICS KINEMATIC SYSTEM

This article summarizes the uniformity law of spatial physics kinematic systems, and studies the chaos and bifurcation behavior of the uniform system in space. In particular, it also fully explains the relation among the uniform system, the coupled map lattice model which has attracted considerable interest currently, and one-dimensional nonlinear dynamical systems.

Gap solitons and Bloch waves in nonlinear periodic systems

We comprehensively investigate gap solitons and Bloch waves in one-dimensional nonlinear periodic systems. Our results show that there exists a composition relation between them: Bloch waves at either the center or edge of the Brillouin zone are infinite chains composed of fundamental gap solitons(FGSs). We argue that such a relation is related to the exact relation between nonlinear Bloch waves and nonlinear Wannier functions. With this composition relation, many conclusions can be drawn for gap solitons without any computation. For example, for the defocusing nonlinearity, there are $n$ families of FGS in the $n$th linear Bloch band gap; for the focusing case, there are infinite number of families of FGSs in the semi-infinite gap and other gaps. In addition, the stability of gap solitons is analyzed. In literature there are numerical results showing that some FGSs have cutoffs on propagation constant (or chemical potential), i.e. these FGSs do not exist for all values of propagation constant (or chemical potential) in the linear band gap. We offer an explanation for this cutoff.

Lagrangian dynamics on an infinite-dimensional torus; a Weak KAM theorem

The space L 2 ( 0 , 1 ) has a natural Riemannian structure on the basis of which we introduce an L 2 ( 0 , 1 ) -infinite-dimensional torus T . For a class of Hamiltonians defined on its cotangent bundle we establish existence of a viscosity solution for the cell problem on T or, equivalently, we prove a Weak KAM theorem. As an application, we obtain existence of absolute action-minimizing solutions of prescribed rotation number for the one-dimensional nonlinear Vlasov system with periodic potential.

Energy transport in a one-dimensional granular gas

We study heat conduction in one-dimensional granular gases. In particular, we consider two mechanisms of viscous dissipation during intergrain collisions. In one, the dissipative force is proportional to the grain's velocity and dissipates not only energy but also momentum. In the other, the dissipative force is proportional to the relative velocity of the grains and therefore conserves momentum even while dissipating energy. This allows us to explore the role of momentum conservation in the heat conduction properties of this one-dimensional nonlinear system. We find normal thermal conduction whether or not momentum is conserved.

Global attractor for the Ginzburg–Landau thermoviscoelastic systems with hinged boundary conditions

This paper is concerned with the following one-dimensional nonlinear system of equations: (0.1) u t t − P ( θ , ε ) x − ν u x x t + R u x x x x = f , (0.2) C V θ t − κ θ x x − θ P θ ε t − ν ε t 2 = g , where both ν and R are positive constants. The corresponding free energy density is assumed to be in Ginzburg–Landau form and nonconvex as a function of the order parameter. Results concerning the existence and uniqueness of the global solution, the asymptotic behavior of the solution as time tends to infinity and the compactness of the orbit are obtained. Furthermore, we investigate dynamics of the system and prove the existence of global attractor.

Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems

We give a new sufficient condition on the boundary conditions for the exponential stability of one-dimensional nonlinear hyperbolic systems on a bounded interval. Our proof relies on the construction of an explicit strict Lyapunov function. We compare our sufficient condition with other known sufficient conditions for nonlinear and linear one-dimensional hyperbolic systems.

Large-time behaviour of solutions to the equations of one-dimensional nonlinear thermoviscoelasticity with memory

This paper is concerned with the large-time behaviour of globally defined smooth solutions of the initial-boundary value problem for the one-dimensional nonlinear thermoviscoelasticity system with memory.

SUDDEN OCCURRENCE OF CHAOS IN NONSMOOTH MAPS

This paper introduces a type of one-dimensional nonsmooth nonlinear discrete dynamic system. We find a direct route to chaos from stable period-two point, and this is called Sudden Occurrence of Chaos. It is completely different from the three routes from regular motion to chaos — period-doubling bifurcation chaos, intermittency and quasi-periodicity chaos. Furthermore, we present some examples of sudden occurrence of chaos from m-period directly to chaos.

Derivation and analysis of a fluid-dynamical model in thin and long elastic vessels

Starting from the three-dimensional Newtonian and incompressible Navier-Stokes equations in a compliant straight vessel, we derive a reduced one-dimensional model by an averaging procedure which takes into consideration the elastic properties of the wall structure. In particular, we neglect terms of the first order with respect to the ratio between the vessel radius and length. Furthermore, we consider that the viscous effects are negligible with respect to the propagative phenomena. The result is a one-dimensional nonlinear hyperbolic system of two equations in one space dimension, which describes the mean longitudinal velocity of the flow and the radial wall displacement. The modelling technique here applied to straight cylindrical vessels may be generalized to account for curvature and torsion. An analysis of well posedness is presented which demonstrates, under reasonable hypotheses, the global in time existence of regular solutions.

Nonlinear equations for p-adic open, closed, and open-closed strings

We investigate the structure of solutions of boundary value problems for a one-dimensional nonlinear system of pseudodifferential equations describing the dynamics (rolling) of p-adic open, closed, and open-closed strings for a scalar tachyon field using the method of successive approximations. For an open-closed string, we prove that the method converges for odd values of p of the form p=4n+1 under the condition that the solution for the closed string is known. For p=2, we discuss the questions of the existence and the nonexistence of solutions of boundary value problems and indicate the possibility of discontinuous solutions appearing.

Excitation-Induced Stability and Phase Transition: A Review

Dynamical systems may experience undesirable behavior or instability, which can be eliminated using feedback control means. However, in the absence of feedback control, the stability of some systems may be increased by imposing parametric excitation. In other cases, the exit time of the system response from the stable to the unstable domain may be prolonged by imposing external noise, a phenomenon termed noise-enhanced stability (NES). This article presents an assessment of the mechanisms of stabilization via multiplicative noise and noise-enhanced stability. The first part deals with stabilization via deterministic parametric excitation of gravity-defying systems such as the inverted simple and spherical pendulums, aeroelastic structures, human walking, and quantum nonlinear couplers. The second part introduces the concept of noise-induced transition (NIT) in one-dimensional nonlinear systems and ship roll motion. Stabilization of originally unstable systems via multiplicative noise is treated in the third part. The fourth part addresses the influence of additive noise in delaying the exit time of system response to an unstable domain. This topic is related to the phenomenon of stochastic resonance (SR) and NES of systems with one or more metastable states and fluctuating potential. Finally, this review article also introduces some applications in other fields such as the Ising model, ecosystems, and tumor-immune system models.

Numerical solutions of inverse spatial Lotka-Volterra systems

A numerical marching scheme is introduced for the recovery of the solutions and gradient distributions in a one-dimensional nonlinear system of parabolic partial differential equations with space dependent coefficients. In this inverse problem, only Cauchy noisy data at the active boundary is given and no information about the amount and/or character of the noise in the data is assumed. The error analysis for the algorithm is discussed and several numerical examples are presented.

Novel approximations for inference in nonlinear dynamical systems using expectation propagation

We formulate the problem of inference in nonlinear dynamical systems in the framework of expectation propagation, and propose two novel algorithms. The first algorithm is based on Laplace approximation and allows for iterated forward and backward passes. The second is based on repeated application of the unscented transform. It leads to an unscented Kalman smoother for which the dynamics need not be inverted explicitly. In experiments with a one-dimensional nonlinear dynamical system we show that for relatively low observation noise levels, the Laplace algorithm allows for the best estimates of the state means. The unscented algorithm however is more robust to high observation noise and always outperforms the conventional inference methods against which it was benchmarked.

SOME CONTROL RESULTS FOR SIMPLIFIED ONE-DIMENSIONAL MODELS OF FLUID-SOLID INTERACTION

We analyze the null controllability of a one-dimensional nonlinear system which models the interaction of a fluid and a particle. This can be viewed as a first step in the control analysis of fluid-solid systems. The fluid is governed by the Burgers equation and the control is exerted at the boundary points. We present two main results: the global null controllability of a linearized system and the local null controllability of the nonlinear original model. The proofs rely on appropriate global Carleman inequalities, observability estimates and fixed point arguments.

Finite differences method for shallow water equations in a class of discontinuous functions

In this study a finite difference method for solving initial boundary value problem for the one-dimensional nonlinear system of differential equations describing shallow water flows in a class of discontinuous functions is suggested. In order to find the numerical solution of the problem, known as main problem, a special auxiliary problem is introduced. The degree of smoothness of the solution of the auxiliary problem is higher than the smoothness of the solution of the main problem. Moreover, the suggested auxiliary problem makes it possible to write out the effective and higher order numerical algorithms. Some results of numerical experiments are demonstrated.