Nonlinear Differential-difference System Research Articles

Stability of Delayed Homogeneous Positive Differential–Difference Systems

Stability theory of linear differential-difference system has been well-established, while fewer results can be found for nonlinear differential-difference systems. In this paper, stability and boundedness of homogeneous differential-difference system with bounded delay is studied based on the system positivity. At first, an exponential stability criterion is obtained, which is an extension of an existing result. Next, another boundary is computed under the previous condition, which proves to be tighter than the first result at least in some cases. Then a finite-time stability condition is achieved for the delay-free system, and an upper bound of the settling time and an explicit boundary of the state are derived. Finally, a numerical example is given to verify the results presented in this paper.

Nonlinear differential-difference hierarchy relevant to the Ablowitz-Ladik equation: Lax pair, conservation laws, N-fold Darboux transformation and explicit exact solutions

Nonlinear differential-difference equations appear in optics, condensed matter physics, plasma physics and other fields. In this paper, we investigate a nonlinear differential-difference hierarchy relevant, in the case of θ=0, to the Ablowitz-Ladik equation, where θ=0,1. That hierarchy is obtained via a discrete spectral problem and the associated discrete spectral problem. When θ=1, Lax pair of the first nonlinear differential-difference system in that hierarchy is obtained. When θ=1, conservation laws and N-fold Darboux transformation of the first nonlinear differential-difference system in that hierarchy are derived with the aid of that Lax pair, where N is a positive integer. When θ=1, explicit exact solutions of that system are determined via that N-fold Darboux transformation. Discrete one soliton and interaction between the discrete one soliton and one breather-like wave are graphically depicted.

DECOMPOSING A NEW NONLINEAR DIFFERENTIAL-DIFFERENCE SYSTEM UNDER A BARGMANN IMPLICIT SYMMETRY CONSTRAINT

Firstly, a hierarchy of integrable lattice equations and its bi-Hamilt-onian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.

Control in dormancy or eradication of cancer stem cells: Mathematical modeling and stability issues

Objective: Modeling and analysis of cell population dynamics enhance our understanding of cancer. Here we introduce and explore a new model that may apply to many tissues.Analyses: An age-structured model describing coexistence between mutated and ordinary stem cells is developed and explored. The model is transformed into a nonlinear time-delay system governing the dynamics of healthy cells, coupled to a nonlinear differential-difference system describing dynamics of unhealthy cells. Its main features are highlighted and an advanced stability analysis of several steady states is performed, through specific Lyapunov-like functionals for descriptor-type systems.Results: We propose a biologically based model endowed with rich dynamics. It incorporates a new parameter representing immunoediting processes, including the case where proliferation of cancer cells is locally kept under check by the immune cells. It also considers the overproliferation of cancer stem cells, modeled as a subpopulation of mutated cells that is constantly active in cell division. The analysis that we perform here reveals the conditions of existence of several steady states, including the case of cancer dormancy, in the coupled model of interest. Our study suggests that cancer dormancy may result from a plastic sensitivity of mutated cells to their shared environment, different from that – fixed – of healthy cells, and this is related to an action (or lack of action) of the immune system. Next, the stability analysis that we perform is essentially oriented towards the determination of sufficient conditions, depending on all the model parameters, that ensure either a regionally (i.e., locally) stable dormancy steady state or eradication of unhealthy cells. Finally, we discuss some biological interpretations, with regards to our findings, in light of current and emerging therapeutics. These final insights are particularly formulated in the paradigmatic case of hematopoiesis and acute leukemia, which is one of the best known malignancies for which it is always hard, in presence of a clinical and histological remission, to decide between cure and dormancy of a tumoral clone.

The Impulse-Refractive Mode in the Neural Network with Ring Synaptic Interaction

In the paper, a mathematical model of a neural network with an even number of ring synaptic interaction elements is considered. The model is a system of scalar nonlinear differentialdifference equations, the right parts of which depend on large parameters. The unknown functions included in the system characterize the membrane potentials of the neurons. The search of special impulse-refraction cycles within the system of equations is of interest. The functions with odd numbers of the impulse-refraction cycle have an asymptotically high pulses and the functions with even numbers are asymptotically small. Two changes allow to study a two-dimension nonlinear differential-difference system with two delays instead of the system. Further, a limit object that represents a relay system with two delays is defined by a large parameter tending to infinity. There exists the only periodic solution of the relay system with the initial function from a suitable function class. This is structurally proved, by using the step method. Next, the existence of relaxation periodic solutions of the two-dimension singularly perturbed system is proved by using the Poincare operator and the Schauder principle. The asymptotics of this solution is constructed, and it is proved that the solution is close to the decision of the relay system. Because of the exponential estimate of the Frechet derivative of the Poincare operator it implies the uniqueness and stability of solutions of the two-dimension differential-difference equation with two delays. Furthermore, with the help of reverse replacement the proved result is transferred to the original system.

Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons

Using a comparison theorem technique, we study the long time behavior of certain classes of nonlinear difference-differential systems. Zero is a solution for these systems. We are concerned in this paper with conditions forcing nonconvergence to zero of solutions as time approaches infinity; that is, we obtain threshold properties of the systems. The results parallel results by Aronson and Weinberger on reaction-diffusion equations somewhat, and the study was motivated by consideration of models for myelinated nerve axons.

Behaviour of some models of myelinated axons.

In this paper we discuss several approaches to modelling myelinated axons and examine the qualitative behaviour of the models. To facilitate our goal of understanding in detail the differences in mechanisms modelling myelination, we impose at the nodes the simplest nonlinear current-voltage relation which allows the models to possess appropriate threshold behaviour and propagating action potentials (travelling waves). Our type of model is a nonlinear differential-difference system and the resulting travelling wave must satisfy a nonlinear delay-differential equation of mixed type. Another type of model is a diffusion equation coupled nonlinearly to ordinary differential systems whose solutions represent boundary data for the diffusion equation. We give some threshold results and derive a relationship between conduction speed and various model parameters for a few classes of these models.

Some threshold results for models of myelinated nerves

A model of a myelinated axon is constructed, and some threshold conditions are derived using Lyapunov and comparison theorem techniques. The model takes the form of a nonlinear differential-difference system, and the threshold conditions show the rest state to be locally, but not globally, stable.