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Discrete & Continuous Dynamical Systems - S

December 2014 , Volume 7 , Issue 6

Issue on nonlinear dynamical systems and applications in biology or engineering

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Preface
Zhaosheng Feng and Wei Feng
2014, 7(6): i-i doi: 10.3934/dcdss.2014.7.6i +[Abstract](596) +[PDF](71.8KB)
Abstract:
As we all know, many biological and physical systems, such as neuronal systems and disease systems, are featured by certain nonlinear and complex patterns in their elements and networks. These phenomena carry significant biological and physical information and regulate down-stream mechanism in many instances. This issue of Discrete and Continuous Dynamical Systems, Series S, comprises a collection of recent works in the general area of nonlinear differential equations and dynamical systems, and related applications in mathematical biology and engineering. The common themes of this issue include theoretical analysis, mathematical models, computational and statistical methods on dynamical systems and differential equations, as well as applications in fields of neurodynamics, biology, and engineering etc.
    Research articles contributed to this issue explore a large variety of topics and present many of the advances in the field of differential equations, dynamical systems and mathematical modeling, with emphasis on newly developed theory and techniques on analysis of nonlinear systems, as well as applications in natural science and engineering. These contributions not only present valuable new results, ideas and techniques in nonlinear systems, but also formulate a few open questions which may stimulate further study in this area. We would like to thank the authors for their excellent contributions, the referees for their tireless efforts in reviewing the manuscripts and making suggestions, and the chief editors of DCDS-S for making this issue possible. We hope that these works will help the readers and researchers to understand and make future progress in the field of nonlinear analysis and mathematical modeling.
Fluid structure interaction problem with changing thickness beam and slightly compressible fluid
Eugenio Aulisa, Akif Ibragimov and Emine Yasemen Kaya-Cekin
2014, 7(6): 1133-1148 doi: 10.3934/dcdss.2014.7.1133 +[Abstract](712) +[PDF](473.4KB)
Abstract:
In this work, we consider the dynamical response of a non-linear beam with changing thickness, perturbed in both vertical and axial directions, interacting with a Darcy flow. We explore this fluid-structure interaction problem where the fluid is assumed to be slightly compressible. In an appropriate Sobolev norm, we build an energy functional for the displacement field of the beam and the gradient pressure of the fluid flow. We show that for a class of boundary conditions the energy functional is bounded by the flux of mass through the inlet boundary.
Well-posedness for the BBM-equation in a quarter plane
Jerry L. Bona, Hongqiu Chen and Chun-Hsiung Hsia
2014, 7(6): 1149-1163 doi: 10.3934/dcdss.2014.7.1149 +[Abstract](818) +[PDF](396.1KB)
Abstract:
The so-called wave-maker problem for the $BBM$-equation is studied on the half-line. Improving on earlier results, global well-posedness is established for square-integrable initial data and boundary data that is only assumed to be locally bounded.
Connections of zero curvature and applications to nonlinear partial differential equations
Paul Bracken
2014, 7(6): 1165-1179 doi: 10.3934/dcdss.2014.7.1165 +[Abstract](613) +[PDF](396.0KB)
Abstract:
A general formulation of zero curvature connections in a principle bundle is presented and some applications are discussed. It is proved that a related connection based on a prolongation in an associated bundle remains zero curvature as well. It is also shown that the connection coefficients can be defined so that the partial differential equation to be studied appears as the curvature term in the structure equations. It is discussed how Lax pairs and Bäcklund tranformations can be formulated for such equations that occur as zero curvature terms.
Alternate steady states for classes of reaction diffusion models on exterior domains
Dagny Butler, Eunkyung Ko and R. Shivaji
2014, 7(6): 1181-1191 doi: 10.3934/dcdss.2014.7.1181 +[Abstract](626) +[PDF](451.2KB)
Abstract:
We study positive radial solutions to the problem \begin{equation*} \left\{ \begin{split} -\Delta u &= \lambda K(|x|)f(u), \quad x \in \Omega, \\u(x) &= 0 \qquad
   
   
    \mbox{ if } |x|=r_0, \\u(x) &\rightarrow 0 \qquad
   
    \mbox{ as } |x|\rightarrow\infty, \end{split} \right. \end{equation*} where $\Delta u=div \big(\nabla u\big)$ is the Laplacian of $u$, $\lambda$ is a positive parameter, $\Omega=\{x\in\mathbb{R}^N: |x|>r_0\}$, $r_0>0$, and $N>2$. Here, $f\in C^2[0,\infty)$ and $f(u)>0$ on $(0,\sigma)$ and $f(u)<0$ for $u>\sigma$. Furthermore, $K:[r_0, \infty)\rightarrow(0,\infty)$ is continuous and $\lim_{r\rightarrow\infty}K(r)=0$. We discuss the existence of multiple positive solutions for a certain range of $\lambda$ leading to the occurrence of an S-shaped bifurcation curve when $f$ satisfies some additional assumptions. In particular, the two models we consider are $f_1(u)=u-\frac{u^2}{K}-c\frac{u^2}{1+u^2}$ and $f_2(u)=\tilde{K}-u+\tilde{c}\frac{u^4}{1+u^4}$. We prove our results by the method of sub-super solutions.
Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity
José Caicedo, Alfonso Castro, Rodrigo Duque and Arturo Sanjuán
2014, 7(6): 1193-1202 doi: 10.3934/dcdss.2014.7.1193 +[Abstract](835) +[PDF](366.0KB)
Abstract:
For Dirichlet-periodic and double periodic boundary conditions, we prove the existence of solutions to a forced semilinear wave equation with large forcing terms not flat on characteristics. The nonlinearity is assumed to be non-monotone, asymptotically linear, and not resonanant. We prove that the solutions are in $L^{p}$, $(p\geq 2)$, when the forcing term is in $L^{p}$. This is optimal; even in the linear case there are $L^p$ forcing terms for which the solutions are only in $L^p$. Our results extend those in [9] where the forcing term is assumed to be in $L_{\infty}$, and are in contrast with those in [6] where the non-existence of continuous solutions is established for $C^{\infty}$ forcing terms flat on characteristics. 200 words.
Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms
Zengji Du, Xiao Chen and Zhaosheng Feng
2014, 7(6): 1203-1214 doi: 10.3934/dcdss.2014.7.1203 +[Abstract](699) +[PDF](398.2KB)
Abstract:
In this paper, we study a delayed predator-prey model with the Leslie-Gower Holling-type II functional response and harvesting terms. The existence of multiple positive periodic solutions for the system and the permanence of the predator-prey model are obtained by means of the generalized Mawhin coincidence degree theory.
Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain
Wei Feng, Nicole Rocco, Michael Freeze and Xin Lu
2014, 7(6): 1215-1230 doi: 10.3934/dcdss.2014.7.1215 +[Abstract](663) +[PDF](456.7KB)
Abstract:
In this paper, we study a new model as an extension of the Rosenzweig-MacArthur tritrophic food chain model in which the super-predator consumes both the predator and the prey. We first obtain the ultimate bounds and conditions for exponential convergence for these populations. We also find all possible equilibria and investigate their stability or instability in relation with all the ecological parameters. Our main focus is on the conditions for the existence, uniqueness and stability of a coexistence equilibrium. The complexity of the dynamics in this model is theoretically discussed and graphically demonstrated through various examples and numerical simulations.
Duffing-van der Pol-type oscillator systems
Zhaosheng Feng
2014, 7(6): 1231-1257 doi: 10.3934/dcdss.2014.7.1231 +[Abstract](667) +[PDF](869.4KB)
Abstract:
In this paper, under certain parametric conditions we are concerned with the first integrals of the Duffing-van der Pol-type oscillator system, which include the van der Pol oscillator and the damped Duffing oscillator etc as particular cases. After applying the method of differentiable dynamics to analyze the bifurcation set and bifurcations of equilibrium points, we use the Lie symmetry reduction method to find two nontrivial infinitesimal generators and use them to construct canonical variables. Through the inverse transformations we obtain the first integrals of the original oscillator system under the given parametric conditions, and some particular cases such as the damped Duffing equation and the van der Pol oscillator system are included accordingly.
Dynamics of two phytoplankton species competing for light and nutrient with internal storage
Sze-Bi Hsu and Chiu-Ju Lin
2014, 7(6): 1259-1285 doi: 10.3934/dcdss.2014.7.1259 +[Abstract](610) +[PDF](6273.7KB)
Abstract:
We analyze a competition model of two phytoplankton species for a single nutrient with internal storage and light in a well mixed aquatic environment. We apply the theory of monotone dynamical system to determine the outcomes of competition: extinction of two species, competitive exclusion, stable coexistence and bistability of two species. We also present the graphical presentation to classify the competition outcomes and to compare outcome of models with and without internal storage.
On the arrow of time
Y. Charles Li and Hong Yang
2014, 7(6): 1287-1303 doi: 10.3934/dcdss.2014.7.1287 +[Abstract](694) +[PDF](759.9KB)
Abstract:
We believe the following three ingredients are enough to explain the mystery of the arrow of time: (1). equations of dynamics of gas molecules, (2). chaotic instabilities of the equations of dynamics, (3). unavoidable perturbations to the gas molecules. The level of physical rigor or mathematical rigor that can be reached for such a theory is unclear.
Stoichiometric producer-grazer models with varying nitrogen pools and ammonia toxicity
Hao Liu, Aaron Packer and Yang Kuang
2014, 7(6): 1305-1320 doi: 10.3934/dcdss.2014.7.1305 +[Abstract](592) +[PDF](414.4KB)
Abstract:
We formulate and analyze a stoichiometric model of producer-grazer systems with excess nutrient recycling (waste) that may inhibit grazer survival and growth. Specifically, we model the intoxication dynamics caused by accumulation of grazer waste and dead biomass decay. This system has a range of applications, but we focus on those in which the producers are microalgae and the limiting nutrient is nitrogen. High levels of ammonia (and to a lesser extent nitrite) have been observed to increase grazer death, especially in aquaculture systems. We assume that all nitrification is due to nitrogen uptake and assimilation by the producer; therefore, the model explores systems in which the producer serves the dual role of grazer food and water treatment. The model exhibits three equilibria corresponding to total extinction, grazer-only extinction, and coexistence. While a sufficient condition is found under which grazer extinction equilibrium is globally stable, we propose a conjecture for neccessary and sufficient conditions, which remains an open mathematical problem. Local stability of grazer extinction equilibrium is ensured under a sharp necessary and sufficient condition. Local stability for the coexistence equilibrium is studied algebraically and numerically. Bifurcation diagrams with respect to total nitrogen and its implications are also presented.
Control via decoupling of a class of second order linear hybrid systems
David L. Russell
2014, 7(6): 1321-1334 doi: 10.3934/dcdss.2014.7.1321 +[Abstract](574) +[PDF](538.8KB)
Abstract:
We study a terminal state control (reachability) problem for certain elastic systems of ``hybrid" type consisting of a single space dimension distributed parameter part coupled, at one endpoint of the relevant spatial, $x$, interval, to a lumped mass component. Two such systems are studied in detail. The first is a vibrating string system fixed at $x = 0$ and attached to a point mass at the right hand endpoint $x = L$. The second example concerns an Euler - Bernoulli beam system ``clamped" at $x = 0$ and attached, at $x = L$, to a mass with both translational and rotational inertia. In both cases the controls act on the mass, which is modeled by a finite dimensional system of differential equations. Analysis of the reachability problem is facilitated by a preliminary ``feedback type" transformation of the control variable which decouples the point mass from the distributed system. In both examples a concluding analysis is required to show that the component of the control generated by feedback lies in the same space as the originally applied control.
On the growth of positive entire solutions of elliptic PDEs and their gradients
Antonio Vitolo
2014, 7(6): 1335-1346 doi: 10.3934/dcdss.2014.7.1335 +[Abstract](665) +[PDF](375.4KB)
Abstract:
We investigate the growth of entire positive functions $u(x)$ and their gradients $Du$ in Sobolev spaces when a polynomial growth is assumed for their image $Lu$ through a linear second-order uniform elliptic operator $L$. In particular, under suitable assumptions on the coefficients, we show that if $Lu$ is bounded, then $u(x)$ may grow at most quadratically at infinity. We also discuss, by counterexamples, the optimality of the assumptions and extend the results to viscosity solutions of fully nonlinear equations.
Random attractor for stochastic reversible Schnackenberg equations
Yuncheng You
2014, 7(6): 1347-1362 doi: 10.3934/dcdss.2014.7.1347 +[Abstract](641) +[PDF](445.6KB)
Abstract:
Asymptotic dynamics of stochastic reversible Schnackenberg equations with multiplicative white noise, which is a typical trimolecular autocatalytic reaction-diffusion system on a three-dimensional bounded domain with Dirichlet boundary condition, is investigated in this paper. The existence of a random attractor is proved through uniform grouping estimates showing the pullback absorbing property and the pullback asymptotic compactness.
On bursting solutions near chaotic regimes in a neuron model
Feng Zhang, Alice Lubbe, Qishao Lu and Jianzhong Su
2014, 7(6): 1363-1383 doi: 10.3934/dcdss.2014.7.1363 +[Abstract](899) +[PDF](1116.7KB)
Abstract:
In this paper, we use mathematical analysis to study the transition of dynamic behavior in a system of two synaptically coupled Hindmarsh-Rose (HR) neurons, based on its flow-induced Poincaré map. Numerical simulations have shown that the individual HR neuron has chaotic behavior, but neurons become regularized when coupled. Using a geometric method for dynamical systems, we begin with an investigation of the bifurcation structure of its fast subsystem. We show that the emergence of regular patterns out of chaos is due to a topological change in its underlying bifurcations. Then we focus on the transitional phase of coupling strength, where the bursting solutions need to pass near two homoclinic bifurcation points located on a branch of saddle points, and we study the flow-induced Poincaré maps. We observe that as the gap between the homoclinic points narrows, the image of the return map moves away from chaotic regions where winding numbers vary abruptly. That, along with Lyaponov exponent calculations, reveals the fine dynamics in the pathway across chaotic bursting behavior and regular bursting of coupled HR neurons as the synaptic coupling strength of the neurons increases. The main contribution of this paper is the mathematical description of the transition of synaptically coupled neurons from chaotic trajectories to regular burst phases using dynamical system tools such as Poincaré maps.

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