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Discrete & Continuous Dynamical Systems - B

Open Access Articles

Preface of the special issue
Yihong Du, Je-Chiang Tsai, Feng-Bin Wang and Xiao-Qiang Zhao
2021, 26(4): ⅰ-ⅱ doi: 10.3934/dcdsb.2021071 +[Abstract](360) +[HTML](139) +[PDF](83.14KB)
Preface for the special issue "20 years of DCDS-B"
Peter E. Kloeden and Yuan Lou
2021, 26(1): i-ii doi: 10.3934/dcdsb.2020372 +[Abstract](744) +[HTML](237) +[PDF](85.05KB)
Chaotic dynamics in a simple predator-prey model with discrete delay
Guihong Fan and Gail S. K. Wolkowicz
2021, 26(1): 191-216 doi: 10.3934/dcdsb.2020263 +[Abstract](624) +[HTML](251) +[PDF](5405.39KB)

A discrete delay is included to model the time between the capture of the prey and its conversion to viable biomass in the simplest classical Gause type predator-prey model that has equilibrium dynamics without delay. As the delay increases from zero, the coexistence equilibrium undergoes a supercritical Hopf bifurcation, two saddle-node bifurcations of limit cycles, and a cascade of period doublings, eventually leading to chaos. The resulting periodic orbits and the strange attractor resemble their counterparts for the Mackey-Glass equation. Due to the global stability of the system without delay, this complicated dynamics can be solely attributed to the introduction of the delay. Since many models include predator-prey like interactions as submodels, this study emphasizes the importance of understanding the implications of overlooking delay in such models on the reliability of the model-based predictions, especially since temperature is known to have an effect on the length of certain delays.

Yoshikazu Giga, Peter Kloeden, Irena Lasiecka, Peter Markowich, Elisabetta Rocca, Enrico Valdinoci, Enrique Zuazua and Krzysztof Ciepliński
2020, 25(10): i-ii doi: 10.3934/dcdsb.2020265 +[Abstract](516) +[HTML](145) +[PDF](78.88KB)
From approximate synchronization to identical synchronization in coupled systems
Chih-Wen Shih and Jui-Pin Tseng
2020, 25(9): 3677-3714 doi: 10.3934/dcdsb.2020086 +[Abstract](981) +[HTML](375) +[PDF](970.2KB)

We establish a framework to investigate approximate synchronization of coupled systems under general coupling schemes. The units comprising the coupled systems may be nonidentical and the coupling functions are nonlinear with delays. Both delay-dependent and delay-independent criteria for approximate synchronization are derived, based on an approach termed sequential contracting. It is explored and elucidated that the synchronization error, the distance between the asymptotic state and the synchronous set, decreases with decreasing difference between subsystems, difference between the row sums of connection matrix, and difference of coupling time delays between different units. This error vanishes when these factors decay to zero, and approximate synchronization becomes identical synchronization for the coupled system comprising identical subsystems and connection matrix with identical row sums, and with identical coupling delays. The application of the present theory to nonlinearly coupled heterogeneous FitzHugh-Nagumo neurons is illustrated. We extend the analysis to study approximate synchronization and asymptotic synchronization for coupled Lorenz systems and show that for some coupling schemes, the synchronization error decreases as the coupling strength increases, whereas in another case, the error remains at a substantial level for large coupling strength.

Local orthogonal rectification: Deriving natural coordinates to study flows relative to manifolds
Benjamin Letson and Jonathan E. Rubin
2020, 25(9): 3725-3747 doi: 10.3934/dcdsb.2020088 +[Abstract](886) +[HTML](360) +[PDF](1964.9KB)

We recently derived a method, local orthogonal rectification (LOR), that provides a natural and useful geometric frame for analyzing dynamics relative to a base curve in the phase plane for two-dimensional systems of ODEs (Letson and Rubin, SIAM J. Appl. Dyn. Syst., 2018). This work extends LOR to apply to any embedded base manifold in a system of ODEs of arbitrary dimension and establishes a corresponding system of LOR equations for analyzing dynamics within the LOR frame, which maps naturally back to the original phase space. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. In addition to developing a general theory for LOR that makes use of a given normal frame, we show how to construct a normal frame that conveniently simplifies the computations involved in LOR. Finally, we illustrate the utility of LOR by showing that a blow-up transformation on the LOR equations provides a useful decomposition for studying trajectories' behavior relative to the embedded base manifold and by using LOR to identify canard behavior near a fold of a critical manifold in a two-timescale system.

Preface: Population dynamics in epidemiology and ecology
Luca Gerardo-Giorda, Pierre Magal, Shigui Ruan, Ousmane Seydi and Glenn Webb
2020, 25(6): ⅰ-ⅱ doi: 10.3934/dcdsb.2020125 +[Abstract](1688) +[HTML](325) +[PDF](81.19KB)
Iván Area, Alberto Cabada, José Ángel Cid, Daniel Franco, Eduardo Liz and Rosana Rodríguez-López
2020, 25(2): ⅰ-ⅳ doi: 10.3934/dcdsb.2019269 +[Abstract](1227) +[HTML](240) +[PDF](88.82KB)
Tomas Caraballo, Xiaoying Han and Arnulf Jentzen
2019, 24(8): ⅰ-ⅱ doi: 10.3934/dcdsb.2019184 +[Abstract](1411) +[HTML](187) +[PDF](95.7KB)
Avner Friedman, Urszula Foryś, Maria do Rosário de Pinho and Richard Vinter
2019, 24(5): ⅰ-ⅱ doi: 10.3934/dcdsb.201905i +[Abstract](1776) +[HTML](182) +[PDF](440.81KB)
A sufficient optimality condition for delayed state-linear optimal control problems
Ana P. Lemos-Paião, Cristiana J. Silva and Delfim F. M. Torres
2019, 24(5): 2293-2313 doi: 10.3934/dcdsb.2019096 +[Abstract](2458) +[HTML](660) +[PDF](471.3KB)

We give answer to an open question by proving a sufficient optimality condition for state-linear optimal control problems with time delays in state and control variables. In the proof of our main result, we transform a delayed state-linear optimal control problem to an equivalent non-delayed problem. This allows us to use a well-known theorem that ensures a sufficient optimality condition for non-delayed state-linear optimal control problems. An example is given in order to illustrate the obtained result.

Applications of stochastic semigroups to cell cycle models
Katarzyna Pichór and Ryszard Rudnicki
2019, 24(5): 2365-2381 doi: 10.3934/dcdsb.2019099 +[Abstract](2388) +[HTML](659) +[PDF](440.63KB)

We consider a generational and continuous-time two-phase model of the cell cycle. The first model is given by a stochastic operator, and the second by a piecewise deterministic Markov process. In the second case we also introduce a stochastic semigroup which describes the evolution of densities of the process. We study long-time behaviour of these models. In particular we prove theorems on asymptotic stability and sweeping. We also show the relations between both models.

Preface to the special issue "Dynamics and control in distributed systems: Dedicated to the memory of Valery S. Melnik (1952-2007)"
Tomás Caraballo Garrido, Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero and Michael Zgurovsky
2019, 24(3): ⅰ-ⅴ doi: 10.3934/dcdsb.20193i +[Abstract](2653) +[HTML](237) +[PDF](112.13KB)
Xiaoming He, Eric Kaufmann, Steve Pankavich and Erik Van Vleck
2019, 24(1): ⅰ-ⅰ doi: 10.3934/dcdsb.201901i +[Abstract](1315) +[HTML](224) +[PDF](81.02KB)
Mathematical analysis of macrophage-bacteria interaction in tuberculosis infection
Danyun He, Qian Wang and Wing-Cheong Lo
2018, 23(8): 3387-3413 doi: 10.3934/dcdsb.2018239 +[Abstract](3826) +[HTML](950) +[PDF](1087.46KB)

Tuberculosis (TB) is a leading cause of death from infectious disease. TB is caused mainly by a bacterium called Mycobacterium tuberculosis which often initiates in the respiratory tract. The interaction of macrophages and T cells plays an important role in the immune response during TB infection. Recent experimental results support that active TB infection may be induced by the dysfunction of Treg cell regulation that provides a balance between anti-TB T cell responses and pathology. To better understand the dynamics of TB infection and Treg cell regulation, we build a mathematical model using a system of differential equations that qualitatively and quantitatively characterizes the dynamics of macrophages, Th1 and Treg cells during TB infection. For sufficiently analyzing the interaction between immune response and bacterial infection, we separate our model into several simple subsystems for further steady state and stability studies. Using this system, we explore the conditions of parameters for three situations, recovery, latent disease and active disease, during TB infection. Our numerical simulations support that Th1 cells and Treg cells play critical roles in TB infection: Th1 cells inhibit the number of infected macrophages to reduce the chance of active disease; Treg cell regulation reduces the immune response to stabilize the dynamics of the system.

Preface: "Structural Dynamical Systems: Computational aspects"
Nicoletta Del Buono, Cinzia Elia, Roberto Garrappa and Alessandro Pugliese
2018, 23(7): i-i doi: 10.3934/dcdsb.201807i +[Abstract](2938) +[HTML](155) +[PDF](92.02KB)
Preface: Special issue on dynamical systems on graphs
Stefan Siegmund and Petr Stehlík
2018, 23(5): ⅰ-ⅲ doi: 10.3934/dcdsb.201805i +[Abstract](3735) +[HTML](300) +[PDF](83.21KB)
In memoriam Igor D. Chueshov September 23, 1951 - April 23, 2016
Ludwig Arnold
2018, 23(3): ⅰ-ⅸ doi: 10.3934/dcdsb.201803i +[Abstract](4739) +[HTML](953) +[PDF](6684.8KB)
Does assortative mating lead to a polymorphic population? A toy model justification
Ryszard Rudnicki and Radoslaw Wieczorek
2018, 23(1): 459-472 doi: 10.3934/dcdsb.2018031 +[Abstract](4357) +[HTML](1193) +[PDF](1511.3KB)

We consider a model of phenotypic evolution in populations with assortative mating of individuals. The model is given by a nonlinear operator acting on the space of probability measures and describes the relation between parental and offspring trait distributions. We study long-time behavior of trait distribution and show that it converges to a combination of Dirac measures. This result means that assortative mating can lead to a polymorphic population and sympatric speciation.

Urszula Ledzewicz, Drábek Pavel, Avner Friedman, Marek Galewski, Maria do Rosário de Pinho, Bogdan Przeradzki and Ewa Schmeidel
2018, 23(1): i-ii doi: 10.3934/dcdsb.201801i +[Abstract](2422) +[HTML](202) +[PDF](76.6KB)

2019  Impact Factor: 1.27




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