scholarly journals Multiscale dynamics of an adaptive catalytic network

2019 ◽  
Vol 14 (4) ◽  
pp. 402 ◽  
Author(s):  
Christian Kuehn
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Equilibrium Points ◽  
Network Models ◽  
Multiscale Methods ◽  
Adaptive Network ◽  
Time Dynamics ◽  
Multiscale Dynamics ◽  
Time Graph ◽  
Catalytic Network

We study the multiscale structure of the Jain–Krishna adaptive network model. This model describes the co-evolution of a set of continuous-time autocatalytic ordinary differential equations and its underlying discrete-time graph structure. The graph dynamics is governed by deletion of vertices with asymptotically weak concentrations of prevalence and then re-insertion of vertices with new random connections. In this work, we prove several results about convergence of the continuous-time dynamics to equilibrium points. Furthermore, we motivate via formal asymptotic calculations several conjectures regarding the discrete-time graph updates. In summary, our results clearly show that there are several time scales in the problem depending upon system parameters, and that analysis can be carried out in certain singular limits. This shows that for the Jain–Krishna model, and potentially many other adaptive network models, a mixture of deterministic and/or stochastic multiscale methods is a good approach to work towards a rigorous mathematical analysis.

CHAOS OF DYNAMICAL SYSTEMS ON GENERAL TIME DOMAINS

2009 ◽  
Vol 19 (11) ◽  
pp. 3829-3832
Author(s):  
ABRAHAM BOYARSKY ◽  
PAWEŁ GÓRA
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Continuous Time ◽  
Chaotic Map ◽  
Discrete Time System ◽  
Combined System ◽  
Time Intervals ◽  
Time Dynamics ◽  
Time Component ◽  
Time Domains

We consider dynamical systems on time domains that alternate between continuous time intervals and discrete time intervals. The dynamics on the continuous portions may represent species growth when there is population overlap and are governed by differential or partial differential equations. The dynamics across the discrete time intervals are governed by a chaotic map and may represent population growth which is seasonal. We study the long term dynamics of this combined system. We study various conditions on the continuous time dynamics and discrete time dynamics that produce chaos and alternatively nonchaos for the combined system. When the discrete system alone is chaotic we provide a condition on the continuous dynamical component such that the combined system behaves chaotically. We also provide a condition that ensures that if the discrete time system has an absolutely continuous invariant measure so will the combined system. An example based on the logistic continuous time and logistic discrete time component is worked out.

scholarly journals Evolutionary dynamics in discrete time for the perturbed positive definite replicator equation

2021 ◽  
Vol 62 ◽  
pp. 148-184
Author(s):  
Amie Albrecht ◽  
Konstantin Avrachenkov ◽  
Phil Howlett ◽  
Geetika Verma
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Evolutionary Dynamics ◽  
Positive Definite ◽  
Genotype Distribution ◽  
Replicator Equation ◽  
System Matrix ◽  
Discrete Time System ◽  
Time System ◽  
Time Dynamics

The population dynamics for the replicator equation has been well studied in continuous time, but there is less work that explicitly considers the evolution in discrete time. The discrete-time dynamics can often be justified indirectly by establishing the relevant evolutionary dynamics for the corresponding continuous-time system, and then appealing to an appropriate approximation property. In this paper we study the discrete-time system directly, and establish basic stability results for the evolution of a population defined by a positive definite system matrix, where the population is disrupted by random perturbations to the genotype distribution either through migration or mutation, in each successive generation. doi: 10.1017/S1446181120000140

Discrete-time and continuous-time modelling: some bridges and gaps

2007 ◽  
Vol 17 (2) ◽  
pp. 261-276 ◽  
Author(s):  
HUBERT KRIVINE ◽  
ANNICK LESNE ◽  
JACQUES TREINER
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Logistic Equation ◽  
Planetary Motion ◽  
Natural Phenomena ◽  
Predator Prey ◽  
Time Dynamics ◽  
Predator Prey Model ◽  
The Relationship ◽  
General Duality

The relationship between continuous-time dynamics and the corresponding discrete schemes, and its generally limited validity, is an important and widely acknowledged field within numerical analysis. In this paper, we propose another, more physical, viewpoint on this topic in order to understand the possible failure of discretisation procedures and the way to fix it. Three basic examples, the logistic equation, the Lotka–Volterra predator–prey model and Newton's law for planetary motion, are worked out. They illustrate the deep difference between continuous-time evolutions and discrete-time mappings, hence shedding some light on the more general duality between continuous descriptions of natural phenomena and discrete numerical computations.

scholarly journals EVOLUTIONARY DYNAMICS IN DISCRETE TIME FOR THE PERTURBED POSITIVE DEFINITE REPLICATOR EQUATION

2020 ◽  
Vol 62 (2) ◽  
pp. 148-184
Author(s):  
AMIE ALBRECHT ◽  
KONSTANTIN AVRACHENKOV ◽  
PHIL HOWLETT ◽  
GEETIKA VERMA
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Evolutionary Dynamics ◽  
Positive Definite ◽  
Genotype Distribution ◽  
Replicator Equation ◽  
System Matrix ◽  
Discrete Time System ◽  
Time System ◽  
Time Dynamics

AbstractThe population dynamics for the replicator equation has been well studied in continuous time, but there is less work that explicitly considers the evolution in discrete time. The discrete-time dynamics can often be justified indirectly by establishing the relevant evolutionary dynamics for the corresponding continuous-time system, and then appealing to an appropriate approximation property. In this paper we study the discrete-time system directly, and establish basic stability results for the evolution of a population defined by a positive definite system matrix, where the population is disrupted by random perturbations to the genotype distribution either through migration or mutation, in each successive generation.

scholarly journals Consensus of Continuous-Time Multiagent Systems with General Linear Dynamics and Nonuniform Sampling

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Yanping Gao ◽  
Bo Liu ◽  
Min Zuo ◽  
Tongqiang Jiang ◽  
Junyan Yu
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
General Linear ◽  
Linear Matrix ◽  
Time Intervals ◽  
Time Dynamics ◽  
Controller Gain ◽  
General Linear Dynamics ◽  
Static Output

This paper studies the consensus problem of multiple agents with general linear continuous-time dynamics. It is assumed that the information transmission among agents is intermittent; namely, each agent can only obtain the information of other agents at some discrete times, where the discrete time intervals may not be equal. Some sufficient conditions for consensus in the cases of state feedback and static output feedback are established, and it is shown that if the controller gain and the upper bound of discrete time intervals satisfy certain linear matrix inequality, then consensus can be reached. Simulations are performed to validate the theoretical results.

scholarly journals Qualitative analysis of a discrete-time phytoplankton–zooplankton model with Holling type-II response and toxicity

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Muhammad Salman Khan ◽  
Maria Samreen ◽  
Hassen Aydi ◽  
Manuel De la Sen
Keyword(s):  
Mathematical Models ◽  
Discrete Time ◽  
Hybrid Method ◽  
Continuous Time ◽  
Hybrid Control ◽  
Numerical Study ◽  
Equilibrium Points ◽  
Control Technique ◽  
Positive Equilibrium ◽  
Time Model

AbstractThe interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton–zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark–Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper.

A Discrete-Time Model for Lotka-Volterra Equations With Preserved Stability of Equilibria

2013 ◽  
Author(s):  
Triet Nguyen-Van ◽  
Noriyuki Hori
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Equilibrium Points ◽  
Volterra Equations ◽  
Time System ◽  
Time Model ◽  
Discrete Time Model ◽  
Stability And Instability ◽  
Discrete Time Models ◽  
Continuous Time System

A Lotka-Volterra differential equation is discretized using a method proposed recently by the same authors for nonlinear autonomous systems and the stability of equilibrium points of the resulting discrete-time model is investigated. It is shown that when Jacobian matrix of the nonlinear equation is invertible, the equilibrium points of the model are identical to those of the original continuous-time system, and their asymptotic stability and instability are retained for any sampling period. While the method can be applied to any Lotka-Volterra types, simulation results are presented for a competitive-type example, where the continuous-time system and their discrete-time models obtained by the forward-difference, Mickens’, Kahan’s, and the proposed methods are compared. They illustrate that, in general, the proposed model performs better than other discrete-time models.

scholarly journals On Timescale Separation in Networked Systems With Intermittent Communication

2017 ◽  
Vol 140 (5) ◽  
Author(s):  
Armand Awad ◽  
Anshu Narang-Siddarth
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Order Reduction ◽  
Networked Systems ◽  
Time Dynamics ◽  
Update Rate ◽  
The Stability ◽  
Order Reduction Method ◽  
Isolated Systems ◽  
Time Systems

This paper studies the multiple timescale behavior that is induced by dynamic coupling between continuous-time and discrete-time systems, and that arises naturally in distributed networked systems. An order reduction method is proposed that establishes a mathematically rigorous separation principle between the fast evolution of the continuous-time dynamics and the slow updates of the discrete-time dynamics. Quantitative conditions on the discrete update rate are then derived that ensure the stability of the coupled dynamics based on the behavior of the isolated systems. The results are illustrated for a distributed network of satellites whose attitudes evolve continuously while communicating intermittently over the network.

Design of Programmable Wideband Low Pass Filter with Continuous-Time/Discrete-Time Hybrid Architecture

2017 ◽  
Vol E100.C (10) ◽  
pp. 858-865 ◽  
Author(s):  
Yohei MORISHITA ◽  
Koichi MIZUNO ◽  
Junji SATO ◽  
Koji TAKINAMI ◽  
Kazuaki TAKAHASHI
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Hybrid Architecture ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass
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