nach oben

2019 | Buch

Dynamical Systems, Bifurcation Analysis and Applications

Penang, Malaysia, August 6–13, 2018

herausgegeben von: Prof. Mohd Hafiz Mohd, Prof. Norazrizal Aswad Abdul Rahman, Prof. Nur Nadiah Abd Hamid, Prof. Yazariah Mohd Yatim

Verlag: Springer Singapore

Buchreihe : Springer Proceedings in Mathematics & Statistics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

This book is the result of Southeast Asian Mathematical Society (SEAMS) School 2018 on Dynamical Systems and Bifurcation Analysis (DySBA). It addresses the latest developments in the field of dynamical systems, and highlights the importance of numerical continuation studies in tracking both stable and unstable steady states and bifurcation points to gain better understanding of the dynamics of the systems.
The SEAMS School 2018 on DySBA was held in Penang from 6th to 13th August at the School of Mathematical Sciences, Universiti Sains Malaysia.The SEAMS Schools are part of series of intensive study programs that aim to provide opportunities for an advanced learning experience in mathematics via planned lectures, contributed talks, and hands-on workshop.
This book will appeal to those postgraduates, lecturers and researchers working in the field of dynamical systems and their applications. Senior undergraduates in Mathematics will also find it useful.

Inhaltsverzeichnis

Frontmatter

Ordinary Differential Equations

Frontmatter

Mathematical Modeling and Stability Analysis of Population Dynamics

Abstract

This study provides a brief introduction of important terminologies and methodologies in the mathematical modelling and stability analysis of the population dynamics. As an example, a mathematical model of population dynamics for thyroid disorder during pregnancy is developed and analysed. The disorders are the second most common endocrine disorders among women in childbearing age, where inadequate or excessive amount of thyroid hormones are produced due to various causes. Thyroid disorders during pregnancy and postpartum can be divided into three types: hyperthyroidism, hypothyroidism and postpartum thyroiditis. They may lead to numerous complications to both mothers and foetuses, such as heart failure, pre-eclampsia, miscarriage, premature birth, and perinatal mortality. The model is described using a system of first order linear ordinary differential equations. Its stability is studied using Routh-Hurwitz criteria. It is found that the model has only one non-negative equilibrium, which is locally asymptotically stable.

Auni Aslah Mat Daud

Optimal Harvesting Regions of a Polluted Predator-Prey Fishery System

Abstract

The present paper examines a predator-prey fishery system by taking into account the toxin released by the fish which can lead to polluted system. Both predator and prey fish species obey the logistic population growth with their respective environmental carrying capacities. In the proposed model, both fish species produce toxins that contribute to mutual infection which is detrimental to each other. Different harvesting efforts are applied on the predator and prey fish populations, respectively. The equilibria existed in the model are studied together with the local stability properties. We consider the threshold conditions which trigger the bifurcation that occurred in the steady states. The global stability properties of coexistence equilibrium are studied by constructing an appropriate Lyapunov function. Bendixson-Dulac criterion is applied to rule out the existence of limit cycle in the system. From the bifurcation analysis, the dynamical behaviors of the system are observed as well as the persistence and extinction properties. It is shown that harvesting parameters are most likely to drive a fish population towards extinction compared to toxicant parameters which are less influential. Regions of optimal harvesting strategies were found to guarantee the persistence of both fisheries. Finally, the existence of a bionomic equilibrium solution has been examined with three possible cases.

Tau Keong Ang, Hamizah M. Safuan, Ummu Atiqah M. Roslan, Mohd Hafiz Mohd

Dynamics and Bifurcations in a Dynamical System of a Predator-Prey Type with Nonmonotonic Response Function and Time-Periodic Variation

Abstract

We study a two dimensional system of ordinary differential equations of a predator-prey type. We use the Holling type IV functional response which models the group defence mechanism. For this system we discuss the number of equilibria in the system and prove it using a geometrical approach. Using the classical Lagrange Multiplier method, we compute fold and cusp bifurcations for equilibrium in the system. As we turn on to numerics, we compute the other bifurcations for equilibrium, namely Hopf bifurcations, and homoclinic bifurcations. As for bifurcation of periodic solution we compute the Fold of Limit Cycle bifurcation. We also include time-periodic variation in the system which translates most of the bifurcation sets for equilibria into bifurcation sets for periodic solutions. Furthermore, we found the swallowtail bifurcation for periodic solution in the system.

Johan M. Tuwankotta, Eric Harjanto, Livia Owen

Modeling and Experimental Data on the Dynamics of Predation of Rice Plants and Weeds by Golden Apple Snail (Pomacea Canaliculata)

Abstract

Golden Apple Snail(GAS) (Pomacea canaliculata), popularly known in the Philippines as “golden kuhol” is considered a serious invader in a paddy ecosystem. Rice farmers consider it as a notorious invasive species and a serious pest in several rice farms. In this study, we model the predation of rice plants and weeds by GAS. We formulate three ordinary differential equations to model the simplified dynamics of apple snails, rice plants and weeds in the presence of harvesting on snails. We then investigate the mathematical features of the model and analyze the stability of equilibria. Actual death and harvesting rates on GAS gathered from actual field experiments conducted on the enclosed rice paddies are used for the parameters in the numerical simulations to demonstrate the potential effect of snail harvesting.

Joel Addawe, Zenaida Baoanan, Rizavel Addawe

Fractional Differential Equations

Frontmatter

Analysis of a Discrete-Time Fractional Order SIR Epidemic Model for Childhood Diseases

Abstract

In this paper, a discrete-time fractional order SIR epidemic model for a childhood disease with constant vaccination program is investigated. The local asymptotic stability and bifurcation of the equilibrium points are analyzed using basic reproduction number. Flip and Neimark-Sacker (N-S) bifurcations are investigated for endemic equilibrium point and numerical simulations are carried out to illustrate the dynamical behaviors of the model. Chaos phenomenon is observed through numerical simulation inside the flip and N-S bifurcation regions. Results of the numerical simulations support the theoretical analysis.

Mahmoud A. M. Abdelaziz, Ahmad Izani Ismail, Farah A. Abdullah, Mohd Hafiz Mohd

Delay and Partial Differential Equations

Frontmatter

A Tuberculosis Epidemic Model with Latent and Treatment Period Time Delays

Abstract

In this paper, a Susceptible-Exposed-Infectious-Treated (SEIT) epidemic model with two discrete time delays for the disease transmission of tuberculosis (TB) is proposed and analyzed. The first time delay \(\tau _1\) represents the time of progression of an individual from the latent TB infection to the active TB disease, and the other delay \(\tau _2\) corresponds to the treatment period. We begin our mathematical analysis of the model by establishing the existence, uniqueness, nonnegativity and boundedness of the solutions. We derive the basic reproductive number \(R_0\) for the model. Using LaSalle’s Invariance Principle, we determine the stability of the equilibrium points when the treatment success rate is equal to zero. We prove that if \(R_0<1\), then the disease-free equilibrium is globally asymptotically stable. If \(R_0>1\), then the disease-free equilibrium is unstable and a unique endemic equilibrium exists which is globally asymptotically stable. Numerical simulations are presented to illustrate the theoretical results.

Jay Michael R. Macalalag, Elvira P. De Lara-Tuprio, Timothy Robin Y. Teng

Numerical Bifurcation and Stability Analyses of Partial Differential Equations with Applications to Competitive System in Ecology

Abstract

Bifurcation analysis is a powerful technique for investigating the dynamical behaviours of nonlinear systems. While this approach has been employed extensively in analysing ordinary-differential equations and other deterministic models, the use of bifurcation analysis in studying the dynamics of partial differential equations (PDE) is yet limited. This chapter illustrates the progress on how numerical bifurcation and stability analyses can be used in understanding the overall dynamics of a PDE system under consideration. By considering an ecological example of competitive system with environmental suitability and spatial diffusion terms, distinct behaviours of the model e.g. alternative stable states, multi-species coexistence and extinction phenomena are demonstrated as interspecific competition and dispersal strength change. Further investigation reveals the existence of several threshold values in ecologically-relevant parameters corresponding to distinct bifurcations (e.g. saddle-node and transcritical), which lead to different stability properties of PDE solution branches.

Mohd Hafiz Mohd

Discrete Dynamical Systems

Frontmatter

Global Stability Index for an Attractor with Riddled Basin in a Two-Species Competition System

Abstract

We consider a competition system between two-species containing riddled basin and second basin attractors. To characterize local geometry of riddled basin, we compute a global stability index for the attractor in the system. Our results show that the index varies from \(\infty \) down to positive values within a parameter region. The changes of the index indicates that the attractor looses its stability from asymptotically stable attractor to riddled basin attractor. Thus, the stability index has a great potential to become a new study on bifurcation of dynamical system since it is able to characterize different types of geometry of basins of attraction.

Ummu Atiqah Mohd Roslan, Mohd Tirmizi Mohd Lutfi

Counting Closed Orbits in Discrete Dynamical Systems

Abstract

For a discrete dynamical system, the following functions: (i) prime orbit counting function, (ii) Mertens’ orbit counting function, and (iii) Meissel’s orbit sum, describe the different aspects of the growth in the number of closed orbits of the system. These are analogous to counting functions for primes in number theory. The asymptotic behaviour of those functions can be determined by two approaches: by (i) Artin-Mazur zeta function, or (ii) number of periodic points per period. In the first approach, the analyticity and non-vanishing property of the zeta function lead to the asymptotic equivalence of the prime orbit and Mertens’ orbit counting functions. In the second approach, the estimate on the number of periodic points per period is used to obtain the order of magnitude of all those counting functions. This chapter will introduce the counting functions and demonstrate both approaches in some categories of shift spaces, such as shifts of finite type, countable state Markov shifts, Dyck shifts and Motzkin shifts.

Azmeer Nordin, Mohd Salmi Md Noorani, Syahida Che Dzul-Kifli

Computational Dynamical Systems

Frontmatter

Computational Dynamical Systems Using XPPAUT

Abstract

This article is written as a guide for researchers on how to employ the techniques in numerical continuation and bifurcation analysis using XPPAUT. This is a free software package to solve and analyse dynamical systems numerically. The article starts with a gentle introduction to XPPAUT, how to install this software, and an overview of the numerical routines. By using ordinary differential equations as an example, readers are guided to solve for the steady-states and also perform some graphical analysis, such as phase portraits and time-series plots. Thereafter, the sections gradually increase in complexity, covering general steps in bifurcation analysis and how to produce complete bifurcation diagrams, particularly co-dimension one and co-dimension two bifurcation plots.

Ojonubah James Omaiye, Mohd Hafiz Mohd

A Basic Manual for AUTO-07p in Computing Bifurcation Diagrams of a Predator-Prey Model

Abstract

This is a beginner’s manual of AUTO-07p, a continuation and bifurcation software for ordinary differential equation (ODE). AUTO package is available for Windows, Mac OS, or UNIX/Linux platform. The directory auto/07p/demos has many tutorial demos for algebraic system, ODE and partial differential equation. We focus on the continuation of solutions of system of ODE. In this manual, we will learn the main tools in AUTO by doing cusp and pp2 demos step-by-step. In the first example, we will generate one- and two-parameter bifurcation diagrams. The second example is a 2D predator-prey model with the detection of a Hopf bifurcation. We plot some orbits and time series plot. We provide two options for running AUTO, i.e., by using Unix and Python commands.

Livia Owen, Eric Harjanto

Numerical Continuation and Bifurcation Analysis in a Harvested Predator-Prey Model with Time Delay using DDE-Biftool

Abstract

Time delay has been incorporated in models to reflect certain physical or biological meaning. The theory of delay differential equations (DDEs), which has seen extensive growth in the last seventy years or so, can be used to examine the effects of time delay in the dynamical behaviour of systems being considered. Numerical tools to study DDEs have played a significant role not only in illustrating theoretical results but also in discovering interesting dynamics of the model. DDE-Biftool, which is a Matlab package for numerical continuation and numerical bifurcation analysis of DDEs, is one of the most utilized and popular numerical tools for DDEs. In this paper, we present a guide to using the latest version of DDE-Biftool targeted to researchers who are new to the study of time delay systems. A short discussion of an example application, which is a harvested predator-prey model with a single discrete time delay, will be presented first. We then implement this example model in DDE-Biftool, pointing out features where beginners need to be cautious. We end with a comparison of our theoretical and numerical results.

Juancho A. Collera

Titel: Dynamical Systems, Bifurcation Analysis and Applications
herausgegeben von: Prof. Mohd Hafiz Mohd
Prof. Norazrizal Aswad Abdul Rahman
Prof. Nur Nadiah Abd Hamid
Prof. Yazariah Mohd Yatim
Verlag: Springer Singapore
Electronic ISBN: 978-981-329-832-3
Print ISBN: 978-981-329-831-6
DOI: https://doi.org/10.1007/978-981-32-9832-3