Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis - Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics ...

414 Pages·2013·8.75 MB·English

by  Changpin Li

Recent Advances in
Applied Nonlinear Dynamics
with Numerical Analysis
Fractional Dynamics, Network Dynamics, Classical Dynamics
and Fractal Dynamics with their Numerical Simulations
8637hc_9789814436458_tp.indd 1 28/11/12 4:18 PM
INTERDISCIPLINARY MATHEMATICAL SCIENCES*
Series Editor: Jinqiao Duan (University of California, Los Angeles, USA)
Editorial Board: Ludwig Arnold, Roberto Camassa, Peter Constantin,
Charles Doering, Paul Fischer, Andrei V. Fursikov, Xiaofan Li,
Sergey V. Lototsky, Fred R. McMorris, Daniel Schertzer,
Bjorn Schmalfuss, Yuefei Wang, Xiangdong Ye, and Jerzy Zabczyk
Published
Vol. 5: The Hilbert–Huang Transform and Its Applications
eds. Norden E. Huang & Samuel S. P. Shen
Vol. 6: Meshfree Approximation Methods with MATLAB
Gregory E. Fasshauer
Vol. 7: Variational Methods for Strongly Indefinite Problems
Yanheng Ding
Vol. 8: Recent Development in Stochastic Dynamics and Stochastic Analysis
eds. Jinqiao Duan, Shunlong Luo & Caishi Wang
Vol. 9: Perspectives in Mathematical Sciences
eds. Yisong Yang, Xinchu Fu & Jinqiao Duan
Vol. 10: Ordinal and Relational Clustering (with CD-ROM)
Melvin F. Janowitz
Vol. 11: Advances in Interdisciplinary Applied Discrete Mathematics
eds. Hemanshu Kaul & Henry Martyn Mulder
Vol. 12: New Trends in Stochastic Analysis and Related Topics:
A Volume in Honour of Professor K D Elworthy
eds. Huaizhong Zhao & Aubrey Truman
Vol. 13: Stochastic Analysis and Applications to Finance:
Essays in Honour of Jia-an Yan
eds. Tusheng Zhang & Xunyu Zhou
Vol. 14: Recent Developments in Computational Finance:
Foundations, Algorithms and Applications
eds. Thomas Gerstner & Peter Kloeden
Vol. 15: Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis:
Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal
Dynamics with Their Numerical Simulations
eds. Changpin Li, Yujiang Wu & Ruisong Ye
*For the complete list of titles in this series, please go to
http://www.worldscientific.com/series/ims
RokTing - Recent Advs in Applied Nonlinear.pm1d 11/21/2012, 4:43 PM
Interdisciplinary Mathematical Sciences – Vol. 15
Recent Advances in
Applied Nonlinear Dynamics
with Numerical Analysis
Fractional Dynamics, Network Dynamics, Classical Dynamics
and Fractal Dynamics with their Numerical Simulations
Editors
Changpin Li
Shanghai University, China
Yujiang Wu
Lanzhou University, China
Ruisong Ye
Shantou University, China
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
8637hc_9789814436458_tp.indd 2 28/11/12 4:18 PM
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Interdisciplinary Mathematical Sciences — Vol. 15
RECENT ADVANCES IN APPLIED NONLINEAR DYNAMICS WITH
NUMERICAL ANALYSIS
Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics
with Their Numerical Simulations
Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic
or mechanical, including photocopying, recording or any information storage and retrieval system now known
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For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center,
Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from
the publisher.
ISBN 978-981-4436-45-8
Printed in Singapore.
RokTing - Recent Advs in Applied Nonlinear.pm2d 11/21/2012, 4:43 PM
November21,2012 14:53 WorldScientificBook-9.75inx6.5in ws-book975x65-rev
Professor Zhong-hua Yang
v
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Foreword
Almost no one doubts that Dynamics is always an exciting and serviceable topic in
scienceandengineering. Sincethefounderofdynamicalsystems,H.Poincar´e,there
have been great theoretical achievements and successful applications. Meanwhile,
complexandmultifariousdynamicalevolutionsandnewsocialrequestsproducenew
branches in the field of dynamical systems, such as fractional dynamics, network
dynamics,andvariousgenuineapplicationsinindustrialandagriculturalproduction
as well as national construction.
Although fractional calculus, in allowing integrals and derivatives of any posi-
tive real order (the term “fractional” is kept only for the historical reasons) even
complex number order, has almost the same history as the classical calculus, frac-
tional dynamics is still in the budding stage. As far as we know, the beginning
era of fractional dynamics very possibly originates from a paper on the Lyapunov
exponents of the fractional differential systems published in Chaos in 2010. On the
other hand, there have existed a huge number of publications in network dynamics
albeititappearedin1990’s. Besides,networkdynamicshaspenetratedintovarious
sources and more and more theories and applications will be prominently emerged.
Withtherapiddevelopmentsofthenonlineardynamics,thisvolumetimelycol-
lects contributions of recent advances in fractional dynamics, network dynamics,
fractal dynamics and the classical dynamics. The contents cover applied theories,
numericalalgorithmsandcomputations,andapplicationsinthisregard. Firstchap-
tercontributestosurveysonGronwallinequalitieswherethesingularcasehasbeen
emphasizedwhichareoftenusedinthefractionaldifferentialsystems. Inthesecond
chapter, recentresultsofexistenceanduniquenessofthesolutionstothefractional
differential equations are presented. In the next chapter, the finite element method
and calculation for the fractional differential equations are summarized and in-
troduced. In following three chapters, the numerical method and calculations for
fractional differential equations are proposed and numerically realized, where the
fractional step method, the spectral method, and the discontinuous finite element
method, are used to solve the fractional differential equations, respectively. In the
seventh chapter, recent results on the asymptotic expansion of a singularly per-
turbed problem under curvilinear coordinates are shown with the aid of classical
vii
October5,2012 15:23 WorldScientificBook-9.75inx6.5in ws-book975x65-rev
viii Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis
Laplace transformation. Chapter 8 contributes to investigating the typically dy-
namical numerical solver-incremental unknowns methods under the background of
alternatingdirectionalimplicit(ADI)schemeforaheatconductionequation. Chap-
ter 9 generalizes the sharp estimates of the two-dimensional problems to the sta-
bility analysis of three-dimensional incompressible Navier-Stokes equations solved
numerically by a colocated finite volume scheme. In the tenth chapter, numeri-
cal algorithms for the computation of certain symmetric positive solutions and the
detectionofsymmetry-breakingbifurcationpointsontheseorothersymmetricpos-
itive solutions for p-Henon equation are studied. In the following chapter, recent
results of block incremental unknowns for solving reaction-diffusion equations are
presented. Chapters12,13and19contributetonetworkdynamics,wherethemod-
els and synchronization dynamics are introduced and analyzed in details. Chapter
14focusesonchaoticdynamicalsystemsonfractalsandtheirapplicationstoimage
encryption. Chapter 15 makes contribution to the generation of the planar crystal-
lographic symmetric patterns by discrete systems invariant with respect to planar
crystallographic groups from a dynamical system point of view. Chapter 16 inves-
tigates the complicated dynamics of a simple two-dimensional discrete dynamical
system. Chapter 17 discusses the bifurcations in the delayed ordinary differential
equation and the next chapter introduces the numerical methods for the option
pricing problems.
We are very grateful to all the authors for their contributions to this volume.
WespeciallythankMsTanRokTingforhersparingnopainstoinformus,replying
tousandexplainingvariousdetailsregardingthiseditedvolume. Themostlymen-
tionable question is that the published year of this book happens to be the year of
Professor Zhong-hua Yang’s 70th birthday. We are privileged and honored to dedi-
catethiseditedbooktoProfessorZhong-huaYang,ourteacherandlife-longfriend.
CL acknowledges the financial support of the National Natural Science Foundation
ofChina(grantno. 10872119),theKeyDisciplinesofShanghaiMunicipality(grant
no. S30104), and the Key Program of Shanghai Municipal Education Commission
(grant no. 12ZZ084).
Changpin Li, Shanghai University
Yu-jiang Wu, Lanzhou University
Ruisong Ye, Shantou University
May 28, 2012
October5,2012 15:23 WorldScientificBook-9.75inx6.5in ws-book975x65-rev
Preface
This festschrift volume is dedicated to Professor Zhong-hua Yang on the occasion
of his 70th birthday.
Zhong-huaYangwasbornonOctober5,1942,inShanghai,China. Hegraduated
in1964fromFudanUniversity,aprestigiousuniversityinChina. Aftergraduation,
he was recruited to Shanghai University of Science and Technology (now is called
Shanghai University), as a faculty member at the Department of Mathematics.
In 1982, Yang published his first research paper and in the same year he went
to California Institute of Technology as a senior visiting scholar to work with the
famous mathematician, Professor H.B. Keller, for advanced studies on theory and
computation of bifurcation.
Two years later, he returned to Shanghai University of Science and Technology,
where he spent twenty years as a faculty member. He has published 70 articles
ranging in computational and applied mathematics, especially in computation of
bifurcation. In 1989, he was promoted to full professor and appointed as associate
directorofDepartmentofMathematicsattheuniversity. In1995,hewasappointed
as an advisor of the graduated students for Ph.D. degree.
In 1988, 1992 and 1998, he was granted the Science and Technology Progress
AwardforthreetimesbyMinistryofEducation,China. Inthe1990’s,Yangworked
on bifurcation computation and applications for nonlinear problems, one of the
projects in National “Climbing” Program. He has received special government al-
lowancefromtheStateCouncilofChinasince1992. HewasthenawardedShanghai
splendid educator in 1995.
In1996,hemovedtoShanghaiNormalUniversity,andactedasvicedeanofthe
School of Math. Science (1997-2002). His academic positions and responsibilities
also include: Editor of the journal Numerical Mathematics: A Journal of Chinese
Universities (English Series), Council member of Shanghai Mathematics Society
and reviewer for Mathematical Reviews.
In 2007, his book Nonlinear Bifurcation: Theory and Computation was pub-
lished, which was the first monograph on bifurcation computation in China.
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