Recent advances in applied nonlinear dynamics with numerical analysis

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Fractional Dynamics, Network Dynamics, Classical Dynamics
and Fractal Dynamics with their Numerical Simulations

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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

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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
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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.
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Professor Zhong-hua Yang

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Foreword

Almost no one doubts that Dynamics is always an exciting and serviceable topic in
science and engineering. Since the founder of dynamical systems, H. Poincar´e, there
have been great theoretical achievements and successful applications. Meanwhile,
complex and multifarious dynamical evolutions and new social requests produce new
branches in the field of dynamical systems, such as fractional dynamics, network
dynamics, and various genuine applications in industrial and agricultural production
as well as national construction.
Although fractional calculus, in allowing integrals and derivatives of any positive 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, fractional 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
albeit it appeared in 1990’s. Besides, network dynamics has penetrated into various
sources and more and more theories and applications will be prominently emerged.

With the rapid developments of the nonlinear dynamics, this volume timely collects contributions of recent advances in fractional dynamics, network dynamics,
fractal dynamics and the classical dynamics. The contents cover applied theories,
numerical algorithms and computations, and applications in this regard. First chapter contributes to surveys on Gronwall inequalities where the singular case has been
emphasized which are often used in the fractional differential systems. In the second
chapter, recent results of existence and uniqueness of the solutions to the fractional
differential equations are presented. In the next chapter, the finite element method
and calculation for the fractional differential equations are summarized and introduced. 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 perturbed problem under curvilinear coordinates are shown with the aid of classical
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Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis

Laplace transformation. Chapter 8 contributes to investigating the typically dynamical numerical solver-incremental unknowns methods under the background of
alternating directional implicit (ADI) scheme for a heat conduction equation. Chapter 9 generalizes the sharp estimates of the two-dimensional problems to the stability analysis of three-dimensional incompressible Navier-Stokes equations solved
numerically by a colocated finite volume scheme. In the tenth chapter, numerical algorithms for the computation of certain symmetric positive solutions and the

detection of symmetry-breaking bifurcation points on these or other symmetric positive solutions for p-Henon equation are studied. In the following chapter, recent
results of block incremental unknowns for solving reaction-diffusion equations are
presented. Chapters 12, 13 and 19 contribute to network dynamics, where the models and synchronization dynamics are introduced and analyzed in details. Chapter
14 focuses on chaotic dynamical systems on fractals and their applications to image
encryption. Chapter 15 makes contribution to the generation of the planar crystallographic symmetric patterns by discrete systems invariant with respect to planar
crystallographic groups from a dynamical system point of view. Chapter 16 investigates 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.
We specially thank Ms Tan Rok Ting for her sparing no pains to inform us, replying
to us and explaining various details regarding this edited volume. The mostly mentionable 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 dedicate this edited book to Professor Zhong-hua Yang, our teacher and life-long friend.
CL acknowledges the financial support of the National Natural Science Foundation
of China (grant no. 10872119), the Key Disciplines of Shanghai Municipality (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

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Preface

This festschrift volume is dedicated to Professor Zhong-hua Yang on the occasion
of his 70th birthday.
Zhong-hua Yang was born on October 5, 1942, in Shanghai, China. He graduated
in 1964 from Fudan University, a prestigious university in China. After graduation,
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
director of Department of Mathematics at the university. In 1995, he was appointed
as an advisor of the graduated students for Ph.D. degree.
In 1988, 1992 and 1998, he was granted the Science and Technology Progress
Award for three times by Ministry of Education, China. In the 1990’s, Yang worked
on bifurcation computation and applications for nonlinear problems, one of the
projects in National “Climbing” Program. He has received special government allowance from the State Council of China since 1992. He was then awarded Shanghai
splendid educator in 1995.
In 1996, he moved to Shanghai Normal University, and acted as vice dean of the
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 published, which was the first monograph on bifurcation computation in China.

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Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis

Postgraduate Students under Zhong-hua Yang’s supervision
Jizhong Wang (M. Sc., Shanghai University of Science and Technology, 1991)
Ning Ji (M. Sc., Shanghai University of Science and Technology, 1994)
Ruisong Ye (Ph. D., Shanghai University, 1995)
Yujiang Wu (Ph. D., Shanghai University, 1997)
Changpin Li (Ph. D., Shanghai University, 1998)
Wei Zhou (M. Sc., Shanghai Normal University, 2001)
Ying Zhu (M. Sc., Shanghai Normal University, 2002)
Qian Guo (Ph. D., Shanghai University, 2003)
Bo Xiong (Ph. D., Shanghai Normal University, 2004)
Junqiang Wei (M. Sc., Shanghai Normal University, 2004)

Yezhong Li (M. Sc., Shanghai Normal University, 2005)
Quanbao Ji (M. Sc., Shanghai Normal University, 2006)
Xia Gu (M. Sc., Shanghai Normal University, 2006)
Hailong Zhu (M. Sc., Shanghai Normal University, 2007)
Jian Shen (M. Sc., Shanghai Normal University, 2007)
Zhaoxiang Li (Ph. D., Shanghai Normal University, 2008)
Xiaojuan Xi (M. Sc., Shanghai Normal University, 2008)
Yuanyuan Song (M. Sc., Shanghai Normal University, 2008)
Publications Since 1982
Books
(1) Introduction to Numerical Approximation (with De-ren Wang), Higher Education Press, Beijing 1990, in Chinese.
(2) Nonlinear Bifurcation: Theory and Computation, Science Press, Beijing 2007,
in Chinese.
Selected Papers
(1) Yang, Z. H. (1982). Continuation Newton method for boundary value problems of nonlinear elliptic differential equations, (in Chinese) Numer. Math. J.
Chinese Univ. 4, pp. 28–37.
(2) Yang, Z. H. (1984). Several abstract iterative schemes for solving the bifurcation
at simple eigenvalues, J. Comput. Math. 2, pp. 201–209.
(3) Yang, Z. H and Keller, H. B. (1986). A direct method for computing higher
order folds, SIAM J. Sci. Statist. Comput. 7, pp. 351–361.
(4) Yang, Z. H and Keller, H. B. (1986). Multiple laminar flows through curved
pipes, Appl. Numer. Math. 2, pp. 257–271.
(5) Yang, Z. H. (1987). Folds of degree 4 and swallowtail catastrophe. Numerical

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methods for partial differential equations (Shanghai, 1987), 171–183, Lecture
Notes in Math., 1297, Springer, Berlin, 1987.
Yang, Z. H. (1987). Steady problems in thermal ignition. (in Chinese) Comm.
Appl. Math. Comput. 1, pp. 8–21.
Yang, Z. H. (1988). An acceleration method in the homotopy Newton’s continuation for nonlinear singular problems, J. Comput. Math. 6, pp. 1–6.
Yang, Z. H. (1988). The application of the continuation method in the direct
method for computing higher order folds, (in Chinese) Math. Numer. Sinica
10, pp. 6–17.
Yang, Z. H. (1988). Approximation to cusp catastrophe. BAIL V (Shanghai,
1988), 411–416, Boole Press Conf. Ser., 12, Boole, D´
un Laoghaire, 1988.
Yang, Z. H. (1988). Global asymptotic behavior of solutions to nonsteady state
thermal ignition problems, (in Chinese) Comm. Appl. Math. Comput. 2, pp.
67–73.
Yang, Z. H. (1989). Higher order folds in nonlinear problems with several
parameters, J. Comput. Math. 7(3), pp. 262–278.
Yang, Z. H. (1989). Classification of pitchfork bifurcations and their computation, Sci. China Ser. A 32(5), pp. 537–549.
Yang, Z. H and Sleeman, B. D. (1989). Hopf bifurcation in wave solutions
of FitzHugh-Nagumo equation, Proceedings of the International Conference on
Bifurcation Theory and its Numerical Analysis (Xi’an, 1988), 115–125, Xi’an
Jiaotong Univ. Press, Xi’an, 1989.
Yang, Z. H. (1990). An improved scheme for chord methods at singular points,
(in Chinese) Numer. Math. J. Chinese Univ. 12(2), pp. 151–157.
Yang, Z. H. (1991). A direct method for pitchfork bifurcation points, J. Comput.
Math. 9(2), pp. 149–153.
Yang, Z. H. (1991). Approximation of catastrophe points of cusp form, (in
Chinese) Gaoxiao Yingyong Shuxue Xuebao 6(1), pp. 1–12.
Yang, Z. H. and Li, Z. L. (1992). Bifurcation study on the laminar flow in the
coiled tube with the triangular cross section. Numerical methods for partial

differential equations (Tianjin, 1991), 126–138, World Sci. Publ., River Edge,
NJ, 1992.
Yang, Z. H. (1992). Detecting codimension two bifurcations with a pure imaginary and a simple zero eigenvalue, J. Comput. Math. 10, pp. 204–208.
Yang, Z. H. (1992). Symmetry-breaking in two-cell exothermic reaction problems, (in Chinese) Shanghai Keji Daxue Xuebao 15, pp. 44–54.
Ye, R. S. and Yang, Z. H. (1995). The computation of symmetry-breaking
bifurcation points in Z2 × Z2 -symmetric nonlinear problems, Appl. Math. J.
Chinese Univ. Ser. B 10, pp. 179–194.
Ye, R. S. and Yang, Z. H, Mahmood, A. (1995). Extended systems for multiple
S-breaking turning points in Z2 × Z2 -symmetric nonlinear problems, Numer.
Math. J. Chinese Univ. (English Ser.) 4, pp. 119–132.


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Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis

(22) Ye, R. S. and Yang, Z. H. (1996). Double S-breaking cubic turning points and
their computation, J. Comput. Math. 14, pp. 8–22.
(23) Yang, Z. H and Ye, R. S. (1996). A numerical method for solving nonlinear
singular problems and application to bifurcation problems. World Congress of
Nonlinear Analysts’92, Vol. I-IV (Tampa, FL, 1992), 1619–1626, de Gruyter,

Berlin, 1996.
(24) Yang, Z. H and Ye, R. S. (1996). Double high order S-breaking bifurcation
points and their numerical determination, Appl. Math. Mech. (English Ed.)
17, pp. 633–646
(25) Yang, Z. H. and Ye, R. S. (1996). Symmetry-breaking and bifurcation study on
the laminar flows through curved pipes with a circular cross section, J. Comput.
Phys. 127, pp. 73–87
(26) Yang, Z. H, Mahmood, A. and Ye, R. S. (1997). Fully discrete nonlinear
Galerkin methods for Kuramoto-Sivashinsky equation and their error estimates,
J. Shanghai Univ. 1, pp. 20–27.
(27) Li, C. P., Yang, Z. H and Wu, Y. J. (1997). Bifurcation and stability of nontrivial solution to Kuramoto-Sivashinsky equation, J. Shanghai Univ. 1, pp.
95–97.
(28) Ye, R. S. and Yang, Z. H. (1997). Classification of simple higher-order
symmetry-breaking bifurcations and their computation, J. Shanghai Univ. 1,
pp. 175–183.
(29) Li, C. P. and Yang, Z. H. (1998). Bifurcation of two-dimensional KuramotoSivashinsky equation, Appl. Math. J. Chinese Univ. Ser. B 13, pp. 263–270.
(30) Yang, Z. H and Li, C. P. (1998). A numerical approach to Hopf bifurcation
points, J. Shanghai Univ. 2, pp. 182–185.
(31) Mahmood, A. and Yang, Z. H. (1998). Numerical results of Galerkin and nonlinear Galerkin methods for one-dimensional Kurmoto-Sivashinsky equation,
Proc. Pakistan Acad. Sci. 35, pp. 33–37.
(32) Yang, Z. H, Wu, Y. J. and Guo, B. Y. (1999). Computation of nonlinear
Galerkin methods with variable modes for 2-D K-S equations. Advances in computational mathematics (Guangzhou, 1997), 545–563, Lecture Notes in Pure
and Appl. Math., 202, Dekker, New York, 1999.
(33) Li, C. P. and Yang, Z. H. (2000). A note of nonlinear Galerkin method for
steady state Kuramoto-Sivashinsky equation, Math. Appl. (Wuhan) 13, pp.
46–51.
(34) Yang, Z. H and Zhou, W. (2000). A computational method for D6 equivariant
nonlinear bifurcation problems, (in Chinese) Comm. Appl. Math. Comput. 14,
pp. 1–13.
(35) Li, C. P. and Yang, Z. H. (2001). A nonlinear Galerkin method for K-S equation,

Math. Appl. (Wuhan) 14, pp. 22–27.
(36) Li, C. P. and Yang, Z. H. (2001). Remark on Galerkin method for twodimensional steady state Kuramoto-Sivashinsky equation, Numer. Math. J.

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Chinese Univ. (English Ser.) 10, pp. 161–169.
Yang, Z. H and Zhu, Y. (2001). Numerical determination of symmetryincreasing bifurcation of chaotic attractors in a class of planar D3 -equivariant
mappings, (in Chinese) Comm. Appl. Math. Comput. 15(2), pp. 1–8.
Li, C. P. and Yang, Z. H. (2002). Error estimates of Galerkin method for high
dimensional steady state Kuramoto-Sivashinsky equation, Numer. Math. J.
Chinese Univ. (English Ser.) 11, pp. 129–136.
Guo, Q. and Yang, Z. H. (2002). Dynamics of methods for delay differential
equations, (in Chinese) Comm. Appl. Math. Comput. 16, pp. 7–14.
Wu, Y. J. and Yang, Z. H. (2002). On the error estimates of the fully discrete
nonlinear Galerkin method with variable modes to Kuramoto-Sivashinsky equation. Recent progress in computational and applied PDEs (Zhangjiajie, 2001),
383–397, Kluwer/Plenum, New York, 2002.
Yang, Z. H, Wei, J. Q. and Xiong, B. (2003). Computation of higher-order singular points in nonlinear problems with single parameter. (in Chinese) Comm.
Appl. Math. Comput. 17, pp. 1–6.
Li, C. P. and Yang, Z. H. (2004). Symmetry-breaking bifurcation in O(2) ×
O(2)-symmetric nonlinear large problems and its application to the KuramotoSivashinsky equation in two spatial dimensions, Chaos Solitons Fractals 22, pp.
451–468.
Yang, Z. H. and Guo, Q. (2005). Bifurcation analysis of delayed logistic equation, Appl. Math. Comput. 167, pp. 454–476.

Li, C. P., Yang, Z. H. and Chen, G. R. (2005). On bifurcation from steady-state
solutions to rotating waves in the Kuramoto-Sivashinsky equation, J. Shanghai
Univ. 9, pp. 286–291.
Yang, Z. H. and Zhou, W. (2005). Bifurcation analysis and computation of
double Takens-Bogdanov point in Z2 -equivariable nonlinear equations, Numer.
Math. J. Chinese Univ. (English Ser.) 14, pp. 315–324.
Ji, Q. B., Lu, Q. S. and Yang, Z. H. (2007). Computation of D8 -equivariant
nonlinear bifurcation problems, Dyn. Contin. Discrete Impuls. Syst. Ser. B
Appl. Algorithms 14, suppl. S5, pp. 17–20.
Yang, Z. H, Li, Z. X. and Zhu, H. L. (2008). Bifurcation method for solving
multiple positive solutions to Henon equation, Sci. China Ser. A 51, pp. 2330–
2342.
Wei, J. Q. and Yang, Z. H. (2009). Approximation to butterfly catastrophe, (in
Chinese) Gongcheng Shuxue Xuebao 26, pp. 94–98.
Wei, J. Q. and Yang, Z. H. (2009). Fourier collocation method for a class of
reaction-diffusion equations, (in Chinese) Numer. Math. J. Chinese Univ. 31,
pp. 232–239.
Li, Z. X. and Yang, Z. H. (2010). Bifurcation method for solving multiple
positive solutions to boundary value problem of p-Henon equation on the unit
disk, Appl. Math. Mech. (English Ed.) 31, pp. 511–520.


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(51) Li, Z. X., Yang, Z. H. and Zhu, H. L. (2010). Computing multiple positive
solutions to the p-Henon equation on a square, (in Chinese) J. Numer. Methods
Comput. Appl. 31, pp. 161–171.
(52) Li, Z. X., Yang, Z. H. and Zhu, H. L. (2010). Computing the multiple solutions
to boundary value problem of p-Henon equation on the disk of plane, Int. J.
Comp. Math. Sci. 4, pp. 137-139.
(53) Li, Z. X., Zhu, H. L. and Yang, Z. H. (2011). Bifurcation method for solving multiple positive solutions to Henon equation on the unit cube, Commun.
Nonlinear Sci. Numer. Simul. 16, pp. 3673–3683.
(54) Li, Z. X., Yang, Z. H. and Zhu, H. L. (2011). Bifurcation method for solving
multiple positive solutions to boundary value problem of Henon equation on
unit disk, Comput. Math. Appl., 62, pp. 3775-3784.

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Contents

Foreword

vii

Preface

ix

1. Gronwall inequalities

1

Fanhai Zeng, Jianxiong Cao and Changpin Li
1.1
1.2
1.3
1.4
1.5

Introduction . . . . . . . . . . . . . . . . .
The continuous Gronwall inequalities . . .
The discrete Gronwall inequalities . . . .
The weakly singular Gronwall inequalities
Conclusion . . . . . . . . . . . . . . . . .

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2. Existence and uniqueness of the solutions to the fractional differential equations

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Yutian Ma, Fengrong Zhang and Changpin Li
2.1
2.2
2.3

2.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preliminaries and notations . . . . . . . . . . . . . . . . . . . . . .
Existence and uniqueness of initial value problems for fractional
differential equations . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Initial value problems with Riemann-Liouville derivative .
2.3.2 Initial value problems with Caputo derivative . . . . . . .
2.3.3 The positive solution to fractional differential equation . .
Existence and uniqueness of the boundary value problems . . . . .
2.4.1 Boundary value problems with Riemann-Liouville derivative
2.4.2 Boundary value problems with Caputo derivative . . . . .
2.4.3 Fractional differential equations with impulsive boundary
conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv

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2.5
2.6

Existence and uniqueness of the fractional differential equations
with time-delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42
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Bibliography

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3. Finite element methods for fractional differential equations

49

Changpin Li and Fanhai Zeng
3.1
3.2
3.3
3.4

Introduction . . . . . . . . . . . . . .
Preliminaries and notations . . . . .
Finite element methods for fractional
Conclusion . . . . . . . . . . . . . .

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differential equations
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4. Fractional step method for the nonlinear conservation laws with
fractional dissipation

69

Can Li and Weihua Deng
4.1
4.2

4.3
4.4

Introduction . . . . . . . . .
Fractional step algorithm .
4.2.1 Discretization of the
4.2.2 Discretization of the
Numerical results . . . . .
Concluding remarks . . . .

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fractional calculus
conservation law .
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5. Error analysis of spectral method for the space and time fractional
Fokker–Planck equation

83

Tinggang Zhao and Haiyan Xuan
5.1
5.2
5.3
5.4

5.5
5.6

Introduction . . . . . . . . . . . . . . . . . . . . . . .
Preliminaries . . . . . . . . . . . . . . . . . . . . . .
Spectral method . . . . . . . . . . . . . . . . . . . .
Stability and convergence . . . . . . . . . . . . . . .
5.4.1 Semi-discrete of space spectral method . . .
5.4.2 The time discretization of Caputo derivative
Fully discretization and its error analysis . . . . . . .
Conclusion remarks . . . . . . . . . . . . . . . . . . .

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Contents

6. A discontinuous finite element method for a type of fractional
Cauchy problem

105

Yunying Zheng
6.1
6.2
6.3
6.4
6.5
6.6

Introduction . . . . . . . . . . . .
Fractional derivative space . . . .
The discontinuous Galerkin finite
Error estimation . . . . . . . . .
Numerical examples . . . . . . .
Conclusion . . . . . . . . . . . .

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element approximation
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107
114
117
118

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119

7. Asymptotic analysis of a singularly perturbed parabolic problem
in a general smooth domain

121

Yu-Jiang Wu, Na Zhang and Lun-Ji Song
7.1
7.2

7.3

7.4
7.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The curvilinear coordinates . . . . . . . . . . . . . . . . . . . .
Asymptotic expansion . . . . . . . . . . . . . . . . . . . . . . .
7.3.1 Global expansion . . . . . . . . . . . . . . . . . . . . .
7.3.2 Boundary corrector . . . . . . . . . . . . . . . . . . . .
7.3.3 Estimates of the solutions of boundary layer equations
Error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . .
An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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124
124
126
133
137

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141

8. Incremental unknowns methods for the ADI and ADSI schemes

143

Ai-Li Yang, Yu-Jiang Wu and Zhong-Hua Yang
8.1
8.2
8.3

8.4

Introduction . . . . . . . . . . . . . . . . . . . . . . .
Two dimensional heat equation and the AD scheme
ADIUSI scheme and stability . . . . . . . . . . . . .
8.3.1 ADIUSI scheme . . . . . . . . . . . . . . . .

8.3.2 Stability study of the ADIUSI scheme . . . .
Numerical results . . . . . . . . . . . . . . . . . . . .

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143

144
145
145
147
153

Bibliography

157

9. Stability of a colocated FV scheme for the 3D Navier-Stokes equations

159

Xu Li and Shu-qin Wang
9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159


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9.2
9.3

9.4

9.5
9.6

Full discretization: finite volume scheme in space and projection
method in time . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The main result: stability of the scheme . . . . . . . . . . . . . . .
9.3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.2 Discrete weak formulation . . . . . . . . . . . . . . . . . .
9.3.3 Stability result . . . . . . . . . . . . . . . . . . . . . . . . .
Technical lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4.1 The Poincar´e inequality and an inverse inequality . . . . .
9.4.2 Standard lemma . . . . . . . . . . . . . . . . . . . . . . . .
9.4.3 Specific lemmas for the Navier-Stokes equations . . . . . .
Apriori Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Proof of stability . . . . . . . . . . . . . . . . . . . . . . . . . . . .

160
165
165
167
169
170

170
171
173
181
186

Bibliography

189

10. Computing the multiple positive solutions to p-Henon equation
on the unit square

191

Zhaoxiang Li and Zhonghua Yang
10.1
10.2
10.3
10.4
10.5

Introduction . . . . . . . . . . . . . . . . . . . . . .
Computation of D4 symmetric positive solutions .
Computation of the symmetry-breaking bifurcation
Branch switching to Σ symmetric solutions . . . .
Numerical results . . . . . . . . . . . . . . . . . . .

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193
194
197
198

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203

11. Multilevel WBIUs methods for reaction-diffusion equations

205

Yang Wang, Yu-Jiang Wu and Ai-Li Yang
11.1
11.2
11.3

11.4

11.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . .
Multilevel WBIUs method . . . . . . . . . . . . . . . .
Approximate schemes and their equivalent forms . . .

11.3.1 Approximate schemes . . . . . . . . . . . . . .
11.3.2 The equivalent forms of approximate schemes
Stability analysis . . . . . . . . . . . . . . . . . . . . .
11.4.1 Lemmas for new vector norms . . . . . . . . .
11.4.2 Stability analysis . . . . . . . . . . . . . . . .
Numerical results . . . . . . . . . . . . . . . . . . . . .

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12. Models and dynamics of deterministically growing networks

225

Weigang Sun, Jingyuan Zhang and Guanrong Chen
12.1
12.2
12.3

12.4

12.5

12.6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A generation algorithm . . . . . . . . . . . . . . . . . . . . . .
Structural properties . . . . . . . . . . . . . . . . . . . . . . . .
12.3.1 Degree distribution . . . . . . . . . . . . . . . . . . . .
12.3.2 Clustering coefficient . . . . . . . . . . . . . . . . . . .
12.3.3 Average path length . . . . . . . . . . . . . . . . . . . .
12.3.4 Degree correlations . . . . . . . . . . . . . . . . . . . .
Random walks on Koch networks . . . . . . . . . . . . . . . . .
12.4.1 Evolutionary rule for first passage time . . . . . . . . .
12.4.2 Explicit expression for average return time . . . . . . .
12.4.3 Average sending time from a hub node to another node
An exact solution for mean first passage time . . . . . . . . . .
12.5.1 First passage time at the first step . . . . . . . . . . . .
12.5.2 Evolution scaling for the first passage time . . . . . . .
12.5.3 Analytic formula for mean first passage time . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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227
228
229
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230
234
236
236

236
238
242
242
243
244
247

Bibliography

249

13. On different approaches to synchronization of spatiotemporal
chaos in complex networks

251

Yuan Chai and Li-Qun Chen
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8

Introduction . . . . . . . . . . . . . . . .
Design of the synchronization controller
Numerical results . . . . . . . . . . . . .

Active sliding mode controller design . .
Numerical results . . . . . . . . . . . . .
Master stability functions . . . . . . . .
Numerical results . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . .

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251
254
258
261
264
266
269
272

Bibliography

275

14. Chaotic dynamical systems on fractals and their applications to
image encryption

279

Ruisong Ye, Yuru Zou and Jian Lu
14.1
14.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Chaotic dynamical systems on fractals . . . . . . . . . . . . . . . . 283
14.2.1 Iterated function systems . . . . . . . . . . . . . . . . . . . 283


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14.3
14.4

14.5

14.2.2 Chaotic dynamical systems on fractals . . . . . . . . . . .
A special shift dynamical system associated with IFS . . . . . . . .
The image encryption scheme based on the shift dynamical system
associated with IFS . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.4.1 Permutation process . . . . . . . . . . . . . . . . . . . . . .
14.4.2 Diffusion process . . . . . . . . . . . . . . . . . . . . . . . .
14.4.3 Security analysis . . . . . . . . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

284
287
291
291
291
293
300

Bibliography

303

15. Planar crystallographic symmetric tiling patterns generated from
invariant maps

305

Ruisong Ye, Haiying Zhao and Yuanlin Ma
15.1
15.2

15.3

15.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
Planar crystallographic groups . . . . . . . . . . . . . .
15.2.1 Groups p2, pm, pmm . . . . . . . . . . . . . . .
15.2.2 Groups pg, pmg, pgg, cm, cmm . . . . . . . . .
15.2.3 Groups p4, p4g, p4m . . . . . . . . . . . . . . .
15.2.4 Groups p3, p3m1, p31m . . . . . . . . . . . . .
15.2.5 Groups p6, p6m . . . . . . . . . . . . . . . . . .
Rendering method for planar crystallographic symmetric
tiling patterns . . . . . . . . . . . . . . . . . . . . . . . .
15.3.1 Description of colormaps . . . . . . . . . . . . .
15.3.2 Description of orbit trap methods . . . . . . . .
15.3.3 Description of the rendering scheme . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . .

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317
318
319

321

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323

16. Complex dynamics in a simple two-dimensional discrete system

325

Huiqing Huang and Ruisong Ye
16.1
16.2

16.3
16.4

Introduction . . . . . . . . . . . . . . . . . . . . . . .
Fixed points and bifurcations . . . . . . . . . . . . .
16.2.1 The existence of fixed points . . . . . . . . .
16.2.2 The stability of fixed points and bifurcations
Existence of Marotto–Li–Chen chaos . . . . . . . . .
Numerical simulation results . . . . . . . . . . . . .

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17. Approximate periodic solutions of damped harmonic oscillators
with delayed feedback

339

Qian Guo
17.1
17.2
17.3

17.4
17.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hopf bifurcation analysis . . . . . . . . . . . . . . . . . . . .

Lyapunov-Schmidt reduction approach for periodic solutions .
17.3.1 Preliminary: reformulation and projection operators .
17.3.2 Quadratic Taylor polynomial approximation . . . . .
17.3.3 Bifurcation equations . . . . . . . . . . . . . . . . . .
17.3.4 Accuracy of approximation . . . . . . . . . . . . . . .
Multiple scales analysis for periodic solutions . . . . . . . . .
Simulation of period-doubling cascade . . . . . . . . . . . . .

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341
343
343
344
346
347
348
353

Bibliography

359

18. The numerical methods in option pricing problem

361

Xiong Bo
18.1
18.2
18.3

18.4

Introduction . . . . . . . . . . . . . . . . . . . . .
Black–Scholes option pricing theory assumptions
Binomial tree methods . . . . . . . . . . . . . . .
Finite difference method . . . . . . . . . . . . . .

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363
364

366

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371

19. Synchronization and its control between two coupled networks

373

Yongqing Wu and Minghai L¨
u
19.1
19.2

19.3

19.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Anti-synchronization between two coupled networks with nonlinear
signal’s connection and the inter-network actions . . . . . . . . . .
19.2.1 Two coupled networks with nonlinear signals . . . . . . . .
19.2.2 Two coupled networks with reciprocity . . . . . . . . . . .
19.2.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . .
Pinning anti-synchronization between two general complex
dynamical networks . . . . . . . . . . . . . . . . . . . . . . . . . .
19.3.1 Pinning anti-synchronization criterion . . . . . . . . . . . .
19.3.2 Numerical simulations . . . . . . . . . . . . . . . . . . . . .
Generalized synchronization between two networks . . . . . . . . .

19.4.1 Generalized synchronization criterion . . . . . . . . . . . .
19.4.2 Numerical examples . . . . . . . . . . . . . . . . . . . . . .

373
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19.5

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

Bibliography

389

Index

391

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Chapter 1

Gronwall inequalities1

Fanhai Zeng, Jianxiong Cao and Changpin Li∗
Department of Mathematics, Shanghai University, Shanghai 200444, PR China
∗

In this chapter, we display the existing continuous and discrete Gronwall
type inequalities, including their modifications such as the weakly singular
Gronwall inequalities which are very useful to study the fractional integral
equations and the fractional differential equations.
Keywords: Gronwall inequality, weakly singular Gronwall inequality
1.1

Introduction

It is well known that Gronwall–Bellman type integral inequalities play important
roles in the study of existence, uniqueness, continuation, boundedness, oscillation
and stability properties to the solutions of differential and integral equations. In
1919, Gronwall first introduced the famous Gronwall inequality in the study of the
solution of the differential equation. Since then, a lot of contributions have been
achieved by many researchers. The original Gronwall inequality has been extended
to the more general case, including the generalized linear and nonlinear Gronwall
type inequalities [Bihari (1956); Willett (1964); Bainov and Simenov (1992); Pachpatte (2002a)], the two and more variables cases [Beckenbach and Bellman (1961);
Pachpatte (2002a); Snow (1971); Yeh (1980, 1982b); Bondge and Pachpatte (1979)],
and the Gronwall type inequalities for discontinuous functions [Samoilenko and Borysenko (1998); Borysenko and Iovane (2007); Galloa and Piccirillo (2007)]. At the
same time, the discrete analogues have also been derived [Yang (1983, 1988); Zhou
and Zhang (2010); Salem and Raslan (2004)]. Meanwhile, some useful results of the
weakly singular Gronwall inequalities have been established as well [Mckee (1982);
ˇ (1997); Denton and Vatsala (2010)], which are
Dixo and Mckee (1986); Medved
1 The

present work was supported by the National Natural Science Foundation of China (grant
no. 10872119), the Shanghai Leading Academic Discipline Project (grant no. S30104), and the
Key Program of Shanghai Municipal Education Commission (grant no. 12ZZ084).
1

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2

Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis

the powerful tools in the theoretical analysis of the integral equations with weakly
singular kernels and the fractional differential equations.
In the present chapter, we collect almost all the important existing Gronwall
type inequalities, which include the continuous cases, discrete cases, weakly singular
cases and their discrete analogues. If some important references happened not to
be here, we do apologize for these omissions.
The rest of this chapter is outlined as follows. In Section 1.2, we introduce the
continuous Gronwall inequalities. Then we present the discrete Gronwall inequalities in Section 1.3. In Section 1.4, the weakly singular Gronwall integral inequalities
and some of their discrete analogues are displayed. And the conclusions are included
in the last section.

1.2

The continuous Gronwall inequalities

In this section,we state some continuous integral inequalities of Gronwall type, which
can be used in the analysis of various problems in the theory of the nonlinear
differential equations and the integral equations.
In 1919, Gronwall first proved the following famous inequality, which is called
the Gronwall inequality .
Theorem 1.1 (Gronwall Inequality [Gronwall (1919)]). Let u(t) be a continuous function defined on the interval [t0 , t1 ] and
t

u(t) ≤ a + b

u(s) ds,

(1.1)

t0

where a and b are nonnegative constants. Then we have
u(t) ≤ aeb(t−t0 ) , t ∈ [t0 , t1 ].

(1.2)

After more than 20 years, Bellman extended the original Gronwall inequality, which
reads in the following theorem.
Theorem 1.2 (Bellman Inequality [Bellman (1943)]). Let a be a positive
constant, u(t) and b(t), t ∈ [t0 , t1 ] be real-valued continuous functions, b(t) ≥ 0,
satisfying
t

u(t) ≤ a +

b(s)u(s) ds, ∀t ∈ [t0 , t1 ].

(1.3)

t0

Then we have
t

b(s) ds , ∀t ∈ [t0 , t1 ].

u(t) ≤ a exp

(1.4)

t0

Bellman also proved that if u(t) and b(t) are continuous functions, b(t) is a
t
nonnegative function, and u(t) ≤ u(t0 ) + t0 b(s)u(s) ds, t ∈ (t0 , t1 ), then
t

t

u(t0 ) exp

b(s) ds ,

b(s) ds ≤ u(t) ≤ u(t0 ) exp

−
t0

t0

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t ∈ (t0 , t1 ).