Fractional dynamic system research in China

1
Fractional dynamic system research in China
---Symposium of fractional dynamic system and its
applications in China Mechanic conference

• Reporter Hongguang Sun

Center for Self-Organizing and Intelligent
Systems (CSOIS), Utah State University Institute
of Soft Matter Mechanics, Hohai University, China
2
Information about conference

• Name Academic Conference of China Society of
  Mechanics
• Time Aug.22Aug.26, 2009
• Location ZhengZhou, China
• Papers more than 1500
• It is the highest-level conference of China
  Society of Mechanics

3
Here
4
Fractional dynamics and its applications

• Organizers Keqin Zhu, Wen Chen, Changpin Li
• Number of submitted articles 28
• Full-length paper 21

5
Paper classification

• Theoretical research 8
• Numerical methods 6
• Modeling approaches 5
• Application research8
• Others 1

English articles 2 Chinese articles 26
6
Some new and interesting results
7

• Constitutive equation for the generalized upper
  convected Maxwell model

Yang, Pan, Zhu, Keqin (Tsinhua University)
Fractional UCM model
Where is inverse factor of stress
Is inverse factor of strain
l is relaxation time E is displacement
gradient
a, b are fractional differential orders.
8
The integral expression of UCM constitutive model
The expression of the G has the following form
9
Research on dynamical analysis of fractional
nonlinear systems and typical complex behaviors
induced by numerical algorithm
Liu, Jie, Dong, Pengzhen and Shang Gang (Wuhan
University)

• The widely used Charefand, Oustloup methods may
  cause fake chaotic or fake periodic phenomena
  under some wrong conditions
• The modified Oustloup method can prove the
  ability of traditional frequency method

They first considered Liu system (q0.8)
10
Fake
Simulation result of fractional Liu system,
q0.8. Left Charefand method, Right improved
Oustaloup
Moreover, they investigated the Lü system (q0.86)
11
The ADM predictor-corrector scheme may cause fake
complex behaviors while using unsuitable length
of iteration step.
Fake
Simulation result of fractional Lü system with
q0.86, Left ?t0.01 Right ?t0.001
12
Stability analysis of fractional differential
equation and its application in HIV-1 infection
modeling
Kou, Chunhai, Yan, Ye and Liu, Jian (Donghua
University)
Main contributions

• They make an investigation of fractional model of
  HIV-1
• infection. Finally they get some valuable
  results. (Full
• length paper is unavailable)

13
Fractional Order Robust Control of Four Wheel
Steering Vehicle Based on Yaw Rate Tracking
Chen, Ning, Chen, Ye, Chen, Nan (Nanjing Forestry
University)
The fractional order PI?Dµ robust control is
applied in the yaw rate tracking control of
four-wheel-steering vehicle in this paper to
improve the robustness of yaw rate response for
vehicle with uncertain parameters
Control map
14
The function of fractional PID control
The response curves of step inputs of three
types of vehicle angular
15
Time-dependent Fractional Schrödinger Equation
with Moving Boundary
Jiang, Xiaoyun, Xu, Mingyu (Jiang Xiaoyun)
The considered time-dependent fractional spatial
Schrödinger equation
The moving-boundary problem is transformed into a
fixed-boundary problem and a new Hamiltonian is
obtained by means of Generalized Canonical
Transformations
The time evolution behaviors of quantum states
are described by means of time-dependent
perturbation theory
16
Bound estimation of Lyapunov parameter in
fractional system
Li, Changpin, Gong, Ziqing, Qian Deliang
(Shanghai University)
Main contributions

• Bound estimation of Lyapunov parameter in Riemann
  and Caputo types of fractional system
• The upper bound of fractional system is achieved,
  the existence of the lower bound of fractional
  system is illustrated by an example

17
Stationary response of Duffing oscillator with
fractional derivative damping under combined
harmonic and wide band noise excitations
Chen, Lincong, Zhu, Weiqiu (Tsinhua University)
Main contributions

• The averaged Itô stochastic differential
  equations for duffing oscillator with fractional
  derivative damping under combined harmonic and
  wide band noise excitations are obtained by using
  the generalized harmonic functions
• The stationary response is obtained by solving
  the reduced PFK equation

18
Generations and mechanisms of multi-stripes
fractional order chaotic attractors
Yang, Shuiping, Wu, Jianxin, Li, Min, Xiao,
Aiguo, Fu, Jingli (Xiangtan university)
Main contributions

• The original fractional order chaotic attractors
  was turned into a pattern with multiple
  parallel or rectangular stripes by employing
  simple periodic nonlinear functions
• Theoretic analysis about the underlying mechanism
  of generating the parallel stripes in the
  fractional order attractors is given
• Period doubling routes to chaos in the fractional
  order Rossler equation are also found

19
Identification of constitutive parameters of a
visco-elastic rod with generalized Voigt
fractional derivatives constitutive relations
Yang, Haitian and Zhao, Xiao (Dalian University
of Technology)
Main contributions

• The ant colony algorithm is proposed to solve
  constitutive parameters inverse problems of a
  visco-elastic rod with generalized Voigt
  fractional derivatives constitutive relations

20
The anomalous diffusion of calcium sparks in
cardiac cell
Tan, Wenchang (Peking University)
Main contributions

• Cytoplasm belongs to visco-elastic fluid
• Propose the spatial fractional diffusion equation
  model for diffusion process of calcium sparks in
  cardiac cell

21
IFAC Workshop on Fractional Differentiation and
its Application

• First France, 2004
• Second Portugal, 2006
• Third Turkey, 2008
• Fourth Spain, Lisbon, 2010
• Fifth China, Nanjing, 2012.
• Welcome to China in 2012

22
Paper list

1. Shen Shujun, Liu Fawang, Solution techniques for
   the riesz space-time fractional advection
   dispersion equation.
2. Zhang Xiaodi, Wen Chen, A comparison of
   relaxation-oscillation models involving fractal,
   fractional and positive fractional derivatives.
3. Liu QingXia, Finite element approximation for the
   anomalous sub-diffusion process
4. Yang Pan, Zhu Ke Qin, Investigation on the
   constitutive relationship for the generalized UCM
   model
5. Chen Wen, Introduction of FDA 2008
6. Liu Jie, Dong Pengzhen and Shang Gang, Research
   on dynamical analysis of fractional nonlinear
   Systems and typical complex behaviours induced by
   numerical algorithm
7. Hu Kaixin, Zhu Keqin, A Study of Mechanical
   Analogues of Fractional Elements
8. Sun Hongguang and Chen Wen, Fractional
   differential equation models of complex anomalous
   diffusion

23

1. Wang Zaihua, General solution of a linear
   vibration system with fractional-order derivative
2. Chen Ning, Chen Ye, Chen Nan, Fractional order
   robust control of four wheel steering vehicle
   based on yaw rate tracking
3. Li Xicheng, A fractional mathematical model of
   pharmacokinetics for percutaneous absorption
4. Wei Hui, Chen Wen, Sun Hongguang, The inverse
   source problem of space fractional abnormal
   diffusion equations
5. Liang, Yongshun and Zhang, Qi, Definition of
   fractal functions and their fractional integral
6. Jiang Xiaoyun and Xu Mingyu, Time-dependent
   fractional schrödinger equation with moving
   boundary
7. Su Li-juan, A finite difference method for the
   two-sided space-fractional advection-diffusion
   equation
8. Chen Lin-cong and Zhu Wei-qiu, Stationary
   response of Duffing oscillator with fractional
   derivative damping under combined harmonic and
   wide band noise excitations

24

• Yang Shuiping, Wu Jianxin, Li Min, Xiao Aiguo and
  Fu Jingli, Generations and mechanisms of
  multi-stripes fractional order chaotic attractors
• Hao Wuling, Zhang Wei and Hao Junjun, The
  analytic expressions of one class of fractalcurve
• Zhao Xiao, Yang Haitian, Identification of
  constitutive parameters of a visco-elastic rod
  with generalized Voigt fractional derivatives
  constitutive relations
• Liu Yaqing, Zheng Liancun and Zhang Xinxin, Exact
  Solution for MHD flow of a generalized unsteady
  couette fluid
• Tan, Wenchang, The anomalous diffusion of calcium
  sparks in cardiac cell
• Li Changpin, Gong Ziqing and Qian Deliang, Bound
  estimation of Lyapunov parameter in fractional
  system
• Kou, Chunhai, Yan, Ye and Liu, Jian, Stability
  analysis of fractional differential equation and
  its application in HIV-1 infection modeling
• Zhu Keqin, Hu Kaixin and Yang Fan, Viscous and
  elastic effects in the fractional elemental model

25

• Yin Deshen and Chen Wen, The volumetric strain
  research of soil in the conventional three-axis
  experiment based on fractional calculous
• Chen Ning and Tai Yongpeng, fractional
  semi-active suspension control of vehicles based
  on sliding mode observer
• Pan Xinyuan and Xiao Aiguo, Asymptotic stability
  of numerical methods of two types of fractional
  order functional differential equations
• Chen Ming and Yin Xiezhen, Finite element model
  of relaxation properties of fin material

26

• Thank you

HongGuang Sun sunhongguang08_at_gmail.com
Website of fractional dynamics http//www.ismm.ac
.cn/ismmlink/PLFD/index.html