Dynamical Analysis And Computation Of Caputo-Hadamard Fractional Ordinary Differential Equations

Fractional calculus has a history of more than 300 years,it is one of the important branches of mathematical analysis.Because of its distinctive singularity and nonlocality,fractional calculus is suitable for describing practical problems with memory or hereditary properties.It has been proved that fractional calculus is widely used in physics,mechanics,materials science,biology,finance and many other scientific fields.In recent years,Hadamard derivative and its modified form Caputo-Hadamard derivative have attracted extensive attention due to the important applications in practical problems related to engineering and mechanics.They can be used to characterize the fatigue fracture and Lomnitz logarithmic creep law of elasticity materials.Until now,there are few studies on the dynamics of fractional differential equations with Hadamard or CaputoHadamard derivatives.In this paper,we focus on Hadamard and Caputo-Hadamard derivatives,and deal with the dynamics of the fractional differential equations corresponding to these two kinds of derivatives.The innovative research results in this dissertation have three aspects:(?)Based on the idea of finite part integral,the numerical schemes for Hadamard fractional differential equations have been derived.And the corresponding comparison principles have also been established.(?)The Lyapunov exponents for the CaputoHadamard fractional differential systems are firstly defined.And the chaotic attractors in Chen system with Caputo-Hadamard derivative are detected.(?)The normal forms of basic bifurcations for the Caputo-Hadamard fractional differential system with a parameter are established.In the following,we introduce the results of this dissertation in detail.In Chapter 1,we introduce the development of fractional calculus,especially the Hadamard fractional calculus and Caputo-Hadamard derivative.We present the current situation of the dynamics of fractional differential equations with Hadamard or CaputoHadamard derivatives as well.In Chapter 2,we construct the numerical schemes for Hadamard derivatives based on the idea of the finite part integral,such as the fractional rectangular formula and the trapezoidal formula.Then we apply them to the calculation of the Hadamard fractional differential equation.In Chapter 3,we establish the comparison principles for the fractional differential equation with Hadamard-type derivative.By using the fact that the Volterra integral equation is equivalent to the Hadamard-type fractional differential equation,the continuous dependence of the solution on the right side function is proposed.Then,the first and second comparison principles for the Hadamard-type fractional differential equation are built up,respectively.The corresponding numerical examples are provided to verify the theoretical analysis.In Chapter 4,we define the Lyapunov exponents of the Caputo-Hadamard fractional differential system and then estimate their bounds.First,we review the definition of the Lyapunov exponents of the classical ordinary differential system and introduce the estimation of their bounds.Then,we define the Lyapunov exponents of Caputo-Hadamard fractional differential system with the help of variational equation.By using the asymptotic expansion of Mittag-Leffler function,we determine the upper bound of the corresponding Lyapunov exponents.Numerical examples are displayed to verify the validity of the upper bound of the Lyapunov exponents.In Chapter 5,we focus on the detection of the chaotic attractors of the Chen system with incommensurate orders of Caputo-Hadamard derivatives.Through the positivity of the first(leading)Lyapunov exponent,we find the chaotic attractors of the considered fractional Chen system.In Chapter 6,we calculate the normal forms of the fundamental bifurcations of Caputo-Hadamard fractional differential system with a parameter,including the normal forms of fold and pitchfork bifurcations.By using Taylor's expansion and Implicit Function Theorem,we prove that a generic Caputo-Hadamard fractional differential system with a parameter can be simplified into a system with the same equilibrium and bifurcation conditions.Then,by using topological equivalence,it is proved that the simplified system does not depend on the higher-order term,then the simplest normal form of bifurcation for the generic Caputo-Hadamard fractional differential system with a parameter is obtained.The last chapter gives a brief summary of this paper and presents the possible research topics in the future.