Effect Of The Fractional Differentiation On Dynamic Behaviors Of Hamiltonian System

The fractional calculus reflects intrinsic dissipative processes that are sufficiently complicated in nature. The theory of control and synchronization of dynamical systems has been the focus in the study of scientific research. However, the method of the control and synchronization on fractional-order system is limited and at the initial stage of study. Their theoretical relationship with fractional calculus is not yet ascertained in full. In recent years, some researchers have tried to apply fractional order derivative to conservative system, studied the dynamic behaviors of conservative system with fractional order derivative, and made some preliminary results. But this field has just started in the domestic research.We consider Hamilton’s equation where the time derivative is replaced by the Caputo fractional time derivative. These models are called Hamiltonian systems with fractional time derivative. Using the method of numerical simulation, we have studied the influences of the fractional order differential on the dynamical behaviors of Hamiltonian system. Our main work is as follows:Part one(In the second chapter),we have studied dynamics behaviors of the Hamiltonian system with the fractional order derivative. We have investigated three Hamiltonian systems.The dynamics of the Hamiltonian system with fractional order derivative have been compared with Hamiltonian systems with integer order derivative, we have gotten same meaningful results.(1) Studying two coupled fractional Morse oscillators, we have made the numerical simulation of coupled fractional-order oscillators. Using the method of numerical simulation, we have studied the two coupled fractional Morse oscillators. We have gotten the energy of the single vibration, in coupled fractional order differential Morse oscillators. The energy of coupled fractional order differential Morse oscillators is decaying over time.Reducing the order, we have found the energy decays faster. Integer-order coupled Morse oscillators is a conservative system, and the fractional-order coupled Morseoscillators is dissipative system.(2) Studying fractional-order Sprott systems, we found that x-y phase diagram of the fractional-order Sprott system is confined when the order of fractional derivatives is between 0.9106 and 1.0. This is similar to integer order Sprott system behaviors. When the order is between 0.9106 and 0.90, curves of fractional-order Sprott tend to be a point. We have applied theory of the stability of fractional-order system’s equilibrium point to prove what is said above. Observing time-domain graph of x variables of the fractional Sprott system, when the order is between 0.9106 and 0, time-domain graph of x is in a periodical regime. When the order’s value range is less than 0.9106, the time-domain graph of x variable is similar to damping motion.(3) Researching poincare section, energy, momentum and amplitude changes respectively, in fractional-order Henon-Heiles system, We have discovered that the fractional order derivative can damage internal resonance of Henon-Heiles system and reduce the energy.Part two(In the third chapter), we have studied the synchronous behaviors between fractional-order Hamiltonian systems.(1) Using the feedback control, we have realized the synchronization between fractional-order Sprott systems with the different initial conditions. The stability of error system has been proved.(2) We have used feedback control to achieve the measure synchronization and phase synchronization between fractional-order Sprott with different order respectively.(3)The coupled integer-order Morse oscillators can achieve measure synchronization.There is no measure synchronization phenomenon in Coupled fractional-order Morse oscillators at the same coupling intensity with the coupling intensity of coupled integer-order Morse oscillators. Through a period of time, there is a phenomenon which is similar to the phase locking phenomenon between two single oscillators of coupled fractional-order Morse oscillators. The value of phase difference between single fractional-order Morse oscillators fluctuates in a small scope. We guess that there may exist local measure synchronization phenomenon in coupled fractional-order Morse oscillators.