Behavior Analysis And Numerical Simulation Of Nonlinear Differential Power System With Several Classes

In order to study the evolution of celestial bodies in the universe, Physicists have established a number of nonlinear dynamics models. The research of the evolution of celestial bodies is transformed into the research of these models. Facing with cosmological system who have dif-ferent backgrounds and parameters, Astronomy workers are more concerned about the behavior of these complex systems and whether they are ultimately chaotic. In this paper, we consider some typical nonlinear autonomous systems in Astrophysics:the cosmological system in the Dilatonic, Quintessence and Tachyon scalar field; the cosmological system in a power law Kinetic Quintessence scalar field.For the cosmological system in the Dilatonic scalar field, we have got its equilibrium points by solving the system and analyzed the stability of the equilibrium points with the theory of linear stability. In addition, we draw the corresponding attractor using the MATLAB software. By combining results of numerical analysis and simulation methods, we reveal the local behavior of the system on the phase plane. Subsequently, we study the movement of the system again on the orbits between the stable equilibrium points and the unstable equilibrium points, which shows the dynamic state of the system.For the cosmological system in the Quintessence and Tachyon scalar field, we first solve equilibrium points and apply linearization to nonlinear system on equilibrium points. Next, we determine the type of equilibrium points through the eigenvalues of the linear system’s matrix and analyze the stability of the equilibrium points of system with varied parameter according to the linear stability theorem and the center manifold theorem. In addition, We approximate the solution of the center manifold of the system at the equilibrium point, So we can get the local stability of the system and draw the phase diagrams of the corresponding attractor near the equilibrium points. Finally, we propose an algorithm for calculating the Lyapunov exponent and apply to compute all Lyapunov exponents of the systems with varied parameter, which help us to determine whether the system is chaotic.Due to its complication for the cosmological system in the power law Kinetic Quintessence scalar field, we couldn’t judge the stability of the system by analyzing the eigenvalues of the Jacobian matrix at the equilibrium points. In order to analyze the stability of the equilibrium points, we draw the phase diagram of the solutions near the equilibrium points. Similarly, we calculate the Lyapunov exponent of the system by using the similar algorithm as in above, and obtain the figures of Lyapunov exponent with the parameter’s changing, which makes us determine whether the system is chaotic.