عنوان مقاله [English]
During recent decades, much attention has been paid to solving different kinds of dynamic systems, including pendulums, to observe their chaotic behavior. The governing equations of a double pendulum are investigated numerically in this paper. The pivot of the first pendulum is considered to be moving, following a certain mathematical template. Also, the pendulum, as a dynamic system, shows regular and chaotic behavior, under different conditions of its motion, such as initial angles and velocities (as the initial energy injected into the system), and the frequency of the pivot. The Lagrange equation governs this phenomenon.
The proposed solution methods include the modified midpoint, along with fourth-order Runge-Kutta, to check the conditions for probable instability. The total value of the energy of the system is in the form of initial kinetic and potential energies. All numerical evaluations of the mathematical equations were coded. To investigate the chaotic behavior, a three dimensional phase space diagram for characteristic parameters of the system was constructed. The energy of chaos onset for each value of the pivot frequency was obtained utilizing three procedures: 1) The Poincare map, which is a special section in the three-dimensional diagram of phase space, in which one of the system parameters is constant (the angle of the first pendulum with a vertical line is
zero). For a regular system (not chaotic), points in the Poincare map are located in an imaginary circle with limited radius, and they do not exceed this circle, even at very large times, 2) Lyapunov characteristic exponents, which
are the average exponential separation between the nearby phase space trajectories of system motion. A system with one or more positive Lyapunov characteristic exponents is known as chaotic, and 3) The bifurcation diagram for characteristic parameters that shows the period doubling of chaos. It is a form of transition from periodicity to chaos. From this diagram, the onset of chaos is detected. The results are obtained using three methods, which depict good agreement with each other. They indicate that the more the frequency of
the moving pivot, the less the initial energy required by that system to become chaotic. Initial energy decrease, which is due to frequency increase, follows a regular style, except for the natural frequency of the system. It was found that the moving pivot of the double pendulum has a dominant effect on chaos onset under certain circumstances.