4.2 The Pendulum with External Force
4.2.2 Periodic Force and Chaos
In this subsection, we will discussed the equation uθθ+ sin(u) = εsin(θ) and it can be represented as the system
Notice that (4.17) is a non-autonomous system since the system depends on the parameter θ. We can not analyze the solutions by its vector field as before since the vector field changes with θ. Thus, we need a three dimensional picture with u−, v−, and θ− axes to perform the solution curves of this system. But the system (4.17) can be represented as autonomous system by letting τ (θ) = θ. We obtain
Thus, we could draw the vector field of the system (4.18) for an ε. Furthermore, if the system given a initial conditions, we can have a solution curve and the solution curve can be projected on the uθ−, vθ−, and uv−plane. The Figure 4.8 is the system with ε = 0.1 and the initial conditions given u(0) = 0.2, v(0) = 0. The Figure 4.9 shows the system with ε = 0.1 and the initial conditions given u(0) = 3.14, v(0) = 0.
The blue, brown and rainbow curves are represented the solution curve projected on the uθ−, vθ−, and uv−planes respectively. Compare the Figure 4.8 and the Figure 4.9. Though the two cases have the same perturbed coefficient ε = 0.1, their behaviors are quite different with different initial conditions. The initial condition of the former system is near the point (0, 0), the center of the unperturbed system.
The initial condition of the other system is near the point (π, 0), the saddle of the unperturbed system. It seems that the initial condition near the different types of equilibrium of the unperturbed system will influence the behavior of the solution.
Next, we analyze the two different cases by return map [9].
Let us look back to the (4.17). Notice that the external force is εsin(θ), a peri-odic function. Thus, if any two parameter θ1, θ2 with θ1 − θ2 = 2nπ where n ∈ Z, we will get the same system at θ = θ1 and θ = θ2. If we given an initial condition, (u(0), v(0)), there is a solution curve in 3−dimensional space. The solution will intersect the plane θ = 2π at some point (u1, v1) = (u(2π), v(2π)). Since the system is the same at θ = 0 and 2π, the solution travels from (u1, v1) at plane θ = 2π can be regarded as start from (u1, v1) at plane θ = 0. Hence we use a map to translate the point (u1, v1) on the plane θ = 0. The map is call the return map [9].
Figure 4.8: ε = 0.1, u(0) = 0.2, v(0) = 0.
Figure 4.9: ε = 0.1, u(0) = 3.14, v(0) = 0.
Moreover, the points (u(2nπ), v(2nπ)) can be translated on the plane θ = 0 for all n ∈ {0} ∪ N if we keep applying the return map. In other words, we choose some particular points on the last graph of the Figure 4.8 ( or Figure 4.9 ) to analyze the behavior of the solution. The Figure 4.10 and the Figure 4.11 are derived by the Figure 4.8 and the Figure 4.9, respectively.
Figure 4.10: Return map with 1000 iterates for u(0) = 0.2, v(0) = 0.
Figure 4.11: Return map with 500 iterates for u(0) = 3.14, v(0) = 0.
The Figure 4.10 and the Figure 4.11 tell us that the solution with the initial condition near the center of the unperturbed system has more regular behavior and
the behavior of the solution with initial condition near the saddle of the unperturbed system is unpredictable. We can get some clues about the phenomenon from the system (4.17) with ε = 0.1 and the equilibrium points of the unforced system. For the ideal pendulum, the initial condition near the center means that the pendulum given a small displacement and it oscillates forever. Now, we give the pendulum a external force in terms of εsin(θ). The external force will make the pendulum swing higher in some situation and sometimes make it swing less. The result depends on the parameter θ. Since the amplitude of the external force is a small constant ε, the pendulum still oscillate near the center. On the other hand, the initial condi-tions near the saddle is more complicated since the external force, εsin(θ), plays an important roles. Since when the pendulum swing approach to the saddle, the external force will decide the particle rotate in clockwise or counterclockwise. Thus, the behavior of the solution has closely relation with θ. Hence we can not forecast where the solution will approach in long time. The behavior is called the chaos since the solution is incontrollable. Finally, we see some solutions with different initial conditions for the system (4.17) with ε = 0.1 in return map.
Figure 4.12: Return map with 500 iterates for u(0) = −3.14, v(0) = 0.
Figure 4.13: Return map with 1000 iterates for u(0) = 6.28, v(0) = 0.
Figure 4.14: Return map with 500 iterates for u(0) = 9.42, v(0) = 0.
Figure 4.15: Return map with 1000 iterates for u(0) = −4.21, v(0) = 0.
Chapter 5 Conclusion
To summarize, at the beginning, we introduced and classified the linear partial dif-ferential equations where we then focused on the hyperbolic case. Then we presented certain practical problems whose mathematical models are system of the linear hy-perbolic equations and introduced the Sine-Gordon equation roughly. Following, to developed the exact theory of the Sine-Gordon equation, we studied the clas-sical Elliptic functions where one application in solving a nonlinear equation has been presented and gave a mathematical model of the nonlinear vibrating string to practice Jacobian elliptic function. After studying the Jacobian elliptic functions, we applied it to study the systems of the Sine-Gordon equation in detail. In the end, we further studied the perturbations of the Sine-Gordon equation by certain qualitative analysis methods. None of our arguments in this paper are new, yet our effort is. But we will become a more mature researcher for the applied mathematics by engaging a nice and fundamental mathematical model, namely, the Sine-Gordon equation through the hard work in a long period.
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