We consider a simplified version of an oscillating system, namely, we focus on the undamped pendulum...

 

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We consider a simplified version of an oscillating system, namely, we focus on the undamped pendulum where there is no friction or air resistance to slow down the motion of the pendulum. In this case, the non-linear system of differential equations governing the undamped motion of the pendulum of length Lis: dx dt =y :) dy = -2; sinx de where x = 0 is the angular position, y = at is the angular velocity, and g is the acceleration due to gravity (see Figure (left)). (i) Determine all of the critical points for the system. (ii) Determine the linearised system for each critical point in part (i), and find the corresponding eigenvalues of the general solution. (iii) Consider the critical points at the origin and (π, O): ملا i. State the expected type and stability of each critical point, given the eigenvalues of the linearized system ii. Discuss whether we can expect the linearised system to approximate the behaviour of the non-linear system (d) Set and use MATLAB or any other plotter to sketch the global phase portrait of the non-linear system in a region that includes the critical points at the origin and (π, O). Given initial values (xo, yo) = (x (o), y(o)) provide a physical interpretation of the orbits by describing the corresponding oscillations for the case: i. where, and y. are both small; ii. where x. and y. are both just large enough to almost reach the top; iii. where and y, are large enough to swing over the top. Clearly relate your interpretation to the orbits in the phase portrait. We consider a simplified version of an oscillating system, namely, we focus on the undamped pendulum where there is no friction or air resistance to slow down the motion of the pendulum. In this case, the non-linear system of differential equations governing the undamped motion of the pendulum of length Lis: dx dt =y :) dy = -2; sinx de where x = 0 is the angular position, y = at is the angular velocity, and g is the acceleration due to gravity (see Figure (left)). (i) Determine all of the critical points for the system. (ii) Determine the linearised system for each critical point in part (i), and find the corresponding eigenvalues of the general solution. (iii) Consider the critical points at the origin and (π, O): ملا i. State the expected type and stability of each critical point, given the eigenvalues of the linearized system ii. Discuss whether we can expect the linearised system to approximate the behaviour of the non-linear system (d) Set and use MATLAB or any other plotter to sketch the global phase portrait of the non-linear system in a region that includes the critical points at the origin and (π, O). Given initial values (xo, yo) = (x (o), y(o)) provide a physical interpretation of the orbits by describing the corresponding oscillations for the case: i. where, and y. are both small; ii. where x. and y. are both just large enough to almost reach the top; iii. where and y, are large enough to swing over the top. Clearly relate your interpretation to the orbits in the phase portrait.

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Step 11 i Critical points The critical points of the system can be found by setting the righthand si... View the full answer