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Chapter 9: Problem 9

Consider the system $$ d x / d t=a x[1-(y / 2)], \quad d y / d t=b y[-1+(x / 3)] $$ where \(a\) and \(b\) are positive constants. Observe that this system is the same as in the example in the text if \(a=1\) and \(b=0.75 .\) Suppose the initial conditions are \(x(0)=5\) and \(y(0)=2\) (a) Let \(a=1\) and \(b=1 .\) Plot the trajectory in the phase plane and determine (or cstimate) the period of the oscillation. (b) Repeat part (a) for \(a=3\) and \(a=1 / 3,\) with \(b=1\) (c) Repeat part (a) for \(b=3\) and \(b=1 / 3,\) with \(a=1\) (d) Describe how the period and the shape of the trajectory depend on \(a\) and \(b\).

Short Answer

Expert verified
Answer: The period and shape of the trajectory depend on the values of a and b such that varying these parameters results in different oscillation patterns. By plotting the trajectories for different values of a and b, we can observe how the period of oscillation and shape of the trajectory are affected. In general, changing the values of a and b will lead to changes in the period and trajectory shape, highlighting the critical role these parameters play in the oscillating system's behavior.

Step by step solution

01

Part (a) - Plot trajectories and estimate the period of oscillation for a=1 and b=1

To plot the trajectory, we can use a numerical solver to obtain the solution to the system of differential equations for the given initial conditions \(x(0) = 5\) and \(y(0) = 2\) and a =1, b=1. After obtaining the trajectory, we can visualize the result to analyze the period of oscillation.
02

Part (b) - Plot trajectories with a=3 and a=1/3, keeping b=1

To understand the effect of varying \(a\), we need to plot the trajectories in the phase plane for a=3 and a=1/3, making sure we keep b=1. We can repeat the same process as in part (a), but this time with different values of \(a\).
03

Part (c) - Plot trajectories with b=3 and b=1/3, keeping a=1

Now we want to analyze the effect of varying \(b\). To do this, we need to plot trajectories for b=3 and b=1/3, while keeping a=1 constant. We can repeat the same process as in previous parts, just changing the values of \(b\) instead.
04

Part (d) - Describe the dependence of period and trajectory shape on a and b

After plotting the trajectories for different values of \(a\) and \(b\), we can describe how the period and shape of the trajectory depend on these parameters. By analyzing the graphs, it is possible to draw conclusions about how varying \(a\) and \(b\) affect the period of oscillation and the shape of the trajectories.

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Most popular questions from this chapter

In this problem we show how small changes in the coefficients of a system of linear equations can affect the nature of a critical point when the eigenvalues are equal. Consider the system $$ \mathbf{x}^{\prime}=\left(\begin{array}{cc}{-1} & {1} \\ {0} & {-1}\end{array}\right) \mathbf{x} $$ Show that the eigenvalues are \(r_{1}=-1, r_{2}=-1\) so that the critical point \((0,0)\) is an asymptotically stable node. Now consider the system $$ \mathbf{x}^{\prime}=\left(\begin{array}{cc}{-1} & {1} \\ {-\epsilon} & {-1}\end{array}\right) \mathbf{x} $$ where \(|\epsilon|\) is arbitrararily small. Show that if \(\epsilon>0,\) then the eigenvalues are \(-1 \pm i \sqrt{\epsilon}\), so that the asymptotically stable node becomes an asymptotically stable spiral point. If \(\epsilon<0,\) then the roots are \(-1 \pm \sqrt{|\epsilon|},\) and the critical point remains an asymptotically stable node.

(a) Determine all critical points of the given system of equations. (b) Find the corresponding linear system near each critical point. (c) Find the eigenalues of each linear system. What conclusions can you then draw about the nonlinear system? (d) Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system. $$ d x / d t=x-x^{2}-x y, \quad d y / d t=3 y-x y-2 y^{2} $$

(a) Determine all critical points of the given system of equations. (b) Find the corresponding linear system near each critical point. (c) Find the eigenalues of each linear system. What conclusions can you then draw about the nonlinear system? (d) Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system. $$ d x / d t=2 x+y+x y^{3}, \quad d y / d t=x-2 y-x y $$

The equation of motion of an undamped pendulum is \(d^{2} \theta / d t^{2}+\omega^{2} \sin \theta=0,\) where \(\omega^{2}=g / L .\) Let \(x=\theta, y=d \theta / d t\) to obtain the system of equations $$ d x / d t=y, \quad d y / d t=-\omega^{2} \sin x $$ (a) Show that the critical points are \((\pm n \pi, 0), n=0,1,2, \ldots,\) and that the system is almost lincar in the neighborhood of cach critical point. (b) Show that the critical point \((0,0)\) is a (stable) center of the corresponding linear system. Using Theorem 9.3.2 what can be said about the nonlinear system? The situation is similar at the critical points \((\pm 2 n \pi, 0), n=1,2,3, \ldots\) What is the physical interpretation of these critical points? (c) Show that the critical point \((\pi, 0)\) is an (unstable) saddle point of the corresponding linear system. What conclusion can you draw about the nonlinear system? The situation is similar at the critical points \([\pm(2 n-1) \pi, 0], n=1,2,3, \ldots\) What is the physical interpretation of these critical points? (d) Choose a value for \(\omega^{2}\) and plot a few trajectories of the nonlinear system in the neighborhood of the origin. Can you now draw any further conclusion about the nature of the critical point at \((0,0)\) for the nonlinear system? (e) Using the value of \(\omega^{2}\) from part (d) draw a phase portrait for the pendulum. Compare your plot with Figure 9.3 .5 for the damped pendulum.

(a) A special case of the Lienard equation of Problem 8 is $$ \frac{d^{2} u}{d t^{2}}+\frac{d u}{d t}+g(u)=0 $$ where \(g\) satisfies the conditions of Problem 6 . Letting \(x=u, y=d u / d t,\) show that the origin is a critical point of the resulting system. This equation can be interpreted as describing the motion of a spring-mass system with damping proportional to the velocity and a nonlinear restoring force. Using the Liapunov function of Problem \(6,\) show that the origin is a stable critical point, but note that even with damping we cannot conclude asymptotic stability using this Liapunov function. (b) Asymptotic stability of the critical point \((0,0)\) can be shown by constructing a better Liapunov function as was done in part (d) of Problem 7 . However, the analysis for a general function \(g\) is somewhat sophisticated and we only mention that appropriate form for \(V\) is $$ V(x, y)=\frac{1}{2} y^{2}+A y g(x)+\int_{0}^{x} g(s) d s $$ where \(A\) is a positive constant to be chosen so that \(V\) is positive definite and \(\hat{V}\) is negative definite. For the pendulum problem \([g(x)=\sin x]\) use \(V\) as given by the preceding equation with \(A=\frac{1}{2}\) to show that the origin is asymptotically stable. Hint: Use \(\sin x=x-\alpha x^{3} / 3 !\) and \(\cos x=1-\beta x^{2} / 2 !\) where \(\alpha\) and \(\beta\) depend on \(x,\) but \(0<\alpha<1\) and \(0<\beta<1\) for \(-\pi / 2
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