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Chapter 5: Q15RP (page 306)

Sketch some typical trajectories for the given system and by comparing with Figure\({\bf{5}}{\bf{.12}}\), page\({\bf{267}}\), and identify the type of critical point at the origin.

\(\begin{array}{c}{\bf{x}}'{\bf{ = - 2x - y}}\\{\bf{y}}'{\bf{ = 3x - y}}\end{array}\)

Short Answer

Expert verified

The critical point \(\left( {{\bf{0,0}}} \right)\) is an asymptotically stable spiral point.

Step by step solution

01

Finding the value of \({\bf{x,y}}\)

One can solve the critical point. To do so we need to solve the system\({\bf{x}}'{\bf{ = 0,y}}'{\bf{ = 0}}\), so one has

\(\begin{array}{c}{\bf{0 = - 2x - y}}\\{\bf{0 = 3x - y}}\end{array}\)

The first equation gives us that\({\bf{y = - 2 x}}\), so substituting this into the second equation one gets that\({\bf{5 x = 0}}\).

02

Finding the critical point

So, one has that \({\bf{x = 0, y = 0}}\) and the critical point is\(\left( {{\bf{x, y}}} \right){\bf{ = }}\left( {{\bf{0,0}}} \right)\).

Comparing this picture with the Figure \({\bf{5}}{\bf{.12}}\) in the Textbook one can conclude that the critical point \(\left( {{\bf{0,0}}} \right)\) is an asymptotically stable spiral point.

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Most popular questions from this chapter

In Problems 1–7, convert the given initial value problem into an initial value problem for a system in normal form.

3x''+5x-2y=0;x(0)=-1,x'(0)=04y''+2y-6x=0;y(0)=1,y'(0)=2

In Problems 19 – 21, solve the given initial value problem.

d2xdt2=y;    x0=3,    x'0=1,d2ydt2=x;   y0=1,     y'0=-1

Find the critical points and solve the related phase plane differential equation for the system dxdt=(x-1)(y-1),dydt=y(y-1).Describe (without using computer software) the asymptotic behavior of trajectories (as role="math" localid="1664302603010" t→∞) that start at

  1. (3, 2)
  2. (2, 1/2)
  3. ( -2, 1/2)
  4. (3, -2)

In Problems 15–18, find all critical points for the given system. Then use a software package to sketch the direction field in the phase plane and from this description the stability of the critical points (i.e., compare with Figure 5.12).

dxdt=x(7-x-2y),dydt=y(5-x-y)

Rigid Body Nutation. Euler’s equations describe the motion of the principal-axis components of the angular velocity of a freely rotating rigid body (such as a space station), as seen by an observer rotating with the body (the astronauts, for example). This motion is called nutation. If the angular velocity components are denoted by x, y, and z, then an example of Euler’s equations is the three-dimensional autonomous system

\(\begin{array}{l}\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = yz}}\\\frac{{{\bf{dy}}}}{{{\bf{dt}}}}{\bf{ = - 2xz}}\\\frac{{{\bf{dz}}}}{{{\bf{dt}}}}{\bf{ = xy}}\end{array}\)

The trajectory of a solution x(t),y(t), z(t) to these equations is the curve generated by the points (x(t), y(t), z(t) ) in xyz-phase space as t varies over an interval I.

(a) Show that each trajectory of this system lies on the surface of a (possibly degenerate) sphere centered at the origin (0, 0, 0).[Hint: Compute\(\frac{{\bf{d}}}{{{\bf{dt}}}}{\bf{(}}{{\bf{x}}^{\bf{2}}}{\bf{ + }}{{\bf{y}}^{\bf{2}}}{\bf{ + }}{{\bf{z}}^{\bf{2}}}{\bf{)}}\)What does this say about the magnitude of the angular velocity vector?

(b) Find all the critical points of the system, i.e., all points\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{y}}_{\bf{o}}}{\bf{,}}{{\bf{z}}_{\bf{o}}}{\bf{)}}\) such that \({\bf{x(t) = }}{{\bf{x}}_{\bf{o}}}{\bf{,y(t) = }}{{\bf{y}}_{\bf{o}}}{\bf{,z(t) = }}{{\bf{z}}_{\bf{o}}}\) is a solution. For such solutions, the angular velocity vector remains constant in the body system.

(c) Show that the trajectories of the system lie along the intersection of a sphere and an elliptic cylinder of the form\({{\bf{y}}^{\bf{2}}}{\bf{ + 2}}{{\bf{x}}^{\bf{2}}}{\bf{ = C}}\) for some constant C. [Hint: Consider the expression for dy/dx implied by Euler’s equations.]

(d) Using the results of parts (b) and (c), argue that the trajectories of this system are closed curves. What does this say about the corresponding solutions?

(e) Figure 5.19 displays some typical trajectories for this system. Discuss the stability of the three critical points indicated on the positive axes.
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