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Chapter 5: Q16RP (page 306)

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

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

Short Answer

Expert verified

The critical point \(\left( {{\bf{0,0}}} \right)\) is an unstable saddle 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

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

The first equation gives us that\(x{\bf{ = 2 }}y\), so substituting this into the second equation one gets that\(3y{\bf{ = 0}}\).

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 unstable saddle point.

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Most popular questions from this chapter

Find all the critical points of the system

dxdt=x2-1dydt=xy

and the solution curves for the related phase plane differential equation. Thereby proving that two trajectories lie on semicircles. What are the endpoints of the semicircles?

Solve the given initial value problem.

x'=y+z;x(0)=2y'=x+z;y(0)=2z'=x+y;z(0)=-1

Show that the Poincare map for equation (1) is not chaoticby showing that if\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{\nu }}_{\bf{o}}}{\bf{)}}\)and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{o}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{o}}{\bf{)}}\)are two initial values that define the Poincare maps\({\bf{(}}{{\bf{x}}_{\bf{n}}}{\bf{,}}{{\bf{\nu }}_{\bf{n}}}{\bf{)}}\) and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{n}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{n}}{\bf{)}}\), respectively, using the recursive formulas in (3), then one can make the distance between\({\bf{(}}{{\bf{x}}_{\bf{n}}}{\bf{,}}{{\bf{\nu }}_{\bf{n}}}{\bf{)}}\)and\({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{n}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{n}}{\bf{)}}\)small by making the distance between\({\bf{(}}{{\bf{x}}_{\bf{o}}}{\bf{,}}{{\bf{\nu }}_{\bf{o}}}{\bf{)}}\) and \({\bf{(}}{{\bf{x}}^{\bf{*}}}_{\bf{o}}{\bf{,}}{{\bf{\nu }}^{\bf{*}}}_{\bf{o}}{\bf{)}}\)small. (Hint: Let \({\bf{(A,}}\phi {\bf{)}}\)and \({\bf{(}}{{\bf{A}}^{\bf{*}}}{\bf{,}}{\phi ^ * }{\bf{)}}\) be the polar coordinates of two points in the plane. From the law of cosines, it follows that the distance d between them is given by\({{\bf{d}}^{\bf{2}}}{\bf{ = (A - }}{{\bf{A}}^{\bf{*}}}{{\bf{)}}^{\bf{2}}}{\bf{ + 2A}}{{\bf{A}}^{\bf{*}}}{\bf{(1 - cos(}}\phi {\bf{ - }}{\phi ^ * }{\bf{))}}\).)

Find all the critical points of the system

dxdt=x2-1dydt=xy

and the solution curves for the related phase plane differential equation. Thereby proving that two trajectories lie on semicircles. What are the endpoints of the semicircles?

In Problems 19鈥�24, convert the given second-order equation into a first-order system by setting v=y鈥�. Then find all the critical points in the yv-plane. Finally, sketch (by hand or software) the direction fields, and describe the stability of the critical points (i.e., compare with Figure 5.12).

d2ydt2+y3=0.
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