Concept explainers
Interpretation:
To sketch the
Concept Introduction:
The system assigns a vector
Substitute the given system in the equation
The stable manifold of a fixed point is the set of all points in the plane which tends to the fixed point as time goes to positiveinfinity.
The unstable manifold for a fixed point is the set of all points in the plane which tends to the fixed point as time goes to negative infinity.
Answer to Problem 9E
Solution:
a) The vector field of the system
b) It is shown that the trajectories of the system are hyperbolas of the form
c) The equations for the stable and unstable manifolds for origin are
d) The solutions in terms of new variables
e) The equations for the stable and unstable manifolds in terms of
f) The general solution for
Explanation of Solution
a) The vector field of the given system
b) Consider the given system,
It can be shown that the trajectories of the system are hyperbolas of the form
Substituting
Thus, the given system implies
Integrating the above equation,
Thus, the trajectory of the given system is the hyperbola of the form
c) The origin is a saddle point for the given system. From the sketch of the vector field, it is clear that the equation
Also, the equation
This can also be proved by substituting
This linear system gives a stable fixed point.
Similarly, substituting
This linear system gives an unstable fixed point.
Therefore, the equations for the stable and unstable manifolds for origin is
d) Introducing new variables
The system can be rewritten as
Substituting
Again, substituting
The above equation can be also represented as
Solving for the initial conditions,
Substituting
Substituting the value of
Similarly substituting
Again, substituting
The above equation can be also represented as
Solving for the initial conditions,
Substituting
Substituting the value of
Hence, the solutions of the linear equation for the initial condition
e) Since the equations of the stable and unstable manifold are
Therefore, substituting
Similarly, substituting
Thus,
f) Since
Solving the above equations to get the values of
Substituting
Again, substituting
Similarly, substituting
Again, substituting
Therefore, the general solution for
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Chapter 5 Solutions
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