Concept explainers
Interpretation:
To show by linearization the origin is non-isolated fixed point but in fact origin is an isolated fixed point. Classify the stability of the origin and sketch the
Concept Introduction:
The parametric curves traced by solutions of a differential equation are known as trajectories.
The geometrical representation of collection of trajectories in a phase plane is called as phase portrait.
The point which satisfies the condition
Closed Orbit corresponds to periodic solution of the system i.e.
If nearby trajectories moving away from the fixed point then the point is said to be saddle point.
If the trajectories keep swirling around the fixed point, then it is a unstable fixed point.
If nearby trajectories are moving away from the fixed point, then the point is said to be unstable fixed point.
If nearby trajectories are moving towards the fixed point, then the point is said to be stable fixed point.
To check the stability of fixed point, use Jacobian matrix
The point
Isolated Fixed Point: If there is no any other fixed point exists in a region closed to interested fixed point, then it is called Isolated Fixed point.
Non-isolated Fixed point: If there is fixed points in a region close to interested fixed point, then it is called Non-isolated fixed point.
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Nonlinear Dynamics and Chaos
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning