Phase portrait direction clockwise or anticlockwise?

In summary, to determine the direction of an unstable spiral, you need to look at the equations for dy/dt and dx/dt separately. If the phase portraits are identical, they will be traversed in opposite directions. By substituting values for x and y, you can determine the direction of the spiral.
  • #1
pashkar
1
0
hi i would like some steps to help determine whether my unstable spiral is anticlockwise or clockwise.

i have the slope equation of dY/dX=( -4x +y)/y.

iv tried subbing in x = 0 and y = 0, but it really doesn't help me understand the direction of the spiral.
please list some easy steps/ methods to work out direction. i'd really appreciate it. cheers
 
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  • #2
You need to look at dy/dt and dx/dt separately. What are the original equations?

For example dy/dt= -4x+ y, dx/dt= y and dy/dt= 4x- y, dx/dt= -y both give
dy/dx= (-4x+ y)/y- their phase portraits will be identical but they will be "traversed" in opposite directions. Looking at x= 0, y= 1, dy/dt= 1, dx/dt= 1 for the first so the "motion" is clockwise. But dy/dt= -1, dx/dt= -1 for the second so the "motion" is anti-clockwise.
 
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1. How do you determine the direction of a phase portrait?

The direction of a phase portrait can be determined by looking at the arrows or vectors on the graph. If the arrows are pointing in a clockwise direction, the phase portrait is clockwise. If they are pointing in an anticlockwise direction, the phase portrait is anticlockwise.

2. What does a clockwise phase portrait indicate?

A clockwise phase portrait indicates that the system is stable. This means that the system will eventually return to its equilibrium state after being disrupted by a small disturbance. This is commonly seen in physical systems such as pendulums or spring-mass systems.

3. What does an anticlockwise phase portrait indicate?

An anticlockwise phase portrait indicates that the system is unstable. This means that the system will not return to its equilibrium state after being disrupted by a small disturbance. This is often seen in chaotic systems or systems with multiple equilibrium states.

4. Can a phase portrait change direction?

Yes, a phase portrait can change direction depending on the parameters of the system. For example, a small change in the parameters of a stable system can cause it to become unstable, resulting in a change in the direction of the phase portrait.

5. How is the direction of a phase portrait related to the behavior of a system?

The direction of a phase portrait is closely related to the behavior of a system. A clockwise phase portrait indicates stable behavior, while an anticlockwise phase portrait indicates unstable behavior. The direction can also provide information about the magnitude and frequency of oscillations in the system.

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