Nonlinear Dynamics: Nullclines and phase plane of a nonlinear system

In summary, the conversation discusses finding and classifying fixed points using linear analysis, as well as sketching nullclines, vector fields, and a plausible phase portrait. The fixed point at (0,0) is found and the Jacobian matrix is calculated, leading to eigenvalues of 0. However, further analysis is needed to classify the fixed point, and a CAS like Mathematica can be helpful in solving the problem.
  • #1
14
0

Homework Statement



Find the fixed points and classify them using linear analysis. Then sketch the nullclines, the vector field, and a plausible phase portrait.

dx/dt = x(x-y), dy/dt = y(2x-y)

Homework Equations


The Attempt at a Solution



f1(x,y) = x(x-y)

x-nullcline: x(x-y) = 0 [itex]\Rightarrow[/itex] x = 0

f2(x,y) = y(2x-y)

y-nullcline: y(2x-y) = 0 [itex]\Rightarrow[/itex] y = 0

Fixed point: (0,0)

J(x,y) =

(2x-y -x )
(2y 2x-2y)

Therefore,

J(0,0) =

(0 0)
(0 0)

Thus,

0 = |J(0,0) - λI| = (-λ)(-λ) = λ2 = 0 [itex]\Rightarrow[/itex] λ1,2 = 0
___________________________________________________________________

Now this is where I get stuck; I have no idea where to go from here when λ1 = λ2 = 0.

I would really appreciate at least a little nudge in the right direction. Thank you.
 
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  • #2
There are some cases where linearization fails to capture the dynamics of a non-linear system near it's fixed points and I think one case is when the eigenvalues are zero. Need to check out a good reference like "Differential Equations" by Blanchard Hall and Devaney. Also, whenever working on these types of problems, in my opinion, it is essential to have a CAS like Mathematica to help you with it unless you like to suffer. Now unfortunately you can't use StreamPlot with Wolfram Alpha but if you could find a machine at school running Mathematica, you could very easily answer three of the questions with the code:

Show[{StreamPlot[{x^2 - x y, 2 x y - y^2}, {x, -1, 1}, {y, -1, 1}],
Plot[{x, 2 x}, {x, -1, 1}, PlotStyle -> Thickness[0.008]]}]

I don't know how to classify the fixed-point at the origin but if I had to say something, I would say it's some type of sink. Maybe someone else can help us further.
 

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1. What is the significance of nullclines in nonlinear dynamics?

Nullclines are curves in the phase plane of a nonlinear system where one or both variables have zero rates of change. They represent the equilibrium points of the system, where the system is in a steady state. The intersection of the nullclines determines the stability of the equilibrium point.

2. How are nullclines and phase plane related in nonlinear dynamics?

The phase plane is a graphical representation of the dynamics of a system with two state variables. Nullclines are curves on the phase plane that help visualize the behavior of the system by indicating where the state variables are changing or not changing. The phase plane and nullclines together provide a complete picture of the dynamics of the system.

3. Can a system have more than one nullcline?

Yes, a system can have multiple nullclines, depending on the number of state variables. In a system with two state variables, there can be two nullclines, one for each variable. In a system with three state variables, there can be three nullclines, and so on.

4. How do nullclines help in analyzing the stability of a system?

The intersection of nullclines represents the equilibrium points of a system. By analyzing the direction of the nullclines near the intersection, one can determine the stability of the equilibrium point. If the nullclines have opposite directions, the equilibrium point is stable, and if they have the same direction, the equilibrium point is unstable.

5. Can nullclines help in predicting the behavior of a nonlinear system?

Yes, nullclines can provide valuable insights into the behavior of a nonlinear system. By analyzing the shape and direction of the nullclines, one can predict the behavior of the system and understand how changes in the parameters or initial conditions can affect the dynamics of the system.

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