Differential equations equilibrium points

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SUMMARY

The discussion focuses on finding equilibrium points for the system of differential equations defined by x' = x(x² + y²) and y' = y(x² + y²). The equilibrium points are identified as (0,0) and the behavior of the linearized system is analyzed using the Jacobian matrix, which yields eigenvalues of 0. The conversation emphasizes the importance of linearization and the use of Mathematica version 7's "StreamPlot" function for visualizing phase portraits. Additionally, it highlights that linearization may not be applicable for certain types of systems.

PREREQUISITES
  • Understanding of differential equations and equilibrium points
  • Familiarity with Jacobian matrices and eigenvalues
  • Knowledge of phase portraits in dynamical systems
  • Experience with Mathematica version 7 and its functions
NEXT STEPS
  • Study the process of linearization in differential equations
  • Learn how to compute and interpret Jacobian matrices
  • Explore the use of Mathematica's "StreamPlot" for phase portrait visualization
  • Investigate systems where linearization fails and the implications for analysis
USEFUL FOR

Students and researchers in mathematics, particularly those studying dynamical systems, differential equations, and phase portrait analysis. This discussion is also beneficial for anyone using Mathematica for mathematical modeling.

sana2476
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Homework Statement



x'=x(x^2+y^2)
y'=y(x^2+y^2)

i) Find all the equilibrium points and describe the behavior of the associated linearized system
ii) Describe the phase portrait for the nonlinear system
iii) Does the linearized system accurately describe the local behavior near the equilibrium points?

Homework Equations



Solving for equilibrium, you get: x=0 and x^2+y^2=0. Likewise with second function, you get: y=0 and x^2+y^2=0. But I don't know what the phase portrait will behave.
 
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I know that getting the Jacobian matrix and evaluating it at (0,0). You get your eigenvalues as 0 as well.
 
I really need help...can someone try and help me with this please!
 
sana2476 said:
I really need help...can someone try and help me with this please!

Hello sana. I can't help you directly here as I have limited access to the web. But I can give you really good advice in the interim: The book "Differential Equations" by Blanchard, Devaney and Hall does a really good job of working with these problems: First linearize it by calculating the partials and form the Jacobian matrix, find the eigenvalues and determine the types of fixed points based on the kinds of eigenvalues. Draw the phase portrait. Mathematica ver 7 has an excellent function for drawing in an instant what use to take hours: It's called "StreamPlot" (I think, or something like this) and draws the entire phase portrait in a single operation.

Note linearization fails for some systems. Find out which types of systems these are.
 

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