Constructing Phase Portrait of System with Mathematica

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SUMMARY

The discussion focuses on constructing the phase portrait of a dynamical system using Mathematica. The system is defined by the equations \(\dot{x}(t) = y(t)\) and \(\dot{y}(t) = -x^3(t) + 4x(t)y(t)\). The correct method to visualize this phase portrait involves using the StreamDensityPlot function in Mathematica, specifically with the command StreamDensityPlot[{y, -x^3 + 4 x y}, {x, -1, 1}, {y, -1, 1}]. This approach effectively represents the dynamics of the system in a two-dimensional space.

PREREQUISITES
  • Familiarity with Mathematica software
  • Understanding of first-order differential equations
  • Knowledge of phase portraits in dynamical systems
  • Basic skills in using plotting functions in Mathematica
NEXT STEPS
  • Explore advanced plotting techniques in Mathematica for higher-dimensional systems
  • Learn about the stability analysis of dynamical systems
  • Investigate the use of StreamPlot for visualizing vector fields in Mathematica
  • Study the implications of nonlinear dynamics in phase portraits
USEFUL FOR

This discussion is beneficial for mathematicians, physicists, and engineers interested in dynamical systems, as well as students and researchers using Mathematica for modeling and visualization of complex systems.

ManuelF
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How can I construct the phase portrait(with Mathematica) of the following system?
[tex]\dot{x}(t)=y(t)[/tex]
[tex]\dot{y}(t)=-x^3(t)+4x(t)y(t)[/tex]
 
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The phase portrait would be four dimensional, so it would be pretty hard to plot.
 
Oops, my previous response was incorrect. I didn't notice that this is two first order equations instead of two second order equations. The phase portrait can be easily obtained as:

StreamDensityPlot[{y, -x^3 + 4 x y}, {x, -1, 1}, {y, -1, 1}]
 

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