Singular points in 3-dim space

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SUMMARY

The discussion confirms that for a linearized system with eigenvalues \(\lambda_1, \lambda_2 = a \pm bi\) (where \(a > 0\)) and \(\lambda_3 < 0\), the system exhibits characteristics of an unstable spiral point. As time approaches infinity, the trajectory will align with the plane spanned by the eigenvectors \(v_1\) and \(v_2\) corresponding to \(\lambda_1\) and \(\lambda_2\). This conclusion is established as a definitive characteristic of the system's behavior.

PREREQUISITES
  • Understanding of linearized systems in dynamical systems theory
  • Knowledge of eigenvalues and eigenvectors
  • Familiarity with complex numbers and their implications in stability analysis
  • Basic grasp of trajectory behavior in phase space
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  • Study the implications of eigenvalue stability criteria in nonlinear systems
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Mathematicians, physicists, and engineers involved in dynamical systems analysis, particularly those focusing on stability and trajectory behavior in three-dimensional space.

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For a linearized system I have eigenvalues [tex]\lambda_1, \lambda_2 = a \pm bi \;(a>0)[/tex] and [tex]\lambda_3 < 0[/tex],
then it should be an unstable spiral point. As [tex]t \to +\infty[/tex] the trajectory will lie in the plane which is parallel with the plane spanned by eigenvectors [tex]v_1,v_2[/tex] corresponding to [tex]\lambda_1, \lambda_2[/tex].

Right? I am just not very sure.
 
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Yes, that is correct.
 
Thanks:smile:
 

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