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Complex dynamical system: check stability of point numerically

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1

$begingroup$

Given a dynamical system described by the equations (for i=1,...,N)
$$fracd y_idt = P_i - by_i + K underseti neq jsum_i=1^N sin(x_i-x_j)$$
$$fracd x_idt = y_i$$
Say that I have found a stable point in the $R^2n$ dimensional space $((x_1,y_1),(x_2,y_2),...,(x_n,y_n))$ such that $fracdx_idt = fracdy_idt=0$ for all $i =1,...,N$.

How do I check numerically, if this point is stable or unstable? (Preferably without checking the eigenvalues of the jacobian matrix)

numerical-methods dynamical-systems stability-theory

share|cite|improve this question

asked Mar 11 at 11:38

FabricFabric

478

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  You could test if it is a local minimum of the energy function $$sum_ifrac12y_i^2-P_ix_i+Ksum_j< icos(x_i-x_j),$$ if you know a derivative-free method to do that.
  $endgroup$
  – LutzL
  Mar 11 at 12:08
  

  
  
  
  


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  What's wrong with checking eigenvalues numerically? You can find analytical expression for derivatives using any symbolic engine, plug in equilibrium coordinates and find eigenvalues quite reliably. Another crudest way to check stability is just to pick a random point quite close to the equilibrium and integrate trajectory for a moderate amount of time. If it is a saddle, most likely you won't pick a point from its stable manifold and trajectory abruptly goes somewhere else, not to the equilibrium. P.S. Kuramoto with an inertia? :)
  $endgroup$
  – Evgeny
  Mar 11 at 12:29
  

  
  
  
  


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  Will that work? Picking a 'random' point nearby in 2n dimensional space? Wouldn't it by chance be a direction which exhibits 'directional' stability while others are unstable? And yes! It is indeed Kuramoto with inertia :-)
  $endgroup$
  – Fabric
  2 days ago
  

  
  
  
  

add a comment | 

1

$begingroup$

Given a dynamical system described by the equations (for i=1,...,N)
$$fracd y_idt = P_i - by_i + K underseti neq jsum_i=1^N sin(x_i-x_j)$$
$$fracd x_idt = y_i$$
Say that I have found a stable point in the $R^2n$ dimensional space $((x_1,y_1),(x_2,y_2),...,(x_n,y_n))$ such that $fracdx_idt = fracdy_idt=0$ for all $i =1,...,N$.

How do I check numerically, if this point is stable or unstable? (Preferably without checking the eigenvalues of the jacobian matrix)

numerical-methods dynamical-systems stability-theory

share|cite|improve this question

asked Mar 11 at 11:38

FabricFabric

478

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You could test if it is a local minimum of the energy function $$sum_ifrac12y_i^2-P_ix_i+Ksum_j< icos(x_i-x_j),$$ if you know a derivative-free method to do that.
  $endgroup$
  – LutzL
  Mar 11 at 12:08
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What's wrong with checking eigenvalues numerically? You can find analytical expression for derivatives using any symbolic engine, plug in equilibrium coordinates and find eigenvalues quite reliably. Another crudest way to check stability is just to pick a random point quite close to the equilibrium and integrate trajectory for a moderate amount of time. If it is a saddle, most likely you won't pick a point from its stable manifold and trajectory abruptly goes somewhere else, not to the equilibrium. P.S. Kuramoto with an inertia? :)
  $endgroup$
  – Evgeny
  Mar 11 at 12:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Will that work? Picking a 'random' point nearby in 2n dimensional space? Wouldn't it by chance be a direction which exhibits 'directional' stability while others are unstable? And yes! It is indeed Kuramoto with inertia :-)
  $endgroup$
  – Fabric
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

Given a dynamical system described by the equations (for i=1,...,N)
$$fracd y_idt = P_i - by_i + K underseti neq jsum_i=1^N sin(x_i-x_j)$$
$$fracd x_idt = y_i$$
Say that I have found a stable point in the $R^2n$ dimensional space $((x_1,y_1),(x_2,y_2),...,(x_n,y_n))$ such that $fracdx_idt = fracdy_idt=0$ for all $i =1,...,N$.

How do I check numerically, if this point is stable or unstable? (Preferably without checking the eigenvalues of the jacobian matrix)

numerical-methods dynamical-systems stability-theory

share|cite|improve this question

asked Mar 11 at 11:38

FabricFabric

478

$endgroup$

share|cite|improve this question

asked Mar 11 at 11:38

FabricFabric

478

share|cite|improve this question

asked Mar 11 at 11:38

FabricFabric

478

asked Mar 11 at 11:38

FabricFabric

478

asked Mar 11 at 11:38

FabricFabric

478