Lyapunov Indirect Method
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I have searched a reference (book/paper) where I could find a theorem related to the Lyapunov Indirect Method for any equilibrium point, but I have not found yet. I only found for the zero equilibrium point. Somebody can help me?
I am searching the following Theorem:
Let us consider the non-linear system $dot{x}=f(x)$ with an arbitrary equilibrium point $bar{x}$. This equilibrium point $bar{x}$ is:
- asymptotically stable if all eigenvalues of $Df(bar{x})$ have negative real part;
- unstable if at least one eigenvalue of $Df(bar{x})$ has positive real part;
where $Df(bar{x})$ is the Jacobin matrix of $f$ applied in the equilibrium point $bar{x}$.
Thank you very much!
Ana
stability-in-odes non-linear-dynamics
asked Jan 23 at 12:23
AnaAna
284
$endgroup$
add a comment |
$begingroup$
I have searched a reference (book/paper) where I could find a theorem related to the Lyapunov Indirect Method for any equilibrium point, but I have not found yet. I only found for the zero equilibrium point. Somebody can help me?
I am searching the following Theorem:
Let us consider the non-linear system $dot{x}=f(x)$ with an arbitrary equilibrium point $bar{x}$. This equilibrium point $bar{x}$ is:
- asymptotically stable if all eigenvalues of $Df(bar{x})$ have negative real part;
- unstable if at least one eigenvalue of $Df(bar{x})$ has positive real part;
where $Df(bar{x})$ is the Jacobin matrix of $f$ applied in the equilibrium point $bar{x}$.
Thank you very much!
Ana
stability-in-odes non-linear-dynamics
asked Jan 23 at 12:23
AnaAna
284
$endgroup$
2
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You can always make a coordinate transformation $y = x - overline{x}$ and get a system with an equilibrium at the origin. This transformation preserves stability properties (as any orientation-preserving diffeomorphism, really), thus any conclusions that you draw for an equilibrium at the origin are valid for an equilibrium point of original system.
$endgroup$
– Evgeny
Jan 23 at 12:30
add a comment |
$begingroup$
I have searched a reference (book/paper) where I could find a theorem related to the Lyapunov Indirect Method for any equilibrium point, but I have not found yet. I only found for the zero equilibrium point. Somebody can help me?
I am searching the following Theorem:
Let us consider the non-linear system $dot{x}=f(x)$ with an arbitrary equilibrium point $bar{x}$. This equilibrium point $bar{x}$ is:
- asymptotically stable if all eigenvalues of $Df(bar{x})$ have negative real part;
- unstable if at least one eigenvalue of $Df(bar{x})$ has positive real part;
where $Df(bar{x})$ is the Jacobin matrix of $f$ applied in the equilibrium point $bar{x}$.
Thank you very much!
Ana
stability-in-odes non-linear-dynamics
asked Jan 23 at 12:23
AnaAna
284
$endgroup$
I have searched a reference (book/paper) where I could find a theorem related to the Lyapunov Indirect Method for any equilibrium point, but I have not found yet. I only found for the zero equilibrium point. Somebody can help me?
I am searching the following Theorem:
Let us consider the non-linear system $dot{x}=f(x)$ with an arbitrary equilibrium point $bar{x}$. This equilibrium point $bar{x}$ is:
- asymptotically stable if all eigenvalues of $Df(bar{x})$ have negative real part;
- unstable if at least one eigenvalue of $Df(bar{x})$ has positive real part;
where $Df(bar{x})$ is the Jacobin matrix of $f$ applied in the equilibrium point $bar{x}$.
Thank you very much!
Ana
stability-in-odes non-linear-dynamics
stability-in-odes non-linear-dynamics
asked Jan 23 at 12:23
AnaAna
284
asked Jan 23 at 12:23
AnaAna
284
edited Jan 23 at 12:29
Ana
asked Jan 23 at 12:23
AnaAna
284
asked Jan 23 at 12:23
AnaAna
284
asked Jan 23 at 12:23
AnaAna
284
284
2
$begingroup$
You can always make a coordinate transformation $y = x - overline{x}$ and get a system with an equilibrium at the origin. This transformation preserves stability properties (as any orientation-preserving diffeomorphism, really), thus any conclusions that you draw for an equilibrium at the origin are valid for an equilibrium point of original system.
$endgroup$
– Evgeny
Jan 23 at 12:30
add a comment |
2
$begingroup$
You can always make a coordinate transformation $y = x - overline{x}$ and get a system with an equilibrium at the origin. This transformation preserves stability properties (as any orientation-preserving diffeomorphism, really), thus any conclusions that you draw for an equilibrium at the origin are valid for an equilibrium point of original system.
$endgroup$
– Evgeny
Jan 23 at 12:30
2
2
$begingroup$
You can always make a coordinate transformation $y = x - overline{x}$ and get a system with an equilibrium at the origin. This transformation preserves stability properties (as any orientation-preserving diffeomorphism, really), thus any conclusions that you draw for an equilibrium at the origin are valid for an equilibrium point of original system.
$endgroup$
– Evgeny
Jan 23 at 12:30
$begingroup$
You can always make a coordinate transformation $y = x - overline{x}$ and get a system with an equilibrium at the origin. This transformation preserves stability properties (as any orientation-preserving diffeomorphism, really), thus any conclusions that you draw for an equilibrium at the origin are valid for an equilibrium point of original system.
$endgroup$
– Evgeny
Jan 23 at 12:30
add a comment |
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$begingroup$
You can always make a coordinate transformation $y = x - overline{x}$ and get a system with an equilibrium at the origin. This transformation preserves stability properties (as any orientation-preserving diffeomorphism, really), thus any conclusions that you draw for an equilibrium at the origin are valid for an equilibrium point of original system.
$endgroup$
– Evgeny
Jan 23 at 12:30