Instability of a system subject to periodic perturbation

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Consider the following 2-dimensional system
$$
dotx(t) = A(t)x(t) quad x(0)inmathbbR^2,
$$
where $A(t)$ is a 2-dimensional time-varying matrix. Suppose that the origin of the above system is an unstable equilibrium.

Now consider the following "perturbed" system
$$
dotz(t) = (A(t)+Delta(t))z(t) quad z(0)inmathbbR^2,
$$
where $Delta(t)$ is a $2times 2$ matrix whose entries are zero-mean periodic functions of $t$.

My question: Is the origin of the "perturbed" system unstable for every choice of $Delta(t)$ as above?

My feeling is that the answer is no, but I couldn't find so far an explicit counterexample.

analysis dynamical-systems stability-theory

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edited Jul 26 at 23:25

asked Jul 26 at 21:25

Ludwig

737613

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up vote
2
down vote

favorite

2

Consider the following 2-dimensional system
$$
dotx(t) = A(t)x(t) quad x(0)inmathbbR^2,
$$
where $A(t)$ is a 2-dimensional time-varying matrix. Suppose that the origin of the above system is an unstable equilibrium.

Now consider the following "perturbed" system
$$
dotz(t) = (A(t)+Delta(t))z(t) quad z(0)inmathbbR^2,
$$
where $Delta(t)$ is a $2times 2$ matrix whose entries are zero-mean periodic functions of $t$.

My question: Is the origin of the "perturbed" system unstable for every choice of $Delta(t)$ as above?

My feeling is that the answer is no, but I couldn't find so far an explicit counterexample.

analysis dynamical-systems stability-theory

share|cite|improve this question

edited Jul 26 at 23:25

asked Jul 26 at 21:25

Ludwig

737613

add a comment | 

up vote
2
down vote

favorite

2

up vote
2
down vote

favorite

2

2

Consider the following 2-dimensional system
$$
dotx(t) = A(t)x(t) quad x(0)inmathbbR^2,
$$
where $A(t)$ is a 2-dimensional time-varying matrix. Suppose that the origin of the above system is an unstable equilibrium.

Now consider the following "perturbed" system
$$
dotz(t) = (A(t)+Delta(t))z(t) quad z(0)inmathbbR^2,
$$
where $Delta(t)$ is a $2times 2$ matrix whose entries are zero-mean periodic functions of $t$.

My question: Is the origin of the "perturbed" system unstable for every choice of $Delta(t)$ as above?

My feeling is that the answer is no, but I couldn't find so far an explicit counterexample.

analysis dynamical-systems stability-theory

share|cite|improve this question

edited Jul 26 at 23:25

asked Jul 26 at 21:25

Ludwig

737613

Consider the following 2-dimensional system
$$
dotx(t) = A(t)x(t) quad x(0)inmathbbR^2,
$$
where $A(t)$ is a 2-dimensional time-varying matrix. Suppose that the origin of the above system is an unstable equilibrium.

Now consider the following "perturbed" system
$$
dotz(t) = (A(t)+Delta(t))z(t) quad z(0)inmathbbR^2,
$$
where $Delta(t)$ is a $2times 2$ matrix whose entries are zero-mean periodic functions of $t$.

My question: Is the origin of the "perturbed" system unstable for every choice of $Delta(t)$ as above?

My feeling is that the answer is no, but I couldn't find so far an explicit counterexample.

analysis dynamical-systems stability-theory

share|cite|improve this question

edited Jul 26 at 23:25

asked Jul 26 at 21:25

Ludwig

737613

share|cite|improve this question

share|cite|improve this question

edited Jul 26 at 23:25

edited Jul 26 at 23:25

edited Jul 26 at 23:25

asked Jul 26 at 21:25

Ludwig

737613

asked Jul 26 at 21:25

Ludwig

737613

asked Jul 26 at 21:25

Ludwig

737613

737613

add a comment | 

add a comment | 

1 Answer
1

active

oldest

votes

up vote
2
down vote



accepted

Markus and Yamabe showed that the origin in the system
$$
dotx(t) = beginpmatrix -1 + frac32 cos^2(t) & 1 - frac32 sin(t) cos(t) \ -1 - frac32 sin(t) cos(t) & -1 + frac32 sin^2(t) endpmatrix x(t)
$$
is unstable (see, e.g., p. 121 in Hale Ordinary Differential Equations). If we take
$$
A(t) = beginpmatrix -1 + frac32 cos^2(t) & 1 - frac32 sin(t) cos(t) \ -1 - frac32 sin(t) cos(t) & -1 + frac32 sin^2(t) endpmatrix, quad Delta(t) = beginpmatrix - frac34 cos(2t) & frac34 sin(2t) \ frac34 sin(2t) & frac34 cos(2t) endpmatrix
$$
($Delta(t)$ has zero-mean periodic entries) then the autonomous system
$$
dotz(t) = beginpmatrix -frac34 & 1 \ -1 & -frac34 endpmatrix z(t),
$$
having eigenvalues with negative real parts, is asymptottically stable.

share|cite|improve this answer

answered Jul 27 at 11:04

user539887

1,4461313

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks! Perhaps do you have also in mind a counterexample in which $Delta(t) = f(t)C$ with $f(t)$ a periodic zero-mean scalar function and $C$ a constant $2times 2$ matrix?
  – Ludwig
  Jul 27 at 14:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  No, I don't have at the moment. It think that such problems are a staple in Control Theory, but, alas, I'm no expert.
  – user539887
  Jul 28 at 7:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Okay, thanks anyway. I think I’ll post a follow-up question concerning instability in this particular case, which looks interesting to me.
  – Ludwig
  Jul 28 at 16:20
  

  
  
  
  

add a comment | 

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
2
down vote



accepted

Markus and Yamabe showed that the origin in the system
$$
dotx(t) = beginpmatrix -1 + frac32 cos^2(t) & 1 - frac32 sin(t) cos(t) \ -1 - frac32 sin(t) cos(t) & -1 + frac32 sin^2(t) endpmatrix x(t)
$$
is unstable (see, e.g., p. 121 in Hale Ordinary Differential Equations). If we take
$$
A(t) = beginpmatrix -1 + frac32 cos^2(t) & 1 - frac32 sin(t) cos(t) \ -1 - frac32 sin(t) cos(t) & -1 + frac32 sin^2(t) endpmatrix, quad Delta(t) = beginpmatrix - frac34 cos(2t) & frac34 sin(2t) \ frac34 sin(2t) & frac34 cos(2t) endpmatrix
$$
($Delta(t)$ has zero-mean periodic entries) then the autonomous system
$$
dotz(t) = beginpmatrix -frac34 & 1 \ -1 & -frac34 endpmatrix z(t),
$$
having eigenvalues with negative real parts, is asymptottically stable.

share|cite|improve this answer

answered Jul 27 at 11:04

user539887

1,4461313

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks! Perhaps do you have also in mind a counterexample in which $Delta(t) = f(t)C$ with $f(t)$ a periodic zero-mean scalar function and $C$ a constant $2times 2$ matrix?
  – Ludwig
  Jul 27 at 14:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  No, I don't have at the moment. It think that such problems are a staple in Control Theory, but, alas, I'm no expert.
  – user539887
  Jul 28 at 7:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Okay, thanks anyway. I think I’ll post a follow-up question concerning instability in this particular case, which looks interesting to me.
  – Ludwig
  Jul 28 at 16:20
  

  
  
  
  

add a comment | 

up vote
2
down vote



accepted

Markus and Yamabe showed that the origin in the system
$$
dotx(t) = beginpmatrix -1 + frac32 cos^2(t) & 1 - frac32 sin(t) cos(t) \ -1 - frac32 sin(t) cos(t) & -1 + frac32 sin^2(t) endpmatrix x(t)
$$
is unstable (see, e.g., p. 121 in Hale Ordinary Differential Equations). If we take
$$
A(t) = beginpmatrix -1 + frac32 cos^2(t) & 1 - frac32 sin(t) cos(t) \ -1 - frac32 sin(t) cos(t) & -1 + frac32 sin^2(t) endpmatrix, quad Delta(t) = beginpmatrix - frac34 cos(2t) & frac34 sin(2t) \ frac34 sin(2t) & frac34 cos(2t) endpmatrix
$$
($Delta(t)$ has zero-mean periodic entries) then the autonomous system
$$
dotz(t) = beginpmatrix -frac34 & 1 \ -1 & -frac34 endpmatrix z(t),
$$
having eigenvalues with negative real parts, is asymptottically stable.

share|cite|improve this answer

answered Jul 27 at 11:04

user539887

1,4461313

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks! Perhaps do you have also in mind a counterexample in which $Delta(t) = f(t)C$ with $f(t)$ a periodic zero-mean scalar function and $C$ a constant $2times 2$ matrix?
  – Ludwig
  Jul 27 at 14:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  No, I don't have at the moment. It think that such problems are a staple in Control Theory, but, alas, I'm no expert.
  – user539887
  Jul 28 at 7:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Okay, thanks anyway. I think I’ll post a follow-up question concerning instability in this particular case, which looks interesting to me.
  – Ludwig
  Jul 28 at 16:20
  

  
  
  
  

add a comment | 

up vote
2
down vote



accepted

up vote
2
down vote



accepted

Markus and Yamabe showed that the origin in the system
$$
dotx(t) = beginpmatrix -1 + frac32 cos^2(t) & 1 - frac32 sin(t) cos(t) \ -1 - frac32 sin(t) cos(t) & -1 + frac32 sin^2(t) endpmatrix x(t)
$$
is unstable (see, e.g., p. 121 in Hale Ordinary Differential Equations). If we take
$$
A(t) = beginpmatrix -1 + frac32 cos^2(t) & 1 - frac32 sin(t) cos(t) \ -1 - frac32 sin(t) cos(t) & -1 + frac32 sin^2(t) endpmatrix, quad Delta(t) = beginpmatrix - frac34 cos(2t) & frac34 sin(2t) \ frac34 sin(2t) & frac34 cos(2t) endpmatrix
$$
($Delta(t)$ has zero-mean periodic entries) then the autonomous system
$$
dotz(t) = beginpmatrix -frac34 & 1 \ -1 & -frac34 endpmatrix z(t),
$$
having eigenvalues with negative real parts, is asymptottically stable.

share|cite|improve this answer

answered Jul 27 at 11:04

user539887

1,4461313

Markus and Yamabe showed that the origin in the system
$$
dotx(t) = beginpmatrix -1 + frac32 cos^2(t) & 1 - frac32 sin(t) cos(t) \ -1 - frac32 sin(t) cos(t) & -1 + frac32 sin^2(t) endpmatrix x(t)
$$
is unstable (see, e.g., p. 121 in Hale Ordinary Differential Equations). If we take
$$
A(t) = beginpmatrix -1 + frac32 cos^2(t) & 1 - frac32 sin(t) cos(t) \ -1 - frac32 sin(t) cos(t) & -1 + frac32 sin^2(t) endpmatrix, quad Delta(t) = beginpmatrix - frac34 cos(2t) & frac34 sin(2t) \ frac34 sin(2t) & frac34 cos(2t) endpmatrix
$$
($Delta(t)$ has zero-mean periodic entries) then the autonomous system
$$
dotz(t) = beginpmatrix -frac34 & 1 \ -1 & -frac34 endpmatrix z(t),
$$
having eigenvalues with negative real parts, is asymptottically stable.

share|cite|improve this answer

answered Jul 27 at 11:04

user539887

1,4461313

share|cite|improve this answer

share|cite|improve this answer

answered Jul 27 at 11:04

user539887

1,4461313

answered Jul 27 at 11:04

user539887

1,4461313

answered Jul 27 at 11:04

user539887

1,4461313

1,4461313

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks! Perhaps do you have also in mind a counterexample in which $Delta(t) = f(t)C$ with $f(t)$ a periodic zero-mean scalar function and $C$ a constant $2times 2$ matrix?
  – Ludwig
  Jul 27 at 14:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  No, I don't have at the moment. It think that such problems are a staple in Control Theory, but, alas, I'm no expert.
  – user539887
  Jul 28 at 7:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Okay, thanks anyway. I think I’ll post a follow-up question concerning instability in this particular case, which looks interesting to me.
  – Ludwig
  Jul 28 at 16:20
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks! Perhaps do you have also in mind a counterexample in which $Delta(t) = f(t)C$ with $f(t)$ a periodic zero-mean scalar function and $C$ a constant $2times 2$ matrix?
  – Ludwig
  Jul 27 at 14:56
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  No, I don't have at the moment. It think that such problems are a staple in Control Theory, but, alas, I'm no expert.
  – user539887
  Jul 28 at 7:19
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Okay, thanks anyway. I think I’ll post a follow-up question concerning instability in this particular case, which looks interesting to me.
  – Ludwig
  Jul 28 at 16:20
  

  
  
  
  

Thanks! Perhaps do you have also in mind a counterexample in which $Delta(t) = f(t)C$ with $f(t)$ a periodic zero-mean scalar function and $C$ a constant $2times 2$ matrix?
– Ludwig
Jul 27 at 14:56

Thanks! Perhaps do you have also in mind a counterexample in which $Delta(t) = f(t)C$ with $f(t)$ a periodic zero-mean scalar function and $C$ a constant $2times 2$ matrix?
– Ludwig
Jul 27 at 14:56

No, I don't have at the moment. It think that such problems are a staple in Control Theory, but, alas, I'm no expert.
– user539887
Jul 28 at 7:19

No, I don't have at the moment. It think that such problems are a staple in Control Theory, but, alas, I'm no expert.
– user539887
Jul 28 at 7:19

Okay, thanks anyway. I think I’ll post a follow-up question concerning instability in this particular case, which looks interesting to me.
– Ludwig
Jul 28 at 16:20

Okay, thanks anyway. I think I’ll post a follow-up question concerning instability in this particular case, which looks interesting to me.
– Ludwig
Jul 28 at 16:20

add a comment | 

 

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