What defines a chaotic system?
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Suppose I have an equation of motion of some sort. All I know about chaotic systems is that small changes in initial conditions will increase the outcome at $t$ by exceedingly large amounts. My question is, how do we actually prove this? Also, is it necessary for a chaotic system to be chaotic in any coordinate system, or do we have to prove it for every coordinate system?
I'd guess that if a system is chaotic for one set of coordinates, it'll be chaotic for almost every other coordinate system - except for when you specifically choose to fix the equation.
mathematical-physics classical-mechanics chaos-theory
asked Sep 1 '17 at 14:52
Dis-integratingDis-integrating
1,036426
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add a comment |
$begingroup$
Suppose I have an equation of motion of some sort. All I know about chaotic systems is that small changes in initial conditions will increase the outcome at $t$ by exceedingly large amounts. My question is, how do we actually prove this? Also, is it necessary for a chaotic system to be chaotic in any coordinate system, or do we have to prove it for every coordinate system?
I'd guess that if a system is chaotic for one set of coordinates, it'll be chaotic for almost every other coordinate system - except for when you specifically choose to fix the equation.
mathematical-physics classical-mechanics chaos-theory
asked Sep 1 '17 at 14:52
Dis-integratingDis-integrating
1,036426
$endgroup$
$begingroup$
You can compute the Lyapunov exponent of your system to see if it is chaotic.
$endgroup$
– citronrose
Sep 1 '17 at 15:02
2
$begingroup$
@citronrose By computing the Lyapunov exponents, you can prove that a dynamical system is not chaotic, or that it may have a chance to be chaotic. There are no general tools to prove that a system is chaotic (see e.g. this post). Note: one consequence of the Poincaré-Bendixson theorem is that chaotic behavior can only arise in continuous dynamical systems whose phase space has 3 or more dimensions.
$endgroup$
– Harry49
Sep 1 '17 at 15:31
add a comment |
$begingroup$
Suppose I have an equation of motion of some sort. All I know about chaotic systems is that small changes in initial conditions will increase the outcome at $t$ by exceedingly large amounts. My question is, how do we actually prove this? Also, is it necessary for a chaotic system to be chaotic in any coordinate system, or do we have to prove it for every coordinate system?
I'd guess that if a system is chaotic for one set of coordinates, it'll be chaotic for almost every other coordinate system - except for when you specifically choose to fix the equation.
mathematical-physics classical-mechanics chaos-theory
asked Sep 1 '17 at 14:52
Dis-integratingDis-integrating
1,036426
$endgroup$
Suppose I have an equation of motion of some sort. All I know about chaotic systems is that small changes in initial conditions will increase the outcome at $t$ by exceedingly large amounts. My question is, how do we actually prove this? Also, is it necessary for a chaotic system to be chaotic in any coordinate system, or do we have to prove it for every coordinate system?
I'd guess that if a system is chaotic for one set of coordinates, it'll be chaotic for almost every other coordinate system - except for when you specifically choose to fix the equation.
mathematical-physics classical-mechanics chaos-theory
mathematical-physics classical-mechanics chaos-theory
asked Sep 1 '17 at 14:52
Dis-integratingDis-integrating
1,036426
asked Sep 1 '17 at 14:52
Dis-integratingDis-integrating
1,036426
asked Sep 1 '17 at 14:52
Dis-integratingDis-integrating
1,036426
asked Sep 1 '17 at 14:52
Dis-integratingDis-integrating
1,036426
asked Sep 1 '17 at 14:52
Dis-integratingDis-integrating
1,036426
1,036426
$begingroup$
You can compute the Lyapunov exponent of your system to see if it is chaotic.
$endgroup$
– citronrose
Sep 1 '17 at 15:02
2
$begingroup$
@citronrose By computing the Lyapunov exponents, you can prove that a dynamical system is not chaotic, or that it may have a chance to be chaotic. There are no general tools to prove that a system is chaotic (see e.g. this post). Note: one consequence of the Poincaré-Bendixson theorem is that chaotic behavior can only arise in continuous dynamical systems whose phase space has 3 or more dimensions.
$endgroup$
– Harry49
Sep 1 '17 at 15:31
add a comment |
$begingroup$
You can compute the Lyapunov exponent of your system to see if it is chaotic.
$endgroup$
– citronrose
Sep 1 '17 at 15:02
2
$begingroup$
@citronrose By computing the Lyapunov exponents, you can prove that a dynamical system is not chaotic, or that it may have a chance to be chaotic. There are no general tools to prove that a system is chaotic (see e.g. this post). Note: one consequence of the Poincaré-Bendixson theorem is that chaotic behavior can only arise in continuous dynamical systems whose phase space has 3 or more dimensions.
$endgroup$
– Harry49
Sep 1 '17 at 15:31
$begingroup$
You can compute the Lyapunov exponent of your system to see if it is chaotic.
$endgroup$
– citronrose
Sep 1 '17 at 15:02
$begingroup$
You can compute the Lyapunov exponent of your system to see if it is chaotic.
$endgroup$
– citronrose
Sep 1 '17 at 15:02
2
2
$begingroup$
@citronrose By computing the Lyapunov exponents, you can prove that a dynamical system is not chaotic, or that it may have a chance to be chaotic. There are no general tools to prove that a system is chaotic (see e.g. this post). Note: one consequence of the Poincaré-Bendixson theorem is that chaotic behavior can only arise in continuous dynamical systems whose phase space has 3 or more dimensions.
$endgroup$
– Harry49
Sep 1 '17 at 15:31
$begingroup$
@citronrose By computing the Lyapunov exponents, you can prove that a dynamical system is not chaotic, or that it may have a chance to be chaotic. There are no general tools to prove that a system is chaotic (see e.g. this post). Note: one consequence of the Poincaré-Bendixson theorem is that chaotic behavior can only arise in continuous dynamical systems whose phase space has 3 or more dimensions.
$endgroup$
– Harry49
Sep 1 '17 at 15:31
add a comment |
1 Answer
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There is no real agreement on the definition of chaotic continuous dynamical systems (see e.g. [1]), other that they can exhibit a chaotic behavior. In other words, chaos is a consequence that is observed for some nonlinear first-order ODE systems, and some sets of initial conditions. A simple and widely used definition of chaotic systems by R.L. Devaney is that a chaotic system
- has sensitive dependence on initial conditions
- is topologically transitive (for any two open sets, some points from one set will eventually hit the other set)
- its periodic orbits form a dense set
It was found later that 1. is a consequence of 2. and 3.
Of course, this type of dynamics should not be affected by any ($C^1$-diffeomorphic) change of coordinates, or I would be happy if somebody can provide a counter-example.
answered Sep 1 '17 at 15:59
Harry49Harry49
7,26831240
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add a comment |
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$begingroup$
There is no real agreement on the definition of chaotic continuous dynamical systems (see e.g. [1]), other that they can exhibit a chaotic behavior. In other words, chaos is a consequence that is observed for some nonlinear first-order ODE systems, and some sets of initial conditions. A simple and widely used definition of chaotic systems by R.L. Devaney is that a chaotic system
- has sensitive dependence on initial conditions
- is topologically transitive (for any two open sets, some points from one set will eventually hit the other set)
- its periodic orbits form a dense set
It was found later that 1. is a consequence of 2. and 3.
Of course, this type of dynamics should not be affected by any ($C^1$-diffeomorphic) change of coordinates, or I would be happy if somebody can provide a counter-example.
answered Sep 1 '17 at 15:59
Harry49Harry49
7,26831240
$endgroup$
add a comment |
$begingroup$
There is no real agreement on the definition of chaotic continuous dynamical systems (see e.g. [1]), other that they can exhibit a chaotic behavior. In other words, chaos is a consequence that is observed for some nonlinear first-order ODE systems, and some sets of initial conditions. A simple and widely used definition of chaotic systems by R.L. Devaney is that a chaotic system
- has sensitive dependence on initial conditions
- is topologically transitive (for any two open sets, some points from one set will eventually hit the other set)
- its periodic orbits form a dense set
It was found later that 1. is a consequence of 2. and 3.
Of course, this type of dynamics should not be affected by any ($C^1$-diffeomorphic) change of coordinates, or I would be happy if somebody can provide a counter-example.
answered Sep 1 '17 at 15:59
Harry49Harry49
7,26831240
$endgroup$
add a comment |
$begingroup$
There is no real agreement on the definition of chaotic continuous dynamical systems (see e.g. [1]), other that they can exhibit a chaotic behavior. In other words, chaos is a consequence that is observed for some nonlinear first-order ODE systems, and some sets of initial conditions. A simple and widely used definition of chaotic systems by R.L. Devaney is that a chaotic system
- has sensitive dependence on initial conditions
- is topologically transitive (for any two open sets, some points from one set will eventually hit the other set)
- its periodic orbits form a dense set
It was found later that 1. is a consequence of 2. and 3.
Of course, this type of dynamics should not be affected by any ($C^1$-diffeomorphic) change of coordinates, or I would be happy if somebody can provide a counter-example.
answered Sep 1 '17 at 15:59
Harry49Harry49
7,26831240
$endgroup$
There is no real agreement on the definition of chaotic continuous dynamical systems (see e.g. [1]), other that they can exhibit a chaotic behavior. In other words, chaos is a consequence that is observed for some nonlinear first-order ODE systems, and some sets of initial conditions. A simple and widely used definition of chaotic systems by R.L. Devaney is that a chaotic system
- has sensitive dependence on initial conditions
- is topologically transitive (for any two open sets, some points from one set will eventually hit the other set)
- its periodic orbits form a dense set
It was found later that 1. is a consequence of 2. and 3.
Of course, this type of dynamics should not be affected by any ($C^1$-diffeomorphic) change of coordinates, or I would be happy if somebody can provide a counter-example.
answered Sep 1 '17 at 15:59
Harry49Harry49
7,26831240
edited Jan 8 at 12:53
answered Sep 1 '17 at 15:59
Harry49Harry49
7,26831240
answered Sep 1 '17 at 15:59
Harry49Harry49
7,26831240
answered Sep 1 '17 at 15:59
Harry49Harry49
7,26831240
7,26831240
add a comment |
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$begingroup$
You can compute the Lyapunov exponent of your system to see if it is chaotic.
$endgroup$
– citronrose
Sep 1 '17 at 15:02
2
$begingroup$
@citronrose By computing the Lyapunov exponents, you can prove that a dynamical system is not chaotic, or that it may have a chance to be chaotic. There are no general tools to prove that a system is chaotic (see e.g. this post). Note: one consequence of the Poincaré-Bendixson theorem is that chaotic behavior can only arise in continuous dynamical systems whose phase space has 3 or more dimensions.
$endgroup$
– Harry49
Sep 1 '17 at 15:31