Complex Equilibrium Points in Dynamical Systems?
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Why do we not study the behavior of a system of ODEs around complex equilibrium points? How does their existence influence the flow?
I have studied the stability and bifurcation analysis for small systems. Perhaps the "Ghosts" in bifurcation diagrams from Strogatz's book may give some context.
dynamical-systems systems-of-equations stability-in-odes
asked Sep 18 '15 at 16:02
squeak
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Why do we not study the behavior of a system of ODEs around complex equilibrium points? How does their existence influence the flow?
I have studied the stability and bifurcation analysis for small systems. Perhaps the "Ghosts" in bifurcation diagrams from Strogatz's book may give some context.
dynamical-systems systems-of-equations stability-in-odes
asked Sep 18 '15 at 16:02
squeak
196
1
You've said that you've studied bifurcation analysis. But bifurcation theory is the theory that answers what happens first at bifurcation point and how you can transform this picture if you perturb the flow. Maybe I've misunderstood your question?
– Evgeny
Sep 18 '15 at 16:55
If you define your dynamical systems to live in real space (say $R^n$), you cannot talk about complex equilibrium points. On the other hand, in the domain of complex dynamical systems, we do study complex equilibrium points.
– nonlinearism
Sep 18 '15 at 22:50
@nonlinearism No-no-no, I think he meant equilibria that can split or disappear after perturbation, i.e. ones with null eigenvalues. I still can't remember the right term for them.
– Evgeny
Sep 19 '15 at 7:01
@nonlinearism I wanted to know why we can't talk about complex equilibria when things like "Ghosts" exist. From what little I understand it seems like complex equilibrium affect the flow for when the equilibrium becomes complex, it vanishes from the picture and yet the phase diagram doesn't change as if it had some latent effect. In brief, if we do talk of complex equilibria for complex valued dynamical systems what information do we miss out when we don't consider them?
– squeak
Sep 19 '15 at 18:39
It's not clear what exactly are you asking. What do you mean by "we can't talk about complex equilibria when things like 'Ghosts' exist" ? And phase diagram of course changes. Bottleneck has different effect than equilibrium.
– Evgeny
Sep 19 '15 at 19:15
|
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Why do we not study the behavior of a system of ODEs around complex equilibrium points? How does their existence influence the flow?
I have studied the stability and bifurcation analysis for small systems. Perhaps the "Ghosts" in bifurcation diagrams from Strogatz's book may give some context.
dynamical-systems systems-of-equations stability-in-odes
asked Sep 18 '15 at 16:02
squeak
196
Why do we not study the behavior of a system of ODEs around complex equilibrium points? How does their existence influence the flow?
I have studied the stability and bifurcation analysis for small systems. Perhaps the "Ghosts" in bifurcation diagrams from Strogatz's book may give some context.
dynamical-systems systems-of-equations stability-in-odes
dynamical-systems systems-of-equations stability-in-odes
asked Sep 18 '15 at 16:02
squeak
196
asked Sep 18 '15 at 16:02
squeak
196
asked Sep 18 '15 at 16:02
squeak
196
asked Sep 18 '15 at 16:02
squeak
196
asked Sep 18 '15 at 16:02
squeak
196
196
1
You've said that you've studied bifurcation analysis. But bifurcation theory is the theory that answers what happens first at bifurcation point and how you can transform this picture if you perturb the flow. Maybe I've misunderstood your question?
– Evgeny
Sep 18 '15 at 16:55
If you define your dynamical systems to live in real space (say $R^n$), you cannot talk about complex equilibrium points. On the other hand, in the domain of complex dynamical systems, we do study complex equilibrium points.
– nonlinearism
Sep 18 '15 at 22:50
@nonlinearism No-no-no, I think he meant equilibria that can split or disappear after perturbation, i.e. ones with null eigenvalues. I still can't remember the right term for them.
– Evgeny
Sep 19 '15 at 7:01
@nonlinearism I wanted to know why we can't talk about complex equilibria when things like "Ghosts" exist. From what little I understand it seems like complex equilibrium affect the flow for when the equilibrium becomes complex, it vanishes from the picture and yet the phase diagram doesn't change as if it had some latent effect. In brief, if we do talk of complex equilibria for complex valued dynamical systems what information do we miss out when we don't consider them?
– squeak
Sep 19 '15 at 18:39
It's not clear what exactly are you asking. What do you mean by "we can't talk about complex equilibria when things like 'Ghosts' exist" ? And phase diagram of course changes. Bottleneck has different effect than equilibrium.
– Evgeny
Sep 19 '15 at 19:15
|
show 2 more comments
1
You've said that you've studied bifurcation analysis. But bifurcation theory is the theory that answers what happens first at bifurcation point and how you can transform this picture if you perturb the flow. Maybe I've misunderstood your question?
– Evgeny
Sep 18 '15 at 16:55
If you define your dynamical systems to live in real space (say $R^n$), you cannot talk about complex equilibrium points. On the other hand, in the domain of complex dynamical systems, we do study complex equilibrium points.
– nonlinearism
Sep 18 '15 at 22:50
@nonlinearism No-no-no, I think he meant equilibria that can split or disappear after perturbation, i.e. ones with null eigenvalues. I still can't remember the right term for them.
– Evgeny
Sep 19 '15 at 7:01
@nonlinearism I wanted to know why we can't talk about complex equilibria when things like "Ghosts" exist. From what little I understand it seems like complex equilibrium affect the flow for when the equilibrium becomes complex, it vanishes from the picture and yet the phase diagram doesn't change as if it had some latent effect. In brief, if we do talk of complex equilibria for complex valued dynamical systems what information do we miss out when we don't consider them?
– squeak
Sep 19 '15 at 18:39
It's not clear what exactly are you asking. What do you mean by "we can't talk about complex equilibria when things like 'Ghosts' exist" ? And phase diagram of course changes. Bottleneck has different effect than equilibrium.
– Evgeny
Sep 19 '15 at 19:15
1
1
You've said that you've studied bifurcation analysis. But bifurcation theory is the theory that answers what happens first at bifurcation point and how you can transform this picture if you perturb the flow. Maybe I've misunderstood your question?
– Evgeny
Sep 18 '15 at 16:55
You've said that you've studied bifurcation analysis. But bifurcation theory is the theory that answers what happens first at bifurcation point and how you can transform this picture if you perturb the flow. Maybe I've misunderstood your question?
– Evgeny
Sep 18 '15 at 16:55
If you define your dynamical systems to live in real space (say $R^n$), you cannot talk about complex equilibrium points. On the other hand, in the domain of complex dynamical systems, we do study complex equilibrium points.
– nonlinearism
Sep 18 '15 at 22:50
If you define your dynamical systems to live in real space (say $R^n$), you cannot talk about complex equilibrium points. On the other hand, in the domain of complex dynamical systems, we do study complex equilibrium points.
– nonlinearism
Sep 18 '15 at 22:50
@nonlinearism No-no-no, I think he meant equilibria that can split or disappear after perturbation, i.e. ones with null eigenvalues. I still can't remember the right term for them.
– Evgeny
Sep 19 '15 at 7:01
@nonlinearism No-no-no, I think he meant equilibria that can split or disappear after perturbation, i.e. ones with null eigenvalues. I still can't remember the right term for them.
– Evgeny
Sep 19 '15 at 7:01
@nonlinearism I wanted to know why we can't talk about complex equilibria when things like "Ghosts" exist. From what little I understand it seems like complex equilibrium affect the flow for when the equilibrium becomes complex, it vanishes from the picture and yet the phase diagram doesn't change as if it had some latent effect. In brief, if we do talk of complex equilibria for complex valued dynamical systems what information do we miss out when we don't consider them?
– squeak
Sep 19 '15 at 18:39
@nonlinearism I wanted to know why we can't talk about complex equilibria when things like "Ghosts" exist. From what little I understand it seems like complex equilibrium affect the flow for when the equilibrium becomes complex, it vanishes from the picture and yet the phase diagram doesn't change as if it had some latent effect. In brief, if we do talk of complex equilibria for complex valued dynamical systems what information do we miss out when we don't consider them?
– squeak
Sep 19 '15 at 18:39
It's not clear what exactly are you asking. What do you mean by "we can't talk about complex equilibria when things like 'Ghosts' exist" ? And phase diagram of course changes. Bottleneck has different effect than equilibrium.
– Evgeny
Sep 19 '15 at 19:15
It's not clear what exactly are you asking. What do you mean by "we can't talk about complex equilibria when things like 'Ghosts' exist" ? And phase diagram of course changes. Bottleneck has different effect than equilibrium.
– Evgeny
Sep 19 '15 at 19:15
|
show 2 more comments
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Practically speaking, we don't care about the existence of complex equilibrium points because all the realizable models have real signals or states which means that the states can not reach these equilibrium points if they are not real. There are no any kind of complex signals despite of the fact that we represent some concepts in electrical analysis using imaginary number as the reactive power but this still be a representation of a concept that is not a signal or a state of the model.
Also there is a big difference between the existence of the complex eigenvalues of the state matrix and the equilibrium because the eigenvalues are used to analyse the behavior of the dynamical system and they are not values of the states in the system. So, the usage of complex analysis is just to analyze systems.
On the other hand; from the definition of the equilibrium point: it is the point that if the states at that point stay forever, so the states must be complex and this is impossible in practice.
answered Feb 9 '16 at 22:34
Bilal Jafar Karaki
253112
add a comment |
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Practically speaking, we don't care about the existence of complex equilibrium points because all the realizable models have real signals or states which means that the states can not reach these equilibrium points if they are not real. There are no any kind of complex signals despite of the fact that we represent some concepts in electrical analysis using imaginary number as the reactive power but this still be a representation of a concept that is not a signal or a state of the model.
Also there is a big difference between the existence of the complex eigenvalues of the state matrix and the equilibrium because the eigenvalues are used to analyse the behavior of the dynamical system and they are not values of the states in the system. So, the usage of complex analysis is just to analyze systems.
On the other hand; from the definition of the equilibrium point: it is the point that if the states at that point stay forever, so the states must be complex and this is impossible in practice.
answered Feb 9 '16 at 22:34
Bilal Jafar Karaki
253112
add a comment |
up vote
0
down vote
Practically speaking, we don't care about the existence of complex equilibrium points because all the realizable models have real signals or states which means that the states can not reach these equilibrium points if they are not real. There are no any kind of complex signals despite of the fact that we represent some concepts in electrical analysis using imaginary number as the reactive power but this still be a representation of a concept that is not a signal or a state of the model.
Also there is a big difference between the existence of the complex eigenvalues of the state matrix and the equilibrium because the eigenvalues are used to analyse the behavior of the dynamical system and they are not values of the states in the system. So, the usage of complex analysis is just to analyze systems.
On the other hand; from the definition of the equilibrium point: it is the point that if the states at that point stay forever, so the states must be complex and this is impossible in practice.
answered Feb 9 '16 at 22:34
Bilal Jafar Karaki
253112
add a comment |
up vote
0
down vote
up vote
0
down vote
Practically speaking, we don't care about the existence of complex equilibrium points because all the realizable models have real signals or states which means that the states can not reach these equilibrium points if they are not real. There are no any kind of complex signals despite of the fact that we represent some concepts in electrical analysis using imaginary number as the reactive power but this still be a representation of a concept that is not a signal or a state of the model.
Also there is a big difference between the existence of the complex eigenvalues of the state matrix and the equilibrium because the eigenvalues are used to analyse the behavior of the dynamical system and they are not values of the states in the system. So, the usage of complex analysis is just to analyze systems.
On the other hand; from the definition of the equilibrium point: it is the point that if the states at that point stay forever, so the states must be complex and this is impossible in practice.
answered Feb 9 '16 at 22:34
Bilal Jafar Karaki
253112
Practically speaking, we don't care about the existence of complex equilibrium points because all the realizable models have real signals or states which means that the states can not reach these equilibrium points if they are not real. There are no any kind of complex signals despite of the fact that we represent some concepts in electrical analysis using imaginary number as the reactive power but this still be a representation of a concept that is not a signal or a state of the model.
Also there is a big difference between the existence of the complex eigenvalues of the state matrix and the equilibrium because the eigenvalues are used to analyse the behavior of the dynamical system and they are not values of the states in the system. So, the usage of complex analysis is just to analyze systems.
On the other hand; from the definition of the equilibrium point: it is the point that if the states at that point stay forever, so the states must be complex and this is impossible in practice.
answered Feb 9 '16 at 22:34
Bilal Jafar Karaki
253112
edited Aug 4 '16 at 22:03
answered Feb 9 '16 at 22:34
Bilal Jafar Karaki
253112
answered Feb 9 '16 at 22:34
Bilal Jafar Karaki
253112
answered Feb 9 '16 at 22:34
Bilal Jafar Karaki
253112
253112
add a comment |
add a comment |
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You've said that you've studied bifurcation analysis. But bifurcation theory is the theory that answers what happens first at bifurcation point and how you can transform this picture if you perturb the flow. Maybe I've misunderstood your question?
– Evgeny
Sep 18 '15 at 16:55
If you define your dynamical systems to live in real space (say $R^n$), you cannot talk about complex equilibrium points. On the other hand, in the domain of complex dynamical systems, we do study complex equilibrium points.
– nonlinearism
Sep 18 '15 at 22:50
@nonlinearism No-no-no, I think he meant equilibria that can split or disappear after perturbation, i.e. ones with null eigenvalues. I still can't remember the right term for them.
– Evgeny
Sep 19 '15 at 7:01
@nonlinearism I wanted to know why we can't talk about complex equilibria when things like "Ghosts" exist. From what little I understand it seems like complex equilibrium affect the flow for when the equilibrium becomes complex, it vanishes from the picture and yet the phase diagram doesn't change as if it had some latent effect. In brief, if we do talk of complex equilibria for complex valued dynamical systems what information do we miss out when we don't consider them?
– squeak
Sep 19 '15 at 18:39
It's not clear what exactly are you asking. What do you mean by "we can't talk about complex equilibria when things like 'Ghosts' exist" ? And phase diagram of course changes. Bottleneck has different effect than equilibrium.
– Evgeny
Sep 19 '15 at 19:15