What is a saddle periodic orbit?
$begingroup$
I have seen saddle fixed point/ equilibrium, but I am trying to understand the meaning of Saddle periodic orbit as I encounter them in many research articles like a google search gives me - saddle periodic orbits.
I am interested to know the meaning and significance of a saddle periodic orbit.
terminology dynamical-systems
asked Dec 20 '18 at 21:13
BAYMAXBAYMAX
2,89121123
$endgroup$
add a comment |
$begingroup$
I have seen saddle fixed point/ equilibrium, but I am trying to understand the meaning of Saddle periodic orbit as I encounter them in many research articles like a google search gives me - saddle periodic orbits.
I am interested to know the meaning and significance of a saddle periodic orbit.
terminology dynamical-systems
asked Dec 20 '18 at 21:13
BAYMAXBAYMAX
2,89121123
$endgroup$
$begingroup$
Yes, but still I am trying to understand it, any reference where there is introductory treatment of "Saddle periodic orbit"?
$endgroup$
– BAYMAX
Dec 20 '18 at 22:43
$begingroup$
Related: Can a limit cycle be stable and unstable and the same time?
$endgroup$
– Wrzlprmft
Dec 22 '18 at 7:04
add a comment |
$begingroup$
I have seen saddle fixed point/ equilibrium, but I am trying to understand the meaning of Saddle periodic orbit as I encounter them in many research articles like a google search gives me - saddle periodic orbits.
I am interested to know the meaning and significance of a saddle periodic orbit.
terminology dynamical-systems
asked Dec 20 '18 at 21:13
BAYMAXBAYMAX
2,89121123
$endgroup$
I have seen saddle fixed point/ equilibrium, but I am trying to understand the meaning of Saddle periodic orbit as I encounter them in many research articles like a google search gives me - saddle periodic orbits.
I am interested to know the meaning and significance of a saddle periodic orbit.
terminology dynamical-systems
terminology dynamical-systems
asked Dec 20 '18 at 21:13
BAYMAXBAYMAX
2,89121123
asked Dec 20 '18 at 21:13
BAYMAXBAYMAX
2,89121123
asked Dec 20 '18 at 21:13
BAYMAXBAYMAX
2,89121123
asked Dec 20 '18 at 21:13
BAYMAXBAYMAX
2,89121123
asked Dec 20 '18 at 21:13
BAYMAXBAYMAX
2,89121123
2,89121123
$begingroup$
Yes, but still I am trying to understand it, any reference where there is introductory treatment of "Saddle periodic orbit"?
$endgroup$
– BAYMAX
Dec 20 '18 at 22:43
$begingroup$
Related: Can a limit cycle be stable and unstable and the same time?
$endgroup$
– Wrzlprmft
Dec 22 '18 at 7:04
add a comment |
$begingroup$
Yes, but still I am trying to understand it, any reference where there is introductory treatment of "Saddle periodic orbit"?
$endgroup$
– BAYMAX
Dec 20 '18 at 22:43
$begingroup$
Related: Can a limit cycle be stable and unstable and the same time?
$endgroup$
– Wrzlprmft
Dec 22 '18 at 7:04
$begingroup$
Yes, but still I am trying to understand it, any reference where there is introductory treatment of "Saddle periodic orbit"?
$endgroup$
– BAYMAX
Dec 20 '18 at 22:43
$begingroup$
Yes, but still I am trying to understand it, any reference where there is introductory treatment of "Saddle periodic orbit"?
$endgroup$
– BAYMAX
Dec 20 '18 at 22:43
$begingroup$
Related: Can a limit cycle be stable and unstable and the same time?
$endgroup$
– Wrzlprmft
Dec 22 '18 at 7:04
$begingroup$
Related: Can a limit cycle be stable and unstable and the same time?
$endgroup$
– Wrzlprmft
Dec 22 '18 at 7:04
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
For discrete-time systems $x_{n+1}=f(x_n)$, an $N$-periodic orbit is called a saddle if the corresponding fixpoint of $f^N$ is a saddle point (Scholarpedia).
But the paper you referred to deals with continuous-time systems (vector fields). They don't really give a precise definition, but from section 3.2 it seems like what they mean is a just a periodic orbit such that some nearby orbits are attracted (those on the stable manifold of that orbit) and others are repelled. Maybe an easy way to state it more precisely would be to say that it's a periodic orbit such that the corresponding fixpoint for the Poincaré map is a saddle point?
answered Dec 21 '18 at 9:57
Hans LundmarkHans Lundmark
35.4k564115
$endgroup$
$begingroup$
Thanks for your answer, seems nice and understandable!!
$endgroup$
– BAYMAX
Dec 22 '18 at 3:16
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
$begingroup$
For discrete-time systems $x_{n+1}=f(x_n)$, an $N$-periodic orbit is called a saddle if the corresponding fixpoint of $f^N$ is a saddle point (Scholarpedia).
But the paper you referred to deals with continuous-time systems (vector fields). They don't really give a precise definition, but from section 3.2 it seems like what they mean is a just a periodic orbit such that some nearby orbits are attracted (those on the stable manifold of that orbit) and others are repelled. Maybe an easy way to state it more precisely would be to say that it's a periodic orbit such that the corresponding fixpoint for the Poincaré map is a saddle point?
answered Dec 21 '18 at 9:57
Hans LundmarkHans Lundmark
35.4k564115
$endgroup$
$begingroup$
Thanks for your answer, seems nice and understandable!!
$endgroup$
– BAYMAX
Dec 22 '18 at 3:16
add a comment |
$begingroup$
For discrete-time systems $x_{n+1}=f(x_n)$, an $N$-periodic orbit is called a saddle if the corresponding fixpoint of $f^N$ is a saddle point (Scholarpedia).
But the paper you referred to deals with continuous-time systems (vector fields). They don't really give a precise definition, but from section 3.2 it seems like what they mean is a just a periodic orbit such that some nearby orbits are attracted (those on the stable manifold of that orbit) and others are repelled. Maybe an easy way to state it more precisely would be to say that it's a periodic orbit such that the corresponding fixpoint for the Poincaré map is a saddle point?
answered Dec 21 '18 at 9:57
Hans LundmarkHans Lundmark
35.4k564115
$endgroup$
$begingroup$
Thanks for your answer, seems nice and understandable!!
$endgroup$
– BAYMAX
Dec 22 '18 at 3:16
add a comment |
$begingroup$
For discrete-time systems $x_{n+1}=f(x_n)$, an $N$-periodic orbit is called a saddle if the corresponding fixpoint of $f^N$ is a saddle point (Scholarpedia).
But the paper you referred to deals with continuous-time systems (vector fields). They don't really give a precise definition, but from section 3.2 it seems like what they mean is a just a periodic orbit such that some nearby orbits are attracted (those on the stable manifold of that orbit) and others are repelled. Maybe an easy way to state it more precisely would be to say that it's a periodic orbit such that the corresponding fixpoint for the Poincaré map is a saddle point?
answered Dec 21 '18 at 9:57
Hans LundmarkHans Lundmark
35.4k564115
$endgroup$
For discrete-time systems $x_{n+1}=f(x_n)$, an $N$-periodic orbit is called a saddle if the corresponding fixpoint of $f^N$ is a saddle point (Scholarpedia).
But the paper you referred to deals with continuous-time systems (vector fields). They don't really give a precise definition, but from section 3.2 it seems like what they mean is a just a periodic orbit such that some nearby orbits are attracted (those on the stable manifold of that orbit) and others are repelled. Maybe an easy way to state it more precisely would be to say that it's a periodic orbit such that the corresponding fixpoint for the Poincaré map is a saddle point?
answered Dec 21 '18 at 9:57
Hans LundmarkHans Lundmark
35.4k564115
answered Dec 21 '18 at 9:57
Hans LundmarkHans Lundmark
35.4k564115
answered Dec 21 '18 at 9:57
Hans LundmarkHans Lundmark
35.4k564115
answered Dec 21 '18 at 9:57
Hans LundmarkHans Lundmark
35.4k564115
35.4k564115
$begingroup$
Thanks for your answer, seems nice and understandable!!
$endgroup$
– BAYMAX
Dec 22 '18 at 3:16
add a comment |
$begingroup$
Thanks for your answer, seems nice and understandable!!
$endgroup$
– BAYMAX
Dec 22 '18 at 3:16
$begingroup$
Thanks for your answer, seems nice and understandable!!
$endgroup$
– BAYMAX
Dec 22 '18 at 3:16
$begingroup$
Thanks for your answer, seems nice and understandable!!
$endgroup$
– BAYMAX
Dec 22 '18 at 3:16
add a comment |
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$begingroup$
Yes, but still I am trying to understand it, any reference where there is introductory treatment of "Saddle periodic orbit"?
$endgroup$
– BAYMAX
Dec 20 '18 at 22:43
$begingroup$
Related: Can a limit cycle be stable and unstable and the same time?
$endgroup$
– Wrzlprmft
Dec 22 '18 at 7:04