Mixing
To classify bifurcations by calculating partial derivatives evaluated at the bifurcation point
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Question: Given a one dimensional system $dotx = f(x;; mu)$ for which a bifurcation of fixed points occurs at $(x^star, mu_c)$, where $x^star$ is a fixed point and $mu_c$ is the corresponding bifurcation parameter value, can we classify the bifurcation point using the same criteria as is used to classify the bifurcation points of iterated maps (see below)?
Context: To classify bifurcations for iterated maps $x_n + 1 = f(x_n;; mu)$ we have the following criteria. In the following the partial derivatives of $f$ are evaluated at $(x^star, mu_c)$,
Saddle node bifurcations: $f_x = 1$, $f_xx neq 0$, $f_mu neq 0$
Transcritical bifurcations: $f_x = 1$, $f_xx neq 0$, $f_mu = 0$, $f_mu x^2 - f_xx f_mu mu neq 0$
Pitchfork bifurcations: $f_x = 1$, $f_xx = 0$, $f_mu = 0$, $f_mu x neq 0$, $f_xxx neq 0$
Period doubling bifurcations: $f_x = -1$, $2f_mu x + f_mu f_xx neq 0$, $f_xx^2/2 + f_xxx/3 neq 0$.
Remarks: Of course, the $f_x = pm 1$ criteria will have to be replaced with $f_x = 0$.
Related question: Do period doubling bifurcations occur in continuous systems of the form $dotx = f(x;; mu)$, or do they only occur in iterated maps?
nonlinear-system chaos-theory nonlinear-analysis non-linear-dynamics
asked Jul 15 at 15:03
LMZ
310110
add a comment |Â
up vote
1
down vote
favorite
Question: Given a one dimensional system $dotx = f(x;; mu)$ for which a bifurcation of fixed points occurs at $(x^star, mu_c)$, where $x^star$ is a fixed point and $mu_c$ is the corresponding bifurcation parameter value, can we classify the bifurcation point using the same criteria as is used to classify the bifurcation points of iterated maps (see below)?
Context: To classify bifurcations for iterated maps $x_n + 1 = f(x_n;; mu)$ we have the following criteria. In the following the partial derivatives of $f$ are evaluated at $(x^star, mu_c)$,
Saddle node bifurcations: $f_x = 1$, $f_xx neq 0$, $f_mu neq 0$
Transcritical bifurcations: $f_x = 1$, $f_xx neq 0$, $f_mu = 0$, $f_mu x^2 - f_xx f_mu mu neq 0$
Pitchfork bifurcations: $f_x = 1$, $f_xx = 0$, $f_mu = 0$, $f_mu x neq 0$, $f_xxx neq 0$
Period doubling bifurcations: $f_x = -1$, $2f_mu x + f_mu f_xx neq 0$, $f_xx^2/2 + f_xxx/3 neq 0$.
Remarks: Of course, the $f_x = pm 1$ criteria will have to be replaced with $f_x = 0$.
Related question: Do period doubling bifurcations occur in continuous systems of the form $dotx = f(x;; mu)$, or do they only occur in iterated maps?
nonlinear-system chaos-theory nonlinear-analysis non-linear-dynamics
asked Jul 15 at 15:03
LMZ
310110
1
Regarding the last question: one-dimensional continuous-time system can't have periodic solutions (except fixed points, if you count that as periodic), so you have to go to higher dimensions to make that question meaningful.
– Hans Lundmark
Jul 15 at 18:25
Thanks! Of course, since one dimensional systems are all conservative by the existence of an anti-derivative.
– LMZ
Jul 15 at 18:28
As stated, provided that $f_x = pm 1$ is substituted for $f_x = 0$, the criteria above (with the exception of the period doubling bifurcation criterion), do indeed apply to continuous one dimensional systems [1]. [1]: personalpages.manchester.ac.uk/staff/yanghong.huang/teaching/…
– LMZ
Jul 15 at 19:43
I've found a reference in the lecture notes above, but the link will one day die. If someone is able to find a textbook (and chapter number) reference, please post it as an answer and I will accept it.
– LMZ
Jul 15 at 19:44
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Question: Given a one dimensional system $dotx = f(x;; mu)$ for which a bifurcation of fixed points occurs at $(x^star, mu_c)$, where $x^star$ is a fixed point and $mu_c$ is the corresponding bifurcation parameter value, can we classify the bifurcation point using the same criteria as is used to classify the bifurcation points of iterated maps (see below)?
Context: To classify bifurcations for iterated maps $x_n + 1 = f(x_n;; mu)$ we have the following criteria. In the following the partial derivatives of $f$ are evaluated at $(x^star, mu_c)$,
Saddle node bifurcations: $f_x = 1$, $f_xx neq 0$, $f_mu neq 0$
Transcritical bifurcations: $f_x = 1$, $f_xx neq 0$, $f_mu = 0$, $f_mu x^2 - f_xx f_mu mu neq 0$
Pitchfork bifurcations: $f_x = 1$, $f_xx = 0$, $f_mu = 0$, $f_mu x neq 0$, $f_xxx neq 0$
Period doubling bifurcations: $f_x = -1$, $2f_mu x + f_mu f_xx neq 0$, $f_xx^2/2 + f_xxx/3 neq 0$.
Remarks: Of course, the $f_x = pm 1$ criteria will have to be replaced with $f_x = 0$.
Related question: Do period doubling bifurcations occur in continuous systems of the form $dotx = f(x;; mu)$, or do they only occur in iterated maps?
nonlinear-system chaos-theory nonlinear-analysis non-linear-dynamics
asked Jul 15 at 15:03
LMZ
310110
Question: Given a one dimensional system $dotx = f(x;; mu)$ for which a bifurcation of fixed points occurs at $(x^star, mu_c)$, where $x^star$ is a fixed point and $mu_c$ is the corresponding bifurcation parameter value, can we classify the bifurcation point using the same criteria as is used to classify the bifurcation points of iterated maps (see below)?
Context: To classify bifurcations for iterated maps $x_n + 1 = f(x_n;; mu)$ we have the following criteria. In the following the partial derivatives of $f$ are evaluated at $(x^star, mu_c)$,
Saddle node bifurcations: $f_x = 1$, $f_xx neq 0$, $f_mu neq 0$
Transcritical bifurcations: $f_x = 1$, $f_xx neq 0$, $f_mu = 0$, $f_mu x^2 - f_xx f_mu mu neq 0$
Pitchfork bifurcations: $f_x = 1$, $f_xx = 0$, $f_mu = 0$, $f_mu x neq 0$, $f_xxx neq 0$
Period doubling bifurcations: $f_x = -1$, $2f_mu x + f_mu f_xx neq 0$, $f_xx^2/2 + f_xxx/3 neq 0$.
Remarks: Of course, the $f_x = pm 1$ criteria will have to be replaced with $f_x = 0$.
Related question: Do period doubling bifurcations occur in continuous systems of the form $dotx = f(x;; mu)$, or do they only occur in iterated maps?
nonlinear-system chaos-theory nonlinear-analysis non-linear-dynamics
asked Jul 15 at 15:03
LMZ
310110
edited Jul 18 at 15:15
asked Jul 15 at 15:03
LMZ
310110
asked Jul 15 at 15:03
LMZ
310110
asked Jul 15 at 15:03
LMZ
310110
310110
1
Regarding the last question: one-dimensional continuous-time system can't have periodic solutions (except fixed points, if you count that as periodic), so you have to go to higher dimensions to make that question meaningful.
– Hans Lundmark
Jul 15 at 18:25
Thanks! Of course, since one dimensional systems are all conservative by the existence of an anti-derivative.
– LMZ
Jul 15 at 18:28
As stated, provided that $f_x = pm 1$ is substituted for $f_x = 0$, the criteria above (with the exception of the period doubling bifurcation criterion), do indeed apply to continuous one dimensional systems [1]. [1]: personalpages.manchester.ac.uk/staff/yanghong.huang/teaching/…
– LMZ
Jul 15 at 19:43
I've found a reference in the lecture notes above, but the link will one day die. If someone is able to find a textbook (and chapter number) reference, please post it as an answer and I will accept it.
– LMZ
Jul 15 at 19:44
add a comment |Â
1
Regarding the last question: one-dimensional continuous-time system can't have periodic solutions (except fixed points, if you count that as periodic), so you have to go to higher dimensions to make that question meaningful.
– Hans Lundmark
Jul 15 at 18:25
Thanks! Of course, since one dimensional systems are all conservative by the existence of an anti-derivative.
– LMZ
Jul 15 at 18:28
As stated, provided that $f_x = pm 1$ is substituted for $f_x = 0$, the criteria above (with the exception of the period doubling bifurcation criterion), do indeed apply to continuous one dimensional systems [1]. [1]: personalpages.manchester.ac.uk/staff/yanghong.huang/teaching/…
– LMZ
Jul 15 at 19:43
I've found a reference in the lecture notes above, but the link will one day die. If someone is able to find a textbook (and chapter number) reference, please post it as an answer and I will accept it.
– LMZ
Jul 15 at 19:44
1
1
Regarding the last question: one-dimensional continuous-time system can't have periodic solutions (except fixed points, if you count that as periodic), so you have to go to higher dimensions to make that question meaningful.
– Hans Lundmark
Jul 15 at 18:25
Regarding the last question: one-dimensional continuous-time system can't have periodic solutions (except fixed points, if you count that as periodic), so you have to go to higher dimensions to make that question meaningful.
– Hans Lundmark
Jul 15 at 18:25
Thanks! Of course, since one dimensional systems are all conservative by the existence of an anti-derivative.
– LMZ
Jul 15 at 18:28
Thanks! Of course, since one dimensional systems are all conservative by the existence of an anti-derivative.
– LMZ
Jul 15 at 18:28
As stated, provided that $f_x = pm 1$ is substituted for $f_x = 0$, the criteria above (with the exception of the period doubling bifurcation criterion), do indeed apply to continuous one dimensional systems [1]. [1]: personalpages.manchester.ac.uk/staff/yanghong.huang/teaching/…
– LMZ
Jul 15 at 19:43
As stated, provided that $f_x = pm 1$ is substituted for $f_x = 0$, the criteria above (with the exception of the period doubling bifurcation criterion), do indeed apply to continuous one dimensional systems [1]. [1]: personalpages.manchester.ac.uk/staff/yanghong.huang/teaching/…
– LMZ
Jul 15 at 19:43
I've found a reference in the lecture notes above, but the link will one day die. If someone is able to find a textbook (and chapter number) reference, please post it as an answer and I will accept it.
– LMZ
Jul 15 at 19:44
I've found a reference in the lecture notes above, but the link will one day die. If someone is able to find a textbook (and chapter number) reference, please post it as an answer and I will accept it.
– LMZ
Jul 15 at 19:44
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2852605%2fto-classify-bifurcations-by-calculating-partial-derivatives-evaluated-at-the-bif%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
Regarding the last question: one-dimensional continuous-time system can't have periodic solutions (except fixed points, if you count that as periodic), so you have to go to higher dimensions to make that question meaningful.
– Hans Lundmark
Jul 15 at 18:25
Thanks! Of course, since one dimensional systems are all conservative by the existence of an anti-derivative.
– LMZ
Jul 15 at 18:28
As stated, provided that $f_x = pm 1$ is substituted for $f_x = 0$, the criteria above (with the exception of the period doubling bifurcation criterion), do indeed apply to continuous one dimensional systems [1]. [1]: personalpages.manchester.ac.uk/staff/yanghong.huang/teaching/…
– LMZ
Jul 15 at 19:43
I've found a reference in the lecture notes above, but the link will one day die. If someone is able to find a textbook (and chapter number) reference, please post it as an answer and I will accept it.
– LMZ
Jul 15 at 19:44