Items tagged with bifurcation

Here, I have a 3D map T=(T1,T2,T3) with

T1(x1,x2,x3)=x1*exp(6-2*x1-x2-0.3*x3),

T2(x1,x2,x3)=x2*exp(5-2*x1-2*x2-*x3),

T3(x1,x2,x3)=x3*exp(8-0.2*x1-x2-x3),

How can I use  IterativeMaps:-Attractor obtain the attractor for T.

Hello,

I would like to study the period doubling bifurcation behaviors of autonomous ODEs.

Although I know how to plot the Poincare section and bifurcation diagram for non-autonomous ODEs, such as Duffing oscillator, I totally stuck at the autonomous ones. Could you please help me.

It could be greatly helpful if you could share me the code of bifurcation diagram for, say, Rossler or Lorenz systems? 

Thank you in advance.

Very kind wishes,

Wang Zhe

How to use IterativeMaps:-Bifurcation for two or more dimensional maps. For example, f(x,y)=x*exp(r1*(1-a11*x-a12*y)), g(x,y)=y*exp(r2*(1-a21*x-a22*y)).

If I set a11=1,a12=2,a21=3,a22=4,r2=1, then how can I get the bifurcation rsp r1. Thanks very much!

with(IterativeMaps);
with(ImageTools);
Logistic := Bifurcation([x], [r*x*(1-x)], [.5], 2.5, 4);
ArrayTools:-Dimensions(Logistic);
ColouringProcedures:-HueToRGB(Logistic);
Embed(Logistic);

This is the code for Bifurcation program of the Logistic map. How can I change the black background color of this figure, and How can I save this figure.

Consider the following dynamical systems with time delay:

diff(x(t), t) = y(t)-bx(t)^3+ax(t)^2-z(t-tau)+I

diff(y(t), t) = c-dx(t)^2-y(t)

diff(z(t), t) = r(s(x-beta)-z(t))

Here the values of the parameters are a = 3, b = 1, c = 1, d = 5, s = 4, beta = 1.6, r = 0.6e-2, I = 3.0

 

Please help me 

How to write code for bifurcation plot for the above differential equations with delay.

Delay as taken as bifurcation parameter.

 

Reply message is very useful.

 

Thanks in Advance

 

How to show what class of bifurcation is in this function :   exp(x2*(a-x)) ?

Hello everyone, 

In Maple8, I tried to plot this logistic map and an error occured (Error, (in Bifurcation) `plots` does not evaluate to a module).

What is wrong into this code?

Thank you

 

restart: with(plots):Warning, the name changecoords has been redefined

> Bifurcation := proc(initialpoint,xexpr,ra,rb,acc)
> local p1,hr,A,L1,i,j,phi:
> global r,L2:
> hr := unapply(xexpr,x);
> A := Vector(600):
> L1 := Vector(acc*500):
> for j from 1 to acc+1 do
> r := (ra + (j-1)*(rb-ra)/acc):
> A[1] := hr(initialpoint):
> for i from 2 to 500 do
> A[i] := evalf(hr(A[i-1])):
> end do:
> for i from 1 to 400 do
> L1[i+400*(j-1)] := [r,A[i+100]]:
> end do:
> end do:
> L2 := {seq(L1[i], i = 1..acc*400)}:
> p1 := plots:-pointplot(L2, 'symbol' = solidcircle, 'symbolsize' = 8, 'color' = blue):
> unassign('r'):
> return(p1):
> end proc:
> P1:= Bifurcation(1/2,r*x*(1-x),2.5,4,250):
>
Error, (in Bifurcation) `plots` does not evaluate to a module

 

Hello ..

 

I need codes bifurcation and cobweb for logistic map by maple ..

 

can you help me ?

 

 

 

Thanks ...

I have a three paramter ode problem that involves three tanks with given initial concentrations.  Overtime the concentration equalizes but one of the steps is to determine all bifurcation values.  Not sure how to continue with this number of variables.

 This is our given system with initial conditions

sys_ode := diff(x(t),t) = (-r*x(t))/100+0+(r*z(t))/50, 
> diff(y(t),t) = (r*x(t))/100+(-r*y(t))/25+0,
> diff(z(t),t) = 0+(r*y(t))/25+(-r*z(t))/50;
> x0:=0; y0 := 200; z0:=0;

Hi,

I have a non linear ode with sinosoial term, (sin(x)).

How can we Analyse the system and plot the bifurcation diagram:

x'=r*x-sin(x);

Thank you very much for your help.

 

Hello,

One again, I have a problem to solve some bifurcation problem using maple.

Discuss the existance of Equilibria and determine any possible bifurcation.

x'=r*x-ln(x+1);

where $r$ is a parameter.

many thinks for your help and suggestion.

 

 

 

Hi.

I need you to answer to thos question using maple,

I have x'(t)=e^{2*x(t)}  + alpha -2*x(t),  where alpha is a constant in R.

I would like to compute the equilibrium points and if there is a saddle-node bifurcation.

Thank you very much.

 

I have the command;

> restart: Digits:=20: N:=10000: M:=100: x_max:=1: r_min:=2.5:
> r_max:=4: for n from 0 to N do r:=r_min+n/N*(r_max-r_min):
> x:=evalf(x_max*rand()/10^12):for m from 0 to M do x:=r*x*(1-x): od:
> X[n]:=x: od:
> with(plots):
> bifpoint:=[seq([r_min+j/N*(r_max-r_min),X[j]],j=0..N)]:
> pitchf:=pointplot(bifpoint,symbol=point):display(pitchf);

 

This plots the bifurcation diagram for the logistic model f(x) = r*x*(1-x).

How do i plot the bifurcation diagram for f(x) = r*(8 - 2*x^2). 

I've tried just replacing the function but it does not work.

 

HI,

I have a system of ODEs like

Here's an example exhibited by Nusc, which I have tweaked slightly to make it look more like your mathematica example.

### Reference: http://www.mapleprimes.com/questions/36580-Bifurcation-Diagram

### xexpr is the logistic function to be iterated (we always start off at x=1/2, which will eventually attract).
### [ra,rb] is the range of the parameter.
### acc is the number of points sampled in [ra,rb]

Bifurcation := proc(initialpoint,xexpr,ra,rb,acc)

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