What is Bifurcation: Definition and 56 Discussions
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations occur in both continuous systems (described by ODEs, DDEs or PDEs) and discrete systems (described by maps). The name "bifurcation" was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior. Henri Poincaré also later named various types of stationary points and classified them with motif.
Here is the Chen System
I am given the initial condition (t=0) that a particle lies on the xyz-plane at a point (-10,0,35). I was notified that if I plugged in a=40, b=5, and c=30, the trajectory of the particle will be chaotic. On the other hand, if I retained the values of a and c, and...
Hello,
I have a question about period doubling and fixed points in the logistic map. Let's say I have a basic logistic map, ##f(x) = 4\lambda x(1-x)##. Let me then compare 1,2 and 4 iterations of this map on fixed points. I assume that ##\lambda## is large enough such that two period doublings...
This is wild.
I was always fascinated with the Mandelbrot set, as well as the bifurcation diagram. I had no idea the Mandelbrot diagram was a different visualization of the bifurcation diagram.
Question: is this video accurate? I always question the veracity of YouTube science videos.
Hello,
I'm lost at where to go after drawing bifurcation diagram of
$$\dot{x} = r + x - x^3.$$ If we also assume our parameter is time dependent such that
$$\dot{r} = -\delta x.$$ How could we use our initial bifurcation diagram to sketch solutions for small δ?
I have to sit an exam on non linear dynamical systems in a couple of weeks. Something that's been asked in the past is to name physical examples of different types of bifurcation. I've consulted Strogatz's book and the internet to try and find some, but I can't seem to find many (or even any)...
Hello my friends,
I am studying excitability in quantum dot lasers and I see a lot of bifurcation diagrams with saddle node bifurcations, Hopf bifurcations, homoclinic bifurcations, PD bifurcations etc. I know some basic things about non-linear systems but I have never met the notion of...
Chaos is when the waveforms become aperiodic. I think bifurcation is the phenomenon inclusive of chaos and in addition, it is also termed for situations in which the waveforms become n-periodic.Does bifurcation include period-n phenomenon as well as chaos? From period-n it means that still the...
I am in the process of writing a lecture out for my Graduate modeling class I teach. I normally don't write lectures out in LaTex or use PDF's because I write on the dry erase board, but if anyone is interested I wouldn't mind spending the time to type out some notes on the topic.
The topic...
Hello,
I hope to find some help.
I have equation: and I need to find the value of parameter A at which the homoclinic bifurcation occurs.
I know, that in this case A=-0.34, but I solution, how to find it.
Can you help me, what numerical method I should use to find A?
Thank you very much.
Hi PF
I am given the ODE ##\ddot{\theta} +q\sin\theta+\mu\cos\theta=0##, where I introduce ##x_1 = \theta## and ##x_2=\dot{\theta}## to yield$$
\dot{x_2}=-q\sin x_1-\mu\cos x_1\\
x_2=\dot{x_1}
$$
subject to ##x_2(0)=x_2(1)=0## and ##\int_0^1\sin x_1\, d t = 0##. I then plug this system into...
Homework Statement
I want to find conditions over A,B,C,D to observe a stable limit cycle in the following system:
\frac{dx}{dt} = x \; (A-B y) \hspace{1cm} \frac{dy}{dt} = -y \; (C-D x)
Homework Equations
Setting dx/dt=0 and dy/dt=0 you can find that (C/D , A/B) is a fixpoint.
The Attempt...
I thought it would be polyfurcation but it s not in dictionary. Also if blood vessels use venturi effect to lower pressure in some places, what is the extra speed used for?
Homework Statement
The equation:
where A is the parameter.
1) Find the value of parameter A at which the homoclinic bifurcation occurs.
2)Find the separatrices
3)Calculate the first integral and find the saddle-saddle connection equation.
Homework EquationsThe Attempt at a Solution
First...
Hello world!
I've done a few simulations of an emulsion droplet which is actuated by a laser beam. The droplet starts to move due to the laser light. I don't want to talk too much about the physics behind this but more discuss the nonlinear dynamics of the trajectories. Depending on a parameter...
Homework Statement
(a): Show the lagrangian derivative in phase space
(b)i: Show how the phase space evolves over time and how they converge
(b)ii: Find the fixed points and stability and sketch phase diagram
(c)i: Find fixed points and stability
(c)ii: Show stable limit cycles exist for T>ga...
I'm trying to solve question 4.12 from Cross and Greenside "pattern formation and dynamics in nonequilibrium systems".
the question is about the equation
\partial_t u = r u - (\partial_x ^2 +1)^2 u - g_2 u - u^3
Part A: with the ansatz u=\sum_{n=0}^\infty a_n cos(nx) show that the...
Homework Statement
bifurcation for the following 2D system:
Homework Equations
x'=ux−y+x^3,y′=bx−y
The Attempt at a Solution
I have got ux−y+x^3=0, y=bx, then x=0 and ±sqrt(b−u).
But I don't how to continue to find the bifurcation?
On Wolfram Alpha it will offend give you a contour plot, and it has similarities to a bifurcation plot with respect to a variable and a parameter.
What is the difference between a contour plot and a bifurcation diagram?
Homework Statement
Find numerically the r values for the first 2 bifurcations.
Homework Equations
xi+1 = f(xi), f(x) = rx(1 − x)
The Attempt at a Solution
To find the values of r, first I set rx(1−x)=0 to find x and then used the x values to find r=0 and r=1. But, I am still...
Homework Statement
What are the bifurcation values for the equation:
dy/dt = y^3 +ay^2
Homework Equations
The Attempt at a Solution
Equilibrium solutions:
y^3 + ay^2 = 0
==> y^2 (y + a) = 0
==> y = 0 (double root), or y = -a.
a = 0 is the sole bifurcation point, since...
Homework Statement
As in title.
Homework Equations
My book has a very shaky definition of what a bifurcation point. Basically, I need to play around with r and see how the system changes.
The Attempt at a Solution
x' = 0 when x = 1/2 - r ± √((r-1/2)2 - 2r)
d/dx (x') = 1/2 -...
Homework Statement
Just making sure I understand this stuff. One question on my homework asks about x' = x(r - ex).
Homework Equations
Definition. From my textbook: "The qualitative structure of the flow can change as parameters are varied. In particular, fixed points can be created or...
Homework Statement
http://www.freeimagehosting.net/t/9369y.jpg
Homework Equations
The Attempt at a Solution
a) is as follows: http://www.freeimagehosting.net/t/4oqft.jpg
Then for b), I have the equilibria as (0,0,0) and (r-1,\frac{r-1}{r},\frac{(r-1)^2}{r})
To...
Question in red at the bottom
For $\mu\ll 1$, there are two time scales.
\begin{alignat*}{3}
x' & = & \mu x - y - xy^2\\
y' & = & x + \mu y - y^3
\end{alignat*}
Let's expand $x,y$.
\begin{alignat*}{3}
x & = & \mu^{1/2}x_0 + \mu^{3/2}x_1 + \cdots\\
y & = & \mu^{1/2}y_0 + \mu^{3/2}y_1 + \cdots...
1. Homework Statement [/b]
Consider the dynamical system
\frac{dx}{dt}=rx-\frac{x}{1+x}
where r>0
Draw the bifurcation diagram for this system.
Homework Equations
The Attempt at a Solution
Well fixed points occur at x=0,\frac{1}{r}-1 and x=0 is stable for 0<r<1 and unstable for...
I'm at a loss on this question...my troubles seem to be algebraic or that I'm simply missing something.x' = \mu - x2 +4x4
my method for these questions has basically been to do everything required to draw bifurcation diagram bar drawing the actual diagram itself (ie, find equilibria, what...
Problem:
The following two-dimensional system of ODEs possesses a limit-cycle solution for certain values of the parameter$a$. What is the nature of the Hopf bifurcation that occurs at the critical value of $a$ and state what the critical value is.
$\dot{x}=-y+x(a+x^2+(3/2)y^2)$...
Homework Statement
The following two-dimensional system of ODEs possesses a limit-cycle solution for certain values of the parameter a. What is the nature of the Hopf bifurcation that occurs at the critical value of a and state what the critical value is.
Homework Equations...
Homework Statement
Consider the ODE
x' = \mux - x2 + x4
where x \in R and \mu \in R is a parameter.
Find and identify all bifurcation points for this equation. Sketch a bifurcation diagram, showing clearly the stability of all equilibria and the location of the bifurcation points.
You may...
Homework Statement
For each of the following equations sketch the bifurcation diagram, determine type of bifurcation, and find the critical value of r.
ẋ = rx + cosh(x)
ẋ = x(r - sinh(x))
ẋ = rx - xe-x2
The attempt at a solution
Fixed points satisfy
f'(x) = r + sinh(x) = 0 ⇒ x* =...
Homework Statement
Consider the system dx/dt = rx + x^3 - x^5, which exhibits a subcritical pitchfork bifurcation.
a) Find algebraic expressions for all the fixed points.
b) Calculate r_s, the parameter value at which the nonzero fixed points are born in a saddle-node bifurcation.
The...
Hello everybody,
Background and problem description
I have derived an analytical expression for an implicit frequency response function. To verify it, I would like to check with a numerical solution. For very weak nonlinearities, congruence is obtained. For weak nonlinearities, the...
Hi everyone,
I am studying a 2-D nonlinear dynamics system, with two key parameters. But I have trouble when I want to locate where the homoclinic bifurcation occurs in the parameter space. Can anyone give me some ideas or reference readings? Thx
The question is as follows: A circular tube of radius 'a' bifurcates into two tubes with equal radii 'ka', where k is a dimensionless coefficient. Derive an expression for the ratio of the pressure gradient in each bifurcated tube to that in the initial tube in terms of 'k'.
I'm not sure...
Can anyone help me find the bifurcation value of dy/dt = y^3 + ay^2 where a is the parameter. I found that the bifurcation value is 0 but at that equilibrium point the phase graph shows a source, not a node, so I'm not totally sure. Someone help please!
Hi, I'm using NSolve in Mathematica but I only want to find the roots that are real numbers as my answer gives a lot of imaginary numbers. How can I do this?
I want to then plot a birfucation diagram using a table of data. Is there a function to do this?
Thanks!
I want to produce a bifurcation diagram using mathematica to represent equilibrium points of p1 using the data obtained from the following code, where pn1, pn2, pn3 are recrusion exquations. i.e. x-axis will be the paramter t (ranging from 0-0.3) and y-axis will be the values of p1 this code...
Homework Statement
Make a bifurcation diagram for the sin map x{sub t+1} = f(x{sub t}) where f(x) = rsin(pi*x).
Homework Equations
The Attempt at a Solution
A bifurcation diagram... I have read and understood what a bifurcation diagram is, and how it is constructed, but to...
I would like to ask for a piece of advice on references on following issue:
Any (macroscopic) bifurcation of dynamical system generates some new information (or loss of information). In particular, when the dynamical system is loosing stability (i.e. stable equilibria becomes unstable) some new...
Homework Statement
The following model describes a fox population:
\frac{dS}{dt} = kS(1 - \frac{S}{N})( \frac{S}{M} - 1)
a) at what value of N does a bifurcation occur?
b) How does the population behave if the parameter N slowly and continouly decreases towards the bifurcation value...
Homework Statement
This is the bouncing ball experiment, a circuit is used as an analog of a ball bouncing on an oscillating table. The amplitude of the tables oscillations is varied and data representing the balls position and velocity is gathered.
I have to plot a poincare section for...
Need urgent help with Bifurcation problem!
Homework Statement
I'm stuck at part (ii) of this question! The question is as follows:
The differential equation is dy/dt = y^2 + (µ + 1)*y.
(i) Using µ = 0 sketch the phase line. Repeat for µ = -1 and µ = -2.
(ii) Calculate the position &...
Homework Statement
Draw the bifurcation diagram for the following equation.
:attached:
Homework Equations
I believe this is a population D.E.
The Attempt at a Solution
I'm ashamed to say I'm at a complete loss with this problem. The only step I'm familiar with is finding the...
Homework Statement
Consider the biased van der Pol oscillator: \frac{d^2x}{dt^2}=u (x^2-1)\frac{dx}{dt} + x = a. Find the curves in (u,a) space at which Hopf bifurcations occur. (Strogatz 8.21)
Homework Equations
The Attempt at a Solution
Not even sure where to start with this...
Hello all,
i was given the following assignment: FitzHugh proposed the dynamical system
\dot{x}=-x(x-a)(x-1)-y+I
\dot{y}=b(x-\gamma y)
to model neurons. I is the input signal, x is the activity, y is the recovery and 0<a<1, \gamma>0, b>0 are constants. Prove that there is a critical...
Let's keep this about the subject. This isn't about religion, but about people believing in one thing (creationism) and pursuing an advanced scientific degree. Some say it is for carrying the "legitimate" degree in order to give credibility to creationism and shouldn't be allowed if they...
Hi all!
I am currently reading the text "Practical Bifurcation and Stability Analysis" by Seyedel for an introduction to practical bifurcation theory.
I do not tatolly understand the theory and hope that some of you could help explain to an idiot like me.
I have read till P.80 of the books...