Bifurcation Definition and 47 Threads

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...

I think that the existence of subharmonics is also bifurcation.Is that true

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...

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.

Why is chaos only more studied in dc-dc converters and not in other nonlinear circuits, such as, rectifiers?

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?

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...

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 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...

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

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...

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 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...

Is there a mathematical way to calculate the bifurcation of a system?

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...

When determining the stability of the equilibria (or, critical points) for our bifurcation diagrams, we have been shown to use phase line diagrams. I understand that if the function is moving away from the equilibria on either side than it is unstable, and i know that if the function is...

I know what Bifurcation Theory is. but What is Perturbation theory? is it similar to bifurcation theory?

I've been reading Wald's book on GR as well as his article "Thermodynamics of Black Holes" in Living Reviews in Relativity about the definitions of mass and energy in GR and the concepts of entropy and temperature of black holes. I keep coming across the words "bifurcation surface" and...

"Bifurcation Points. In many physical problems some observable quantity, such as a velocity, waveform, or chemical reaction, depends on a parameter describing the physical state. As this parameter is increased, a critical value is reached at which the velocity, or waveform, or reaction suddenly...

Could some explain or point me to somewhere explains what a Bifurcation Diagram is and how to draw one given a non-linear function. Got an exam today and nobody knows what one is. If it has already been 2 hours after I've posted this don't worry, I'll be heading off to the exam hehe. Thanks if...

Under what conditions might a star bifurcate?