# 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.

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1. ### Periodic and Chaotic Solutions to Chen System/Attractors

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...
2. ### A Path between fixed points in a logistic map

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...
3. ### B The Mandelbrot Set has a third dimension: the Bifurcation Diagram

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.
4. ### I Using a bifurcation diagrams and time dependent parameters

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?
5. ### I Physical examples of different types of bifurcation

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)...
6. ### Applied Book recommendations for Bifurcation Diagrams

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...
7. S

### Can subharmonics in a system be also termed as bifurcation?

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

### Difference between bifurcation and chaos

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...
9. ### MHB Gathering interest in Dynamical Systems (Bifurcation, Stable Manifolds, Hamiltonian Systems) f

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...
10. ### I Critical value of homoclinic bifurcation

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.
11. S

### Why is chaos more studied in dc-dc converters compared to other circuits?

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

### A How Does the Bifurcation Diagram in AUTO Represent Mathematical Concepts?

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...
13. ### What conditions are needed to get a stable limit cycle here?

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...
14. ### What is the word for bifurcation but with multi-branching?

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...
16. ### Bifurcation between two oscillations

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...
17. ### Pendulum - Stability and fixed points

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...
18. ### Proving a a pitchfork bifurcation: modified swift-hohenberg

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...
19. ### Locate any bifurcation in 2D system

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?
20. ### What is the difference between a contour plot and a bifurcation plot?

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?
21. ### Bifurcation values for logistic map

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...
22. ### Differential equation bifurcation, how to find equilibrium points?

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...
23. ### Bifurcation point of x' = r + x/2 - x/(x+1)

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 -...
24. ### Bifurcation Points and Stability Analysis for x' = x(r - ex)

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...
25. ### Hopf Bifurcation - Nonlinear Diff Eqns

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...
26. ### MHB Answer Hopf Bifurcation Question: What's Wrong?

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...
27. ### Drawing Bifurcation diagrams for a dynamical system

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...
28. ### Find all bifurcation points (ODEs)

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...
29. ### MHB ODE system. Limit cycle; Hopf bifurcation.

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. $\dot{x}=-y+x(a+x^2+(3/2)y^2)$...
30. ### ODE system. Limit cycle; Hopf bifurcation.

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...
31. ### Bifurcation Analysis for the ODE x' = \mux - x2 + x4

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...
32. ### Solve Bifurcation Diagrams: Find Critical Values of r

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* =...
33. ### Subcritical Pitchfork Bifurcation

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...
34. ### Solving nonlinear ODE with ODE45 fails at bifurcation point?

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...
35. ### Nonlinear dynamics bifurcation

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
36. ### Fluid Mechanics: pressure gradient of a tube bifurcation

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...
37. ### Understanding Bifurcation Points in the Function dy/dt = y^3 + ay^2

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!
38. ### Mathematica Solving Real Roots with NSolve and Plotting a Bifurcation Diagram

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!
39. ### Mathematica Produce a bifurcation diagram using mathematica

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...
40. ### Bifurcation diagram of a sin map(plotting it)

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...
41. ### How Does Bifurcation in Dynamical Systems Influence Information Generation?

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...
42. ### Differential Equation - Bifurcation

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...
43. ### Poincare section and bifurcation diagram

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...
44. ### Need help with Bifurcation problem

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 &...
45. ### Bifurcation Diagram <Diff. Eqns>

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...
46. ### Van der Pol oscillator + Hopf bifurcation

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...
47. ### Finding the Hopf Bifurcation in the FitzHugh Model

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...
48. ### Believing Scripture & Pursuing Science: The Bifurcation of the Mind

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

Is there a mathematical way to calculate the bifurcation of a system?
50. ### Convection described by Lorenz's Equation Bifurcation

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...