Bifurcation Definition and 56 Threads

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. HansBu

    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. Decimal

    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. DaveC426913

    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. M

    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. Jezza

    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. U

    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. D

    MHB Interested in Graduate Modeling Lecture Notes on Dynamical Systems?

    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. L

    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. JuanC97

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

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  14. P

    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?
  15. D

    Homoclinic bifurcation -- Saddle-saddle connection, separatrix

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  16. onkel_tuca

    Bifurcation between two oscillations

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  17. U

    Pendulum - Stability and fixed points

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  18. O

    Proving a a pitchfork bifurcation: modified swift-hohenberg

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  19. F

    Locate any bifurcation in 2D system

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  20. J

    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. S

    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. N

    Differential equation bifurcation, how to find equilibrium points?

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  23. J

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

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  24. J

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  25. C

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  26. D

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  27. P

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  28. A

    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. J

    MHB ODE system. Limit cycle; Hopf bifurcation.

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  30. S

    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. S

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

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  32. G

    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. G

    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. J

    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. P

    Where Does Homoclinic Bifurcation Occur in a 2-D Nonlinear Dynamics System?

    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. L

    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. S

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

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  38. J

    Mathematica Solving Real Roots with NSolve and Plotting a Bifurcation Diagram

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  39. J

    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. O

    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. G

    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. C

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  43. P

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  44. S

    Need help with Bifurcation problem

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  45. D

    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. nicksauce

    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. S

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  48. Evo

    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. I

    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. H

    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...
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