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Help Finding Critical Points of a Cubic (with 2 parameters)

I am trying to find bifurcation points in 1 dimension, but am having trouble finding critical points of $x'=\mu x -2x^2-x^3+ \delta$ ( where $x$ is my variable, $\mu$ is a parameter, and $\delta$ ...
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1answer
28 views

Bifurcation flow field?

assuming I have $$x'=5+mx+2x^2$$ how would I find the flow field of this bifurcation with the changing variable m? EDIT: Take for example ...
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1answer
16 views

Onset of n-cycles in the logistic function

The logistic function f(x) = r x (1 - x) is well known in dynamical systems theory. So is the famous bifurcation diagram for it. I recently came upon an article in Mathematics Magazine from 1996 ...
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50 views

AUTO Software for ODE's: references or forums?

I'm learning the AUTO software that does numerical continuation of ODE's by following these two references: The official manual found here www.dam.brown.edu/.../auto07p.pdf Lecture notes found at ...
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2answers
31 views

bifurcation in differential equations

I have an equation $$ y' = (2+y)(k-y^2) $$ and I am asked to find the equilibrium solutions and bifurcation values for all values of $k$. My approach was that... there exists $2$ equlibrium solutions ...
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1answer
43 views

Logistic Map Bifurcation diagram

So in the bifurcation diagram of the logistic map, there is period doubling from about r=3 to about r=3.54409. There are two fluctuation points between r=3 and r=$1+\sqrt{6}$. My question is, how ...
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2answers
106 views

Logistic map bifurcation

Ok I am trying to do this on matlab, but I need to understand how to find the bifurcation values for logistic map by hand first. So here is the logistic map: $$ x_{i+1} = f(x_i) \qquad \text{where} ...
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1answer
58 views

book recommendations for further learning about dynamical system and bifurcation

I have read the book "Introduction to Applied Nonlinear Dynamical Systems and Chaos"by Stephen Wiggins.Could someone recommend books on dynamical systems and bifurcation theory for further learning?
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1answer
84 views

Bifurcation values for logistic map

To find the bifurcation values for $$xi+1=f(xi), f(xi) = rx(1-x)$$first I set $rx(1-x) = 0$ and found the x values and then used the x values to find $r = 0$ and $r = 1$. Do you think what I did ...
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1answer
146 views

Matlab code for logistic map bifurcation

Take the logistic map $$x(i+1) = f(xi), f(x) = rx(1 − x)$$ Find numerically the $r$ values for the first two bifurcations. I know I have to set $f(x)=0$, solve for $x$, and then use the $x$ values ...
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45 views

Is derivation of Feigenbaum constant possible through Mandelbrot set?

this is Mandelbrot set: $z_{n+1}=z_n^2+C$ Is derivation of Feigenbaum constant possible through Mandelbrot set? $$\lim_{n\to\infty}\frac{z_{n+2}-z_{n+1}}{z_{n+1}-z_{n}}=\delta$$
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1answer
48 views

Value of b which moves quadratic from a stable fixed point to a stable period 2 point

I have a question which asks: At what value of b does the quadratic 1.5x^2+bx+2.34 move from having a stable fixed point to having a stable period 2 point? How would I go about solving this?
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14 views

n-body systems and bifurcations

From what I understand, in bifurcation theory, one definition of the equivalence of two dynamical systems is that they are topologically equivalent. However if say proteins A, B start out as straight ...
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0answers
31 views

Bifurcations caused by a single additive parameter

Let's say we have a $n$-dimensional continuous-time dynamical system described by the ODE $$\dot{x} = f(x, \delta) = g(x) + \delta \cdot e_i$$ where $g : \mathbb{R}^n \mapsto \mathbb{R}^n$ is a smooth ...
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16 views

Bifurcation at an undefined point

I have a 6d system of dynamic equations. I am able to calculate numerically the steady states and evaluate their local stability. It turns out that everything depends on one parameter, call it C. If ...
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31 views

Hopf Bifurcation of Reaction-Diffusion System

I'm considering the following reaction-diffusion system: $ \frac{\partial u}{\partial t} = f(u,v)+ D_1 \frac{d^2 u}{dx^2} $ $ \frac{\partial v}{\partial t} = g(u,v)+ D_2 \frac{d^2 v}{dx^2} $ where ...
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1answer
122 views

Bifurcations for 1-dimensional map

Consinder the 1-dim map $F(x,\mu)=\mu-\frac{1}{4} x^2$ as $\mu$ increases from $-\infty$ to $5$ and analyse the bifurcations. I start the analysis by considering $F^2(x,\mu)=\mu^2-\frac{1}{2}\mu ...
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solving for bifurcations in nonlinear 2d system?

If we have this nonlinear system $$ x ̇ = −x + 0.1y + x^2 y $$ $$ y ̇ = b − 0.1y − x^2y $$ What kind of bifurcations happen as you vary b ∈ [0,1]? I managed to solve for equilibriums at the source ...
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1answer
50 views

Physical intuition for period-doubling bifurcations

I'm currently studying period-doubling bifurcations, and being a physicist I want some physical intuition for what they are! From what I've gathered, the system spontaneously undergoes a ...
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2answers
217 views

Finding all bifurcations in a 2D system

I want to find all bifurcations for the system: \begin{align} x' =& -x+y\\ ...
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1answer
60 views

Finding a Hopf Bifucation with eigenvalues

I am trying to show that the following 2D system has a Hopf bifurcation at $\lambda=0$: \begin{align} ...
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3answers
74 views

Generating Bifurcation Animations

https://en.wikipedia.org/wiki/File:Hopf-bif.gif Does anyone know how this animation was produced? I could make it by stitching together snapshots (what I'm doing) but this seems primitive, especially ...
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154 views

Bifurcation of integral curves

Consider the following first order ODE: $$\frac{\operatorname{d}\!y}{\operatorname{d}\!x} = x^2 - y^2$$ Despite the fact that this ODE has a very simple expression, it is not solvable in terms of ...
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0answers
75 views

Does this type of bifurcation exist?

I've been checking out numerically an ODE model of a gene circuit. Just from simulations, it appears that once a parameter passes some critical value a stable fixed point splits into three other fixed ...
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81 views

What is this bifurcation?

I have a discrete dynamical System $x_{n+1}=f(x_{n},x_{n-1},x_{n-2},x_{n-3},x_{n-4},\lambda)$ with a paramteter $\lambda>0$, and where all $x_{n}$s are in [0,1]. $f$ is actually a larger ...
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1answer
121 views

Help with brute force method of producing bifurcation diagrams of discrete-time systems

I have a homework question concerning a brute force method of creating bifurcation diagrams. This seems really abstract for me and would like a clearer description of how the method works. Can someone ...
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1answer
228 views

Relationship between Turing bifurcation, saddle-node bifurcation, and Hopf bifurcation?

Quoting from http://jxshix.people.wm.edu/2009-harbin-course/mississippi-bifurcation-2.pdf a Turing bifurcation occurs when for an ODE and related PDE $u' = f(u,v), v' = g(u,v)$ $u_t = d_1 \nabla ...
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1answer
495 views

Possible ways to do stability analysis of non-linear, three-dimensional Differential Equations

For example Lorenz system, $$ \frac{d}{dt}\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} -\sigma & \sigma & 0\\ \rho & -1 & -x\\ y & 0 & -\beta ...
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2answers
147 views

Chaos without period doubling

I have been studying the Duffing oscillator rather intensively lately, mainly based on the theory in of the book by Guckenheimer and Holmes. From all that I have gathered, it seems that most dynamical ...
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0answers
25 views

Divergent limit cycle frequency along a hopf boundary

I am studying the linear stability of continuous system as a function of two parameters (a and b) and I observe that a hopf bifurcation with frequency w happens along the line described by f(a,b). ...
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1answer
174 views

What is a good text on bifurcation theory?

What is a a good text on bifurcation theory for mathematicians who haven't seen it before? I'm looking to get a feel for the intuition behind the subject, major standard theorems, etc. I do not mind ...
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1answer
59 views

Is there a bifurcation node in the activation energy of chemical reactions?

I was thinking about the activation energy of chemical reactions (obviously), and I was wondering if there exists a bifurcation node somewhere in the transition state. Let me give you a link to an ...
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0answers
86 views

Half-stable fixed point on a circle

On a line graph, it's clear that a half-stable fixed point is the limit of moving the unstable fixed point towards the stable fixed point. Some solutions go to infinity depending on the initial ...
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2answers
305 views

Period doubling is chaos?

I've already read something about chaos and it's origins but I am not sure that this affirmative statement is true for all the cases. Can anyone help me? Thanks, Bruno
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80 views

The equation $2 \cosh(3.1786803659501505 z) = z$?

Let $a$ be a positive real number and $z$ a complex number. I was wondering about the equation $2 \cosh(a z) = z$ where we solve for $z$. Clearly if $z$ is a solution than so is its conjugate. It ...