Questions tagged [bifurcation]

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. (Def: http://en.m.wikipedia.org/wiki/Bifurcation_theory)

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What is the classification of the bifurcation of a tent map?

Considering the tent map where $x_{n+1} = f(x_{n})$ and $f(x)$ is defined as $$ f(x)= \begin{cases} \mu x, & 0 \leq x\leq \frac{1}{2} \\ \mu - \mu x, & \frac{1}{2}\leq ...
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Dynamical System that exhibits a fold bifurcation of Limit Tori?

Fold Bifurcations of a fixed points (i.e. saddle node bifurcations) and Fold bifurcations of limit cycles (i.e. when a stable limit and unstable limit cycle annihilate) are observed in plenty of ...
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Help identifying a bifurcation

I'm investigating a dynamical system and I have come across roots the real part of which looks like this: And imaginary part looks like this: The real part (which I'm primarily interested with) ...
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50 views

Neccesary condition for a family of functions to have a period-doubling bifurcation

I have the following definition of period-doubling bifurcation for a one-parameter family of functions $\left\{{ F_{\lambda} }\right\}$ : Definition. A family of functions $\left\{{ F_{\lambda} ...
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Is the following map topologically transitive?

y_n+1 = 4yn^3 - 6yn^2 + 2y_n + 1/2 I’ve read the material and theory on topological transitivity and it makes sense to me in theory but I am finding it extremely difficult to source any material where ...
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Does the map $y_{n+1} = (r y_n) \bmod 1$ exhibit periodic doubling upon varying the parameter $r$?

My initial thought is that it does from my understanding on reading the material regarding flip bifurcations but the $\bmod 1$ is throwing me off here so I can’t be sure. Can anyone advise?
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How to calculate orbit representatives for the conjugacy classes of isotropy subgroups.

Hello all I shall present my question and then some of the working out I have done. Question Let $\Gamma$ be the group of symmetries of a cube (including reflections) centred at the origin. I have ...
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How to show a bifurcation occurs in the following dynamic problem

I have the following non-dimensional model: $\dot{u}=u(1-u)-X\frac{u}{\alpha+u}$ Now, assuming $\alpha>1$ and $X>0$ are constants, I want to show a bifurcation happens locally near the zero ...
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Dynamic in invariant curve

Suppose I have a map in $X=\mathbb{R} \times \mathbb{S}^1 $ of the form: $$ (\overline{y},\overline{\theta})=f(y,\theta) $$ as regular as you want. If I know the following: There exist an invariant ...
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Brute-force bifurcation diagram

I know there is unaccepted answer in this question on this problem. But let’s make it clear, really. I believe it can serve as great reference starting point. Suppose that I have autonomous system $\...
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Finding the equilibria of a coupled dynamic system in $\mathbb{R}^3$

I've got the following system, $$ \begin{equation} f(\vec{x}) = \begin{cases} \label{basis} \dot{x} = a_1x^3 + a_2x + \phi \\ \dot{y} = b_1z + b_2(\gamma(x) - (y^2 + z^2))y \\ \dot{z} = ...
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Why does ${x}^{x^{x^{x^{\,.^{\,.^{\,.}}}}}}$ bifurcate below $\sim0.065$?

When you calculate what ${x}^{x^{x^{x\cdots }}}$ converges to between $0$ and $1$, before approximately $0.065$ the graph bifurcates. Why does this happen and is there a reason for it happens at that ...
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What are near-identity transformations?

I've started a module on Dynamical systems and we're currently looking at Hopf bifurcations. Below is an exercise in our notes revolving around near identity transformations: The notes seem to assume ...
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Translated systems and bifurcation value

Consider the system $$x'=y$$ $$y'=x+x^2+\mu x-xy$$ we know that $\operatorname{det}(Df(0,0))=-1$, then there is a saddle at the origin. In addition det $(Df (-1,0)) = 1> 0$, so there is a focus or ...
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Saddle node bifurcation

Let $g_\mu:\mathbb R\to \mathbb R$ be a family of smooth maps with bifurcation parameter $\mu\in \mathbb R$. I want to show that if $g_\mu^2$ has a saddle node bifurcation as $\mu$ passes through $\...
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Laplace transform of solution of an integral equation for non-recursive logistic map computations

Consider the logistic map, $x_n = r \: x_{n-1} \left( 1 - x_{n-1} \right)$ If we generalize this to a complex function $f : \mathbb{C} \mapsto \mathbb{C}$, we get, $f \left( z \right) = r \: f \left( ...
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How do I find the bifurcation points of the quadratic map?

I am exploring the quadratic map $x_{n+1} = x_n^2 + c$. If we plot the value of $x$ after many iterations on the vertical axis against the input $c$ value on the horizontal axis we get a classic ...
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Classifying a Bifurcation?

For the system of differential equations $$\dot{x}=y-x, \dot{y} = \mu x - y$$ with Jacobian matrix $$J = \begin{bmatrix}-1 & 1 \\ \mu & -1\end{bmatrix},$$ the point $(x,y) = (0,0)$ is a stable ...
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Equations for Mandelbrot bifurcation diagram

The Mandelbrot set is the set of all complex numbers $c$ that cause the function $z_{n+1} = z_n^2 + c$ to remain bounded within a circle of radius 2 when iterated from $z_0 = 0$. Looking at only the ...
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How to calculate fixed points and plot bifurcation diagram for non-linear ODE system

I am trying to understand how to analyse a system of coupled, non-linear ODEs taken from this paper. I want to perform a fixed point analysis and plot a bifurcation diagram to show how fixed points ...
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Mandelbrot set and logistic map connection

I'm currently writing an undergraduate thesis on chaos theory with a particular focus on the connection between the Mandelbrot set and the logistic map. I have found scattered posts on this site, ...
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66 views

Calculate period doubling bifurcation points

For a given logistic family $f_{\mu}(x)= \mu*x*(1-x),$ where $\mu \in [0, 4]$ and $x \in [0,1].$ This family undergoes the period doubling bifurcation. Let $\mu_{n}$ denote the value of $\mu$ where a $...
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What causes bifurcation?

Suppose let's say we have a recursive function, $$x_{n+1}=rx_{n}(1-x_{n})$$ From what I understand, the $x_{n+1}$ VS r graph starts to split (bifurcate) after a particular value of r (apparently it's ...
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find Bifurcation Diagram

Consider $$x'=\mu x-y+x^2$$ $$y'=x+\mu y+x^2$$ I need to find the bifurcation diagram. I need to find the fork diagram. For this I placed the system in polar coordinates finding $$r'=\mu r+r^2cos^2 \...
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Plotting parametric curve with implicit parameter function

I need to plot a Smale diagram as a parametric curve $(p(x), h(x))$ where $h$ is amended potential and $p = \omega^2$ is parameter of bifurcation: $$h:=V_\omega(x,y)=-2\cos(x)-\cos(y)-\frac{1}{2}\...
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Stability of the fixed point at the bifurcation point; is there a proof of this method?

In my nonlinear dynamics lecture, it has been stated that: In general, close to a bifurcation, we expand $f(x,\nu)$ up to linear in $\nu-\nu_c$ and up to first non-zero term in $x-x_*$, $\newcommand{\...
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What kind of local bifurcation occurs here?

I have encountered a bifurcation diagram for my five-dimensional non-linear ode system? I am really confused to identify the kind of bifurcation that occurs in the knowns forms of local bifurcations ...
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24 views

Identify bifurcation and sketch diagram

I am given the equation $$\dot x = rx + \frac {x}{1+x^2} $$ I believe I've found fixed points at $x=0$ and $x= \sqrt \frac{-1-r}{r}$ and I think its stable for $-1<r<0$ but I'm not sure how to ...
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Forced Duffing equation $\ddot x +x+\varepsilon(bx^3+k \dot x+ax−F\cos(t))=0$ bifurcation analysis

For the forced Duffing oscillator in the limit where the forcing, detuning, damping, and nonlinearity are all weak: $$\ddot x +x+\varepsilon(bx^3+k \dot x+ax−F\cos(t))=0$$ where $0<\varepsilon<&...
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Find the limiting circle of a Hopf Bifurcation point of a non linear oscillator

This problem deals with the nonlinear oscillator $$ y''+(y^2-\lambda)y'+(y+1)(y^2+y-\lambda)=0 $$Find the steady states and determine the values of $\lambda$ at which they are stable. Sketch the ...
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Bios and the creation of complexity

I found some information about Bios series on this PDF file: https://www.societyforchaostheory.org/resources/files/00005/Bios_tutorial.pdf There is an equation and some constant which generates Bios, ...
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Supercritical vs Subcritical Pitchfork Bifurcations

I'm having a really hard time classifying the following bifurcation points. The equation I was given is: $x'=r+x-1/3x^3$ And I found that there are two bifurcation points at: $(x,r) = (1,-2/3)$ and ...
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Bifurcation Diagram- The Logistic Map

I recently got to know about the logistic map, given by $f(x) = rx(1-x)$, and it’s bifurcation diagram when mapped as $r$ along the x-axis. At $r=3.56995 (approx.)$ we enter the chaotic part with some ...
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bifurcation in neural network training?

We use a simple network to approximate a single variable function to be used for trending. Everyday we have new measurement data to train the network so it will follow the trend. This same model is ...
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Continuous map between solution of ODEs

Below is a somehwhat informal question, which may require stronger conditions to make sense. Assume there exists two different first-order ODEs: $$\frac{\mathrm{d}y}{\mathrm{d}x}=F_1(x,y)\tag{1}$$ $$\...
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Sketching vector field for varying $r$ for: $\dot{x} = 1+rx+x^2$

I am examining exercise $3.1.1 $ of Strogatz in which it is asked to sketch the vector fields that occur as $r$ is varied and to show that a saddle-node bifurcation occurs at a critical value of $r$, ...
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1answer
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Is it possible to produce an example where Picard Lindelof existence and uniqueness theorem fails?

Taken from Picard–Lindelöf Theorem (Taken from Hirsch and Smale) (Pretty sure $F$ can be relaxed to locally Lipschitz) Note that the solution is unique on some unspecified interval $(t_0 - a, t_0 + a)...
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Codimensions and Bifurcation Theory

I am working through Introduction to Applied Nonlinear Dynamical Systems and Chaos by Wiggins. Specifically the section on the Idea of the Codimension of a Bifurcation. In the section we talk about ...
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346 views

Drawing bifurcation diagram

I'm considering the ordinary differential equation (ODE) $du/dt = a + u^2 - u^5$. I know that the number of fixed points varies based on the value of $a$, and I've identified the intervals of $a$ ...
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107 views

Periodic solutions of first order ordinary differential equations

Consider a differential equation given by $x'=f(x,t)$ Depending on the parameter $ t $, that is, not autonomous. I want to understand the behavior of the solutions to these equations. For example: ...
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Detecting bifurcations in the logistic map

Having just finished James Gleick's book "Chaos", I thought that I would have a play with examining the behaviour of the Logistic Map myself. I plan to use an FFT to create a spectrogram of ...
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Type of bifurcation when a saddle point collides with a center

Consider the following first-order autonomous system $$ \begin{align} &\dot{x}=y,\\ &\dot{y}=-x^2+\mu, \end{align} $$ where $\mu$ is a parameter. When $\mu>0$, the system has two ...
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Conceptual complex dynamics - Is it reasonable to perform bifurcation diagrams on PDE's?

I am working on a PDE model, the subject has been modeled with ODE's before. The articles usually have a bifurcation diagram, and to be able to validate my model, I want to compare my work with the ...
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Analysis of frequency and amplitude at Hopf bifurcation

I am analyzing the following system, where $I_{in}$ is a scalar parameter: $$ \begin{aligned} &\dot{V} = 10 \left( V - \frac{V^3}{3} - R + I_{in} \right) \\ &\dot{R} = 0.8 \left( -R +1.25V + 1....
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Learning Bifurcation Theory

I'm a physics graduate student. My interests are mainly statistical physics, so I usually deal with non-linear systems (both deterministic and stochastic). I did a dynamical system course, where we ...
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How to find the equation of the bifurcation curve of a cusp catastrophe?

The cusp catastrophe corresponds to the equation $$F(x,a,b)=x^4+ax^2+bx$$ where $a, b$ are the control parameters. The following diagram of cusp catastrophe shows the curves that satisfy $\frac{dF}{dx}...
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Are bifurcation diagrams meaningful without paramaters?

I am given a system of equations: $\dot{x} = -x^3 - y^2$ $\dot{y} = xy - y^3$ .. and asked to draw a bifurcation diagram. I dont think the ask is a valid one because bifurcation diagrams are only ...
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74 views

Find nullclines for nonlinear system [closed]

I have a nonlinear system and need find and plot nullclines: $$ \dot{x}=0.1(-x-1.8*10^{-3}Q(y)+1.3*10^{-3})\\ \dot{y}=0.1(-y-2.1*10^{-3}Q(x)+D) $$ here $Q(x)=\frac{100}{1+e^{(0.01-x)/0.003}}$ and D ...
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1answer
74 views

Find a critical point

I found a method to find a critical control parameter in one article and it worked for other similiar system with saddle-node bifurcation too. In general, we have a 2 equations nonlinear system: $$\...
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1answer
102 views

Find a system point

Here is given nonlinear ODEs system: where $I$ is a control parameter. When $I=I_c$ saddle-node bifurcation is happening. Need to find point $I_c$. By phase plain method via Matlab I ploted ...

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