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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Bifurcation Diagram for system with no bifurcation point

I'm currently studying bifurcation points and diagrams for non-linear dynamical systems. One question that my lecturer has given, clearly does not have a bifurcation point and therefore no bifurcation....
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Eigenvalue problem for linearized reaction-diffusion system

I recently started studying about reaction-diffusion system and Hopf bifurcation theory. I realize that Fourier transform/series is a quite useful tool here but it's been many years since I got ...
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A delayed differential equation system and its characteristic polynomial

I have the following DDE system: \begin{equation} \begin{split} \dot{x}_{1} &= -\mu x_{1}(t) + a_{11}f_{11}(\int_{-\infty}^{t} F(t-s) x_{1}(s-\tau_{1})ds) + a_{12} f_{12}(x_{2}(t-\tau_{1}))) \\...
neuralode's user avatar
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Poincare-Hopf bifurcation

I have the following system $$ X'=a-(b+1)X+X^2Y,\\ Y'=bX-X^2Y. $$ I have to prove that $(a,b/a)$ is an equilibrium point if $a^2+1>1b$ and unstable if $a^2+1<b$. Also i have to prove ...
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Bifurcation Analysis of Non-autonomous system

Suppose we have a matrix equation $$ \frac{d}{dx}\mathbf{u} = \mathbf{A}(x) \mathbf{u} + \mathbf{b}. $$ If $\mathbf{A}(x)=\mathbf{A}$ were constant, then one can inspect the eigenvalues of the ...
AQuestion's user avatar
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Existence of center manifold

I've been working on the following exercise: Prove that the system $$ \begin{cases} \dot{x} = -x^3,\\ \dot{y} = -y + x^2 \end{cases} $$ has no analytic center manifold (supposed in the following way $...
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nonlinear odes: stabilizing terms in a subcritical pitchfork bifurcation

I am reading through Strogatz's book on nonlinear odes and dynamical systems. One thing that is a little confusing is his description of stabilizing higher order terms to control the dynamics of a ...
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Homoclinic, heteroclinic orbits and nonwandering points

I have a relatively well behaved vector field and I can prove that all the nonwandering points along the flow are equilibria. Is this enough to disprove that there is no homoclinic and heteroclinic ...
giangian's user avatar
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How to find a function with several bifurcations of specified type

Suppose I want to construct a function $h(x,r)$ such that it exhibits a bifurcation of type $a$ at the point $(x_{1},r_{1})$ and bifurcation of type $b$ at the point $(x_{2},r_{2})$. One alogirhtm to ...
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Specific bifurcations of 3D ODE system

I'm looking for a list of all types of bifurcation that can exist in a 3D system of differential equations. As it has been well described for 1D and 2D systems, I'm interested in knowing specificities ...
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Unclear simplification step in bifurcation analysis of first-order system from Strogatz

I am going through the examples in "Nonlinear Dynamics and Chaos 2ed" (Strogatz 2015) and I don't understand the simplification step for the solution to example 3.2.1. Why is it that the ...
Jared Frazier's user avatar
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Dynamic system, bifurcation and equilibrium points

I'm a bit stuck in solving the following exercise: consider the dynamical system $$\dot{x} = 2 + 3\mu x - x^3$$ I have to find the equilibrium points and the stability type, and the the bifurcations ...
Martin and Friends's user avatar
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Deriving a Hopf Bifurcation – Perturbation Method vs. Jacobian Matrix

When deriving a Hopf bifurcation of a dynamical system, the usual process is: Find a fixed point $(x_0, y_0)$ Perturb the system about the fixed point $(x_0+\tilde{x}, y_0+\tilde{y})$ Linearize, ...
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Calculating Bifurcation points of trig equations

I am going through Nonlinear Dynamics and Chaos by Strogatz and I am getting stuck on a few questions: To calculate the stability of a fixed point: Work on where it intersects the x axis ( fixed ...
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Identify the Bifurcation of a map

This question is taken from Glendinning's textbook "Stability, instability and chaos": Given the map $x_{n+1} = \mu - x_n^2$, determine the type of bifurcation which occurs at $\mu = -\frac{...
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Bifurcation point at asymptote

I'm trying to find the critical points and bifurcation points of the following ODE: $\frac{\mathrm{d}x}{\mathrm{d}t} = \beta{x} - \frac{x}{1 + x}$ I know that there are two critical points, $x = 0$ ...
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How to compute stable / unstable manifolds when eigenvectors have both non-zero x and y components

In class I was taught to compute stable/unstable manifolds as done in the following example: We have the dynamical system: $$ \dot{x} = x-xy $$ $$ \dot{y} = -y+x^2 $$ We are interested in the ...
Jamal's user avatar
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Where do normal forms of bifurcations come from

In class we have studied $1D$ saddle, transcritical and pitchfork bifurcations. We did this by analysing their normal forms. For some parameter $\mu$, I have that the normal forms are: Saddle Node: $\...
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Are bifurcations preserved under Fourier transforms?

I am interested in studying bifurcation in a specific class of dynamical systems, which follow a change of variables and decoupling by a Fourier transform. In particular, consider the following ...
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Understanding a two-parameter bifurcation

Consider the non-linear system $$ \frac{d\mathbf{x}}{dt} =f\left(\mathbf{x},\mathbf{u} \right) $$ where $f: \mathbb{R}^{n+2}\mapsto\mathbb{R}^n$ and parameters $\mathbf{u}\in\mathbb{R}^2$. I am ...
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Bifurcation in a linearized system

Consider a general non-linear two-dimensional system of the form $$ \begin{aligned} \frac{dx}{dt}&=F(x,y)\\ \frac{dy}{dt}&=G(x,y) \end{aligned} $$ Given I linearize such a system around a ...
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Another derivative of an integral question.

Edit: look at the picture at the bottom first. I'm meddling with exercise 2. Please answer avoiding ODEs, since this course comes before ODEs in my school. So far, my asnwer to ex. 1 is the following: ...
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Bifurication curve and stability

Could you please explain for me this bifurication curve? Which part are stable and why? Many thanks in advance enter image description here
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Amplitude expansion close to bifurcation

Consider a non-linear two-dimensional system of the form $$ \begin{aligned} \frac{dx}{dt}&=F(x,y)\\ \frac{dy}{dt}&=G(x,y) \end{aligned} $$ In the context of stability analysis, what is meant ...
sam wolfe's user avatar
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Simplifying a complicated continued fraction expression.

This probably relates to continued fractions or numerical methods but I do not see how. There is probably some kind of induction or telescoping. $$\sqrt 3 = 1+\dfrac{2+\dfrac{3+\dfrac{4+\cdots}{5+\...
mick's user avatar
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Properties of the bifurcation diagram for the logistic function

Once the bifurcation diagram has been plotted ($x_{n+1}=rx_n(1-x_n)$), there are 3 elements or properties that I don't know haw to explain, and I have not found any article where they are explored. ...
Minerva González García's user avatar
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Periodic orbits of the logistic map

I have a question about the period orbits of the logistic map $f(x)=r \cdot x(1-x), r \in [0,4], x \in [0,1]$. The bifurcation-diagram own for $r<3.5699$ only periodic orbits of period $ p=2^k,k \...
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Showing orbital equivalence of two dynamical systems

I am given a particular normal form for a system with a Bogdanov-Takens bifurcation, namely $\dot{\xi} = f(\beta_1, \beta_2, \xi)$ given by \begin{align} \dot{\xi_1} &= \xi_2 \\ \dot{\xi_2} &= ...
Matthew Neil's user avatar
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Did low-degree nodes tend to connect to other low-degree nodes in networks which follow power law with an exponential cutoff?

Question Take a power-law network with an exponential cutoff as an example: $$P(z)\sim z^{-\alpha} e^{-z/z_c}$$ where $P(z)$ is the degrees of nodes, $z$ is the order of nodes, $\alpha$ is the power ...
LI Bing's user avatar
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How to find liapunov number of system

My book has the given example: The quadratic system \begin{align*}\ \dot{x}&=\mu x-y+x^2\\ \dot{y}&=x+\mu y+x^2 \end{align*} has a weak focus of multiplicity at the origin for $\mu=0$ since ...
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Rescaling one-dimensional map to normal form

(This is part of exercise 10.7.8 in Strogatz's Nonlinear Dynamics and Chaos that I am struggling to solve) QUESTION: Consider the map $x_{n+1} = f(x_n,r)$ where $f(x_n,r)=-r+x-x^2$. We use the ...
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Are there any known/researched bifurcations where time is the control parameter?

Bifurcations are usually determined compared to some external parameter $\lambda$ that can be independently controlled. But what if we take time to be our parameter, that is, the system experiences a ...
agaminon's user avatar
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3 answers
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Need help determining bifurcation points

So I have to find the bifurcation points of the system: $\dot{x}=(ax-x^3+x^5)(x-a+2)$, where $a\in\mathbb{R}$ is a parameter. Attempt: I know that a bifurcation point is the point, where there is a ...
sarah johnson's user avatar
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1 answer
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What is the bifurcation called in this very simple example?

The most simple kind of bifurcation studied in texts is generally a fold, characteristic of equations such as: $$\dot{y}=y^2+\lambda$$ There are two equilibria if $\lambda<0$, and no equilibria if $...
agaminon's user avatar
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2 answers
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Reason behind the names of sub and supercritical bifurcations

What is the reasoning behind the names sub- and super-critical bifurcations that occur in the context of pitchfork and Hopf bifurcations? Textbooks seem to introduce this terminology without any ...
timur's user avatar
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long term behavior of solution to a first order nonlinear differential equation

Consider the following system of first order nonlinear autonomous ODEs (derivatives are taken with respect to $t$): $$ \begin{cases} \dot{x} = -2xy^2+1 \\ \dot{y} = -2x^2y+1 \\ x(0) = x_0,\;y(0)=y_0 \...
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Reference for singular Jacobian being a necessary condition for bifurcation

I am looking for a reference that state and prove that a bifurcation in a system $F(x, \mu) = 0$ can only appear if $\det(J) = 0$, where $J$ is the Jacobi matrix of $F$ with respect to $x$. I looked ...
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Find and classify bifurcation points for a cubic ODE

I have an ODE which can be written as $x' = g(x) = x^3 + px + q$ where $p=-\frac{3c}{A}$ and $q=-\frac{3d}{A}$ (we can assume $c>0,d\neq. 0$) and I am trying to find and classify the bifurcation ...
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A kind of pertubation of laplacian is an Fredholm operator

Let $U, V$ be Banach spaces and $L(U,V)$ be the space of bounded linear operators from $U$ to $V$. An operator $T \in L(U,V)$ is said to be a Fredholm operator if dim $N(T) < +\infty$ and $\mathrm{...
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Variants of "3n + 1 problem" that fall into a 1-node cycle

Background Motivation Based on a previous question, I became interested in "3n + 1 problem" variants using $n \mod 4$ instead of even/odd (i.e. $n \mod 2$). One particular concept is based ...
mattrdowney's user avatar
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$x(x-1)^2+\alpha$ bifurcation diagram [closed]

I have to find all the equilibrium points of $$\dot{x}=x(x-1)^2+\alpha$$ and sketch the corresponding bifurcation diagram, but I don't see how to start, since the roots of this polynomial don't have '...
Daniel Checa's user avatar
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Question studying Hirsch, Smale, and Devaney's "Differential Equations, Dynamical Systems, and an Introduction to Chaos"

I am studying Hirsch, Smale, Devaney; Differential equations, dynamical systems, and an introduction to chaos (2nd edition) and have a couple of questions. First, in Section 3.3 about repeated ...
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Non-hyperbolicity implying bifurcation

I have come to know that a bifurcation point must be non-hyperbolic. But I am not sure whether the converse is also true. As I considered $f(x)=\mu-x^2$, I got $x^2=\mu$ are bifurcation points. i) For ...
Manjoy Das's user avatar
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Plotting the bifurcation diagram for Ikeda map

I'm trying to plot the bifurcation diagram for Ikeda map. I wrote a code in Python to get the points of this diagram, but it seems that for $u > 1$ the points diverge and my code doesn't work ...
Herr Schrödinger's user avatar
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center manifold and bifurcation: 2D Bifurcation system reduction

I have this system to study $$ \left\{ \begin{aligned} \frac{dx}{dt} &= y-x - x^2 \\[5pt] \frac{dy}{dt} &= \mu x - y - y^2 \end{aligned} \right. $$ I have derived the Jacobian around fixed ...
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Period doubling on Möbius strip

I have a hard time understanding a simple argument for the following. Quite often I find in papers that limit cycles before and after period-doubling bifurcations lie on the Möbius strip in the state-...
struct's user avatar
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How to determine bifurcation diagram for following system of difference equations?

So given this system, $\begin{gather*}N_{t+2}=N_t\exp[r(1-\frac{N_t}{K})]\frac{1-e^{-aP_t}}{aP_t}\\ P_{t+1}=N_t[1-\frac{1-e^{-aP_t}}{aP_t}]\end{gather*}$, how do I determine the bifurcation diagram? ...
user5896534's user avatar
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How to find fixed points for an equation with $\ln$ term

Find fixed points $$x'= M + x - \ln(1+x)$$ First step:1 $$0 = M + x - \ln(1+x)$$ I know when $M > 0$, there are no fixed points.            when $M < 0$, there are two fixed points  but I do not ...
Mohammed Alanazi's user avatar
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Taylor expansion of center manifold with respect to a central variable and control parameter

I am being asked to find the Taylor expansion up to order two with respect to $(w, \mu)$ of the center manifold for the following ODE: $\frac{dx^2}{dt^2}+\frac{dx}{dt}-\mu x + x^2=0$, where $\mu \in \...
anna's user avatar
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Examples of Pitchfork Bifurcation in nature?

I was just wondering if anyone had some nice examples of pitchfork bifurcation in nature! For Hopf Bifurcations, for me the classical example is cylinder flow. What is the same for a pitchfork ...
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