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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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Excitability of the FitzHugh-Nagumo model

I have the following variant of the FitzHugh-Nagumo model: $$\dot{u} = u - u^3 - v \\ \dot{v} = \epsilon(u-a)$$ Where $\epsilon>0$ and $a$ is a constant. I need to give a value for $a$ such that ...
CauchyChaos's user avatar
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Characterize Bifurcation in Nonlinear ODE

Consider the mapping $F: H^1_0(\mathbb R_+) \times \mathbb R \to H^1_0(\mathbb R_+)$ for an ODE with solutions $(u_E,E)$ that satisfy $$ F(u_E,E) = 0.$$ Suppose there is a solution iff $E > E_0 >...
jacktrnr's user avatar
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Logistic map: bifurcation and domain of attraction

Let $f(x) = \mu x(1-x)$ be the logistic map, the question is divided into 3 parts: Part (1): what can you say about the domain of attraction of the 2-cycle in $3<\mu<1+\sqrt 6$? My attempt: let $...
vegetandy's user avatar
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Sotomayor's Theorem for Discrete Dynamical Systems?

One of my research projects has recently taken a leap forward using Sotomayor's Theorem, which provides sufficient conditions for a saddle-node bifurcation to occur in an n-dimensional ODE without ...
skier456's user avatar
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Asymptotic methods to find how fold bifurcations evolve

Consider a dynamical system: $\dot{x} = f(x,\lambda,\Lambda)$. For a specific value of $\Lambda=\Lambda_1$, the system admits a fold bifurcation at $(\lambda_1,x_1)$. I want to find out two things, I ...
George's user avatar
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Finding critical value of bifurcation

I have equations such as $\frac{dx}{dt} = y \\ \frac{dy}{dt} = \mu y + x - x^{2} + xy$ This system is known to be homoclinic bifurcation at the origin. To find the critical value of $\mu_{c}$, one ...
이영규's user avatar
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Bifurcation of the dde $x'(t) = r x(t-d) - h x(t)$ wich describes a population model with delay

Consider the following population model: $x'(t) = r x(t-d) - h x(t)$, with growth rate $r > 0$ and death rate $h > 0$, where only adults that have reached the age $d > 0$ can reproduce. I ...
L731289's user avatar
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Birth-death : Always more than 1 bifurcation?

Say I have a (smooth) function $f : \mathbb{R}^n \to \mathbb{R}$, and a critical point $x$ (ie, $f'(x) = 0$). I call this point degenerate if $\det \text{Hess}_x f = 0$ (so, equivalently, if the ...
Azur's user avatar
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Bifurcation in accumulation points of sequence

I stumbled upon what seems like an interesting bifurcation. I wonder if anyone has an explanation for this. Here are the details: Let $a_n\in\mathbb{R}$, $n \in \mathbb{N}$ be a sequence. Define a new ...
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Identifying Bifurcation

I am trying to identify bifurcation of my $3D$ system near $a=6.58$. I am getting trajectories as shown in the picture and I am guessing it is Saddle-Node periodic orbit bifurcation since I am getting ...
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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 ...
kaithkolesidou's user avatar
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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 ...
Gonzalo de Ulloa's user avatar
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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
3 votes
1 answer
186 views

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 $...
Donatello's user avatar
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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 ...
Mani's user avatar
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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 ...
Heidegger'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, ...
Steven Basmith's user avatar
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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 ...
SS1's user avatar
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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{...
Jamal's user avatar
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1 answer
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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: $\...
Jamal's user avatar
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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: ...
agus's user avatar
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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 ...
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5 votes
1 answer
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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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1 vote
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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
2 votes
1 answer
419 views

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 \...
David's user avatar
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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
1 vote
0 answers
56 views

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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-1 votes
1 answer
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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 ...
sarah johnson's user avatar
1 vote
0 answers
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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 ...
tutentaten's user avatar
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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
1 vote
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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1 vote
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 \...
William's user avatar
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0 answers
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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 ...
Kolaru's user avatar
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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 ...
Warrenmovic 's user avatar
3 votes
0 answers
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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{...
Lucas Linhares's user avatar
1 vote
0 answers
174 views

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
0 votes
2 answers
157 views

$x(x-1)^2+\alpha$ 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
1 vote
0 answers
300 views

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 ...
Quiet_waters's user avatar
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1 vote
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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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3 votes
1 answer
218 views

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
2 votes
0 answers
60 views

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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