# 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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### A delayed differential equation system and its characteristic polynomial

I have the following DDE system: \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}))) \\...
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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 ...
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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 ...
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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 ...
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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 '...
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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 ...
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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 ...
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 ...