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)

Filter by
Sorted by
Tagged with
0 votes
0 answers
6 views

Does a spiral fixed point going from stable to unstable always indicate Hopf bifurcation?

Does a spiral fixed point going from stable to unstable (with change in some parameter) always indicate Hopf bifurcation? Could it be a homoclinic bifurcation instead? (Homoclinic bifurcation keeps ...
user avatar
  • 665
3 votes
1 answer
30 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 ...
user avatar
2 votes
0 answers
27 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 ...
user avatar
1 vote
0 answers
25 views

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-...
user avatar
  • 141
0 votes
0 answers
15 views

How to get Poincaré mapping matrix?

I'm researching a system like this: $$ \begin{equation} \begin{cases}\label{eq:main_eq} \dot x_{1} = y_{1} + bx^{2}_{1} - ax^{3}_{1} - z_{1} + I - k_{1}(x_{1} - v_{s})\Gamma(x_{2}) + k(x_{2} - x_{...
user avatar
  • 13
0 votes
0 answers
17 views

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? ...
user avatar
0 votes
1 answer
29 views

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 ...
user avatar
1 vote
0 answers
21 views

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 \...
user avatar
1 vote
0 answers
33 views

Finding the center manifold for a 2D dynamical system

I solved many cases for the following dynamical system $\dot{x} = x (1-x-ay)$ and $\dot{y} = c y (1- b x -y)$. However, I reached the case where $c>0$ and $a>1$, $b=1$ and I ended up with the ...
user avatar
1 vote
1 answer
63 views

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 ...
user avatar
1 vote
1 answer
60 views

The Jacobian and bifurcation

We've learnt in class that if the determinant of the Jacobian matrix of an equation or ODE is zero, the system/equation has a singular point, and this means that it undergoes bifurcation in that point....
user avatar
2 votes
1 answer
58 views

Understanding where does the second (stochastic) attractor of the system come from.

I am currently reading a paper, studying the population dynamics in 3-dimensional Lotka-Volterra model with the following interaction change descriptive system: $$ \begin{equation} \begin{cases} ...
user avatar
3 votes
1 answer
166 views

2D bifurcation problem

I come across this problem which is about bifurcation. I am trying to take all the cases. I am expecting Hopf bifurcation to occur here but the last case I could not find the fixed point. Could you ...
user avatar
1 vote
0 answers
59 views

Analysis of Time-Varying Differential Equation

I'm trying to analyze the a differential equation of the form: $$ \dot{x} = -K_1\sin(\omega t + x) - K_2\sin(x) $$ where $K_1 > 0$, $K_2 > 0$, and $\omega >0$ are positive real constants and $...
user avatar
0 votes
0 answers
12 views

Neutral saddle cycle and its interpretation in codim 2

Suppose the following scenario. 1-parameter limit cycle family at paramater value $\mu_1$ undergo neutral saddle bifurcation, i.e. Floquet multipliers satisfying $\mu_i\mu_j = 1$ and it touches ...
user avatar
  • 141
1 vote
1 answer
37 views

Melnikov's method, homoclinic orbits, and bifurcation values

In nonlinear dynamics, Melnikov's approach provides an intriguing way to detect homoclinic bifurcations and bifurcation values, i.e., the values of the parameter at which a dynamical system exhibits ...
user avatar
2 votes
0 answers
30 views

Show $h$ and $g$ are commutative in the canonical form of ODE.

I come a cross this problem in my nonlinear analysis course. I know how to find the normal forms of any order. However, the commutative isometry! And in the third point the professor put two ...
user avatar
2 votes
0 answers
77 views

I want to understand the characteristics of a particular non-linear hierarchical dynamical system

I would like to study the characteristics of the non-linear dynamical system detailed below; in particular, I would like to find its set of fixed points; and to compute (i) the maximum values of ...
user avatar
  • 867
0 votes
0 answers
25 views

Finding Hopf bifurcation points for a system of 3 ODEs?

I have the following system of ODEs: dx/dt = a/(1 + z) - Q*z dy/dt = Qx - qy dz/dt = qy - cz/(K + z). Assuming K = 1, Q = q < c, and a = c*(Sqrt[c/Q] - 1), how can I show where the Hopf ...
user avatar
0 votes
1 answer
42 views

Bifurcation Diagram and Logistic Map

I want to create a Bifurcation Diagram with the Logistic Map and I have open questions about the correct algorithm. Here is how I understood it so far: logistic map $x_{i+1} = rx_{i}(1-x_{i})$ take a ...
user avatar
  • 101
0 votes
0 answers
44 views

Why does the Mandelbrot Set only bifurcate on real values?

If you take a look at the figure below, the Mandelbrot Set seems to only bifurcate when it collapses into a real value (without an imaginary component). Why does this occur?
user avatar
0 votes
0 answers
16 views

Solving algebraic variety instead of homotopy continuation methods

Consider following ODE system with polynomial nonlinearity \begin{equation} \dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \begin{bmatrix} 0\\ \vdots\\ 0\\ ax_i + bx_i^3 + ax_i + bx_i^5 \\ 0\\ \vdots\\ 0 ...
user avatar
  • 141
1 vote
0 answers
28 views

Limit cycle continuation folding over itself

I was doing some computations in MATCONT. First I continued equlibrium point. Hopf bifurcation occured for $a = 6$ and branching point/pitchfork for $a = -2$. Continuing forward and backward this ...
user avatar
  • 141
0 votes
0 answers
27 views

How to prove that you can ignore the complex coefficient when finding periods in Mandelbrot Set?

So essentially, I am under the impression that if any Ai + B has a period of k, then B will have a period of k as well. For example, -0.2i - 0.5 has a period of 1, hence, -0.5 has a period of 1. ...
user avatar
0 votes
1 answer
63 views

Difference between fixed point and equilibrium point!

Let us consider a first order, autonomous and nonlinear DE: $$\frac{dx}{dt}=f(x)$$ when will we get an equilibrium point and fixed point? Are they the same? I need some clarification.
user avatar
  • 153
0 votes
0 answers
25 views

Is there any set of ODE's with parameters in which I can find saddle-node, transcritical and pitchfork bifurcations all together?

All three bifurcations have different normal form. Saddle-node bifurcation: dx/dt =r − x^2. Transcritical bifurcation : dx/dt = rx- x^2. Pitchfork bifurcation: dx/dt = rx-x^3. I do have a set of 2 ...
user avatar
1 vote
0 answers
24 views

Infinite periods in bifurcation diagram

A slightly more theoretical question for you all. Recently I was looking at the logistic map and the resulting bifurcation diagram (shown). Wikipedia says that prior to roughly r = 3.56995, there is a ...
user avatar
3 votes
0 answers
39 views

Bifurcations where eigenvalues become purely imaginary

Say a system of 2 ODE's has the following Jacobian at a fixed point $(0,0)$, $$ Df_{\mu}(0,0) = \begin{pmatrix} -(1/2 - \mu) & 1/2+\mu \\ -(1/2 + \mu) & 1/2-\mu \end{pmatrix} $$ with ...
user avatar
0 votes
0 answers
19 views

Bifurcation analysis od DAE

Are there any analytical results in the bifurcation theory od algebraic-differential equations similiar to an ODE eigenvalue toolkit? I mean stability testing, limit cycles detection etc. Because I ...
user avatar
  • 141
0 votes
0 answers
57 views

What is wrong with this bifurcation plot?

I am trying to plot the bifurcation diagram based on the steady-state values of the following ODE: $$\frac{dx}{dt}=x(1-x)-rx$$ I take the right hand side $f(x) = x(1-x)-rx$ and I find the fixed points ...
user avatar
  • 51
0 votes
0 answers
56 views

Semi-stable fixed points in plane

Suppose that we have a system of 2 difference equations depending on some parameter and that $(x^*,y^*)$ is non-hyperbolic equilibrium point with one eigenvalue equal to 1 (which appears for the ...
user avatar
  • 415
0 votes
0 answers
40 views

Conditions for a transcritical bifurcation to occur in a 1D dynamical system

Suppose we have a 1D dynamical system $\dot{x}=f(x,\mu)$. What are the most general conditions on $f$ and its derivatives which guarantee that a transcritical bifurcation occurs? I have seen ...
user avatar
3 votes
0 answers
60 views

Bifurcated limit cycles from real eigenvalues

I have never heard of limit cycles bifurcated from real eigenvalues crossing zero. For the below system I believe that is the case. Is this bifurcation known and has a specific name? For $d = b + i c \...
user avatar
  • 417
1 vote
1 answer
91 views

Analyzing bifurcation point of a 2D dynamical system

I have the following nonlinear dynamical system: \begin{align} \frac{dX}{dt} &= Y(\gamma - 2X) \\\ \frac{dY}{dt} &= - \gamma X + X^2- Y^2 + 1 \end{align} And I'm trying to understand what type ...
user avatar
1 vote
0 answers
32 views

Reference for one-dimensional "singular" implicit function theorem (i.e. the shape of a nondegenerate saddle-node bifurcation)

I think I can easily prove the seemingly very basic result below. For a paper, I would like to be able to cite a reference for it, rather than having to either write out a proof [which would be ...
user avatar
2 votes
0 answers
35 views

Equation for curves of high density in bifurcation diagram

Is there an equation for the high density curves in the chaotic regions of the bifurcation diagram for the logistic map? I'm talking about the sinusoidal-looking dark curves in the following picture. ...
user avatar
2 votes
0 answers
68 views

Period-doubling bifurcation theorem

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} ...
user avatar
  • 151
0 votes
0 answers
37 views

What do you mean by bifurcation of the following equations/systems?

I am new to the domain of bifurcation. I encounter the following problems: Analyze the bifurcation of the following equations/systems. Identify the bifurcation values(s), describe the bifurcation and ...
user avatar
  • 11.9k
1 vote
1 answer
104 views

Bifurcation diagram of $\frac{dx}{dt} = r - \cos(x)$

I plotted several vector fields for different values of $r$, specifically: $$r = 1$$ This gave me half-stable fixed points at $2k\pi, k \in \mathbb{Z}$. $$r > 1$$ No fixed points. $$-1<r < 1$$...
user avatar
  • 403
2 votes
2 answers
298 views

Two-parameter bifurcation diagram

In an assignment I am currently doing, we are considering a system of the form $$\dot{x}=f(x,\mu,\delta).$$ In class we have been confronted with systems of the form $$\dot{x}=f(x,\mu),$$ where $\mu,\...
user avatar
  • 563
3 votes
0 answers
40 views

About the stability of cycles for the logistic map $x_{n+1}=rx_n(1-x_n)$

Please indicate a reference (if there exists any) proving that when a cycle appears (i.e. for the minimal value of the logistic parameter $r$ for which a $k$-cycle exists) we also have, "...
user avatar
  • 1,170
0 votes
1 answer
77 views

How to know when a transcritical bifurcation occurs (Example 3.2.1 Strogaz Nonlinear Dynamics and chaos)

Picture of Question + solution Hi, In this question After we expand $\dot{x}$ around $x^*$ = 0 , we get $\dot{x}$ = (1-ab)x + ($\frac{1}{2}$a$b^2$)$x^2$ + O($x^3$). How do we jump from this to knowing ...
user avatar
2 votes
0 answers
39 views

Are 'eye' bifurcations possible?

I am currently studying bifurcations in dynamical systems and have seen saddle-node, transcritical and pitchfork bifurcations. My question is can you have a system which has an 'eye-shaped' ...
user avatar
  • 33
1 vote
0 answers
28 views

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 ...
user avatar
  • 11
3 votes
1 answer
121 views

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 ...
user avatar
  • 193
2 votes
1 answer
89 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} ...
user avatar
  • 151
0 votes
0 answers
49 views

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?
user avatar
  • 105
3 votes
0 answers
44 views

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 ...
user avatar
2 votes
0 answers
49 views

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 ...
user avatar
4 votes
2 answers
142 views

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 $\...
user avatar
  • 141

1
2 3 4 5
7