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1answer
42 views

Hysteresis Model with two real parameters

I would like to ask the following: I am trying to make a throughout analysis of a Hysteresis model in one dimension, with two real parameters: $\frac{dx}{dt}=f(x,ν,μ)=νx-x^3+μ$, where ...
0
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0answers
31 views

Are they stable or unstable limit cycles?

I am using cl_matcont to perform a bifurcation analysis of a dynamical system of ten equations (equations are identical in two blocks, thus 8 of them and 2 of them are the same) During the ...
3
votes
1answer
63 views

Hopf bifurcation and limit cycle

I am studying bifurcation and had a system like this: $$dx/dt=ux-y-x(x^2+y^2),$$ $$dy/dt=x+uy-y(x^2+y^2).$$ I want to determine whether a Hopf bifurcation would occur. I wrote the system into polar ...
0
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1answer
53 views

Manifold projection to 2m+1 dimensional subspace is a manifold.

Let $M \subseteq \mathbb{R}^n$ be a m-dimensional manifold. Suppose $n>2m+1$. Show that there is a projection from $M$ to a (2m+1)-dimensional subspace of $\mathbb{R}^n$ so that the image is ...
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0answers
40 views

Lyapunov-Schmidt reduction.

Use Lyapunov-Schmidt reduction to find an expression, or approximation, of the set of equilibria, as a function of the parameter $\lambda$, of the planar vector field ...
2
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2answers
84 views

What kind of bifurcation occurs for $\mu=-1$ for $f_\mu(x)=\mu+x^2$?

Let $f_\mu(x)=\mu+x^2$. What bifurcation occurs for $\mu=-1$? Pretty straight forward, but I'm having a hard time with this entire section in my book. It's not making any sort of sense and the ...
2
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0answers
75 views

Show that the family $E_\mu$ undergoes tangential bifurcation

Let $E_\mu(x)=\mu e^x$. Show that the family $E_\mu$ undergoes tangential bifurcation at $\mu=1/e$. In particular follow out the following steps: (a) Plot out the diagonal and the graph of $E_\mu ...
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0answers
28 views

Bifurcation Diagram(f '(x)=r-cosh(x))

I am trying to write a bifurcation diagram for f'(x)=r-coshx and I am having a little trouble. I also have to show that a saddle-node bifurcation occurs at a critical value of r. So, I know ...
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0answers
58 views

Bifurcation diagram of periodic function

Let $f(t)$ be a $T$-periodic function. Sketch the bifurcation diagram of the differential equation $$x'=-ax+f(t)$$ Please show all steps.
4
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1answer
88 views

Period doubling bifurcation in a quadratic map

I am attempting the find $\mu$ for which the map $$x_{n+1} = \mu + x_n^2$$ undergoes a period doubling bifurcation. I understand that finding the fixed points of the map is the first step towards ...
1
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0answers
36 views

PDE and Taylor's formula

I'm looking to a prove that a function that satisfies the following equations is actually $f(x,t)=x^3 \pm tx$ after changing coordinates. Here are the equations: 1) $\frac{\partial^3 f}{\partial ...
1
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0answers
26 views

Bifurcation in PDE

How do we characterize bifurcation in nonlinear PDE instead of ODE i.e. ht=f(x,h,hx,hxx,hxxx,...)? For example, study the temporal evolution of a regular pattern into a chaotic one. Can someone please ...
1
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1answer
48 views

Prove there are exactly three solution

In this question, the OP asked to find the solutions of: $$a^{−x}+{\log x \over \log a}=0$$ In my answer, I showed that when $a<e^e$ there can be at most one solution. It is also clear that any ...
3
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0answers
52 views

Period 3 window in chaos

Li and Yorke established the fact that period 3 implies chaos, which implies that if we have a period 3 orbit in a system then we have a chaotic system. I have seen that in bifurcation diagrams there ...
0
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0answers
36 views

Reflection Symmetry for Non-Linear Differential Equations

We are given the equations: \begin{align} \dot{x}& =\mu \, x +y+y^3 \\ \dot{y}& =2x-2y+xy^2+\gamma \, x^2y \end{align} The question at hand is to determine whether there is some sort of ...
1
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0answers
26 views

How to plot a bifurcation graph?

Can anybody tell me how best to plot a bifurcation graph? What steps must I follow? Are there some papers in which I can find some help?
1
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1answer
17 views

What is symmetry in bifurcation analysis?

I did a quick google but I couldn't find much. Could someone please explain when a system has symmetries or link me to some good resources? For example, the system $x'=\mu x-y+x^3$, $y'=bx-y$ has ...
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0answers
26 views

Bifurcation - transforming linearised matrix to normal form

I have the following system \begin{matrix} x′=ux-y+x^3\\ y'=bx-y \end{matrix} I've already determined the position and stability of the critical pts as $u$ and $b$ are varied. I've also worked out ...
1
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1answer
59 views

locate any bifurcation in the $2D$ system?

bifurcation for the following $2D$ system: $$\left\{\begin{matrix} x′=ux-y+x^3\\ y'=bx-y \end{matrix}\right.$$ I have got $ux-y+x^3=0,\ y=bx$, then $x=0\ \ \text{and}\ x = \pm \sqrt{b-u}$. But I ...
1
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0answers
36 views

Help Finding Critical Points of a Cubic (with 2 parameters)

I am trying to find bifurcation points in 1 dimension, but am having trouble finding critical points of $x'=\mu x -2x^2-x^3+ \delta$ ( where $x$ is my variable, $\mu$ is a parameter, and $\delta$ ...
1
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1answer
38 views

Bifurcation flow field?

assuming I have $$x'=5+mx+2x^2$$ how would I find the flow field of this bifurcation with the changing variable m? EDIT: Take for example ...
1
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1answer
45 views

Onset of n-cycles in the logistic function

The logistic function f(x) = r x (1 - x) is well known in dynamical systems theory. So is the famous bifurcation diagram for it. I recently came upon an article in Mathematics Magazine from 1996 ...
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0answers
54 views

AUTO Software for ODE's: references or forums?

I'm learning the AUTO software that does numerical continuation of ODE's by following these two references: The official manual found here www.dam.brown.edu/.../auto07p.pdf Lecture notes found at ...
1
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2answers
57 views

bifurcation in differential equations

I have an equation $$ y' = (2+y)(k-y^2) $$ and I am asked to find the equilibrium solutions and bifurcation values for all values of $k$. My approach was that... there exists $2$ equlibrium solutions ...
1
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1answer
133 views

Logistic Map Bifurcation diagram

So in the bifurcation diagram of the logistic map, there is period doubling from about r=3 to about r=3.54409. There are two fluctuation points between r=3 and r=$1+\sqrt{6}$. My question is, how ...
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2answers
168 views

Logistic map bifurcation

Ok I am trying to do this on matlab, but I need to understand how to find the bifurcation values for logistic map by hand first. So here is the logistic map: $$ x_{i+1} = f(x_i) \qquad \text{where} ...
0
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1answer
100 views

book recommendations for further learning about dynamical system and bifurcation

I have read the book "Introduction to Applied Nonlinear Dynamical Systems and Chaos"by Stephen Wiggins.Could someone recommend books on dynamical systems and bifurcation theory for further learning?
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2answers
122 views

Bifurcation values for logistic map

To find the bifurcation values for $$x_{i+1}=f(x_i) = rx_i(1-x_i)$$first I set $rx(1-x) = 0$ and found the x values and then used the x values to find $r = 0$ and $r = 1$. Do you think what I did ...
0
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1answer
682 views

Matlab code for logistic map bifurcation

Take the logistic map $$x(i+1) = f(xi), f(x) = rx(1 − x)$$ Find numerically the $r$ values for the first two bifurcations. I know I have to set $f(x)=0$, solve for $x$, and then use the $x$ values ...
0
votes
1answer
54 views

Value of b which moves quadratic from a stable fixed point to a stable period 2 point

I have a question which asks: At what value of b does the quadratic 1.5x^2+bx+2.34 move from having a stable fixed point to having a stable period 2 point? How would I go about solving this?
1
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0answers
15 views

n-body systems and bifurcations

From what I understand, in bifurcation theory, one definition of the equivalence of two dynamical systems is that they are topologically equivalent. However if say proteins A, B start out as straight ...
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0answers
70 views

Bifurcations caused by a single additive term

Motivation: A practical dynamical system is often described by an ODE that has a parameter that controls the "power flow" into the system. When no power flows into the system, nothing interesting ...
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0answers
22 views

Bifurcation at an undefined point

I have a 6d system of dynamic equations. I am able to calculate numerically the steady states and evaluate their local stability. It turns out that everything depends on one parameter, call it C. If ...
2
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0answers
60 views

Hopf Bifurcation of Reaction-Diffusion System

I'm considering the following reaction-diffusion system: $ \frac{\partial u}{\partial t} = f(u,v)+ D_1 \frac{d^2 u}{dx^2} $ $ \frac{\partial v}{\partial t} = g(u,v)+ D_2 \frac{d^2 v}{dx^2} $ where ...
4
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1answer
138 views

Bifurcations for 1-dimensional map

Consinder the 1-dim map $F(x,\mu)=\mu-\frac{1}{4} x^2$ as $\mu$ increases from $-\infty$ to $5$ and analyse the bifurcations. I start the analysis by considering $F^2(x,\mu)=\mu^2-\frac{1}{2}\mu ...
2
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0answers
38 views

solving for bifurcations in nonlinear 2d system?

If we have this nonlinear system $$ x ̇ = −x + 0.1y + x^2 y $$ $$ y ̇ = b − 0.1y − x^2y $$ What kind of bifurcations happen as you vary b ∈ [0,1]? I managed to solve for equilibriums at the source ...
1
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1answer
77 views

Physical intuition for period-doubling bifurcations

I'm currently studying period-doubling bifurcations, and being a physicist I want some physical intuition for what they are! From what I've gathered, the system spontaneously undergoes a ...
1
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2answers
414 views

Finding all bifurcations in a 2D system

I want to find all bifurcations for the system: \begin{align} x' =& -x+y\\ ...
2
votes
1answer
75 views

Finding a Hopf Bifucation with eigenvalues

I am trying to show that the following 2D system has a Hopf bifurcation at $\lambda=0$: \begin{align} ...
1
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3answers
116 views

Generating Bifurcation Animations

https://en.wikipedia.org/wiki/File:Hopf-bif.gif Does anyone know how this animation was produced? I could make it by stitching together snapshots (what I'm doing) but this seems primitive, especially ...
5
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0answers
192 views

Bifurcation of integral curves

Consider the following first order ODE: $$\frac{\operatorname{d}\!y}{\operatorname{d}\!x} = x^2 - y^2$$ Despite the fact that this ODE has a very simple expression, it is not solvable in terms of ...
2
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0answers
82 views

Does this type of bifurcation exist?

I've been checking out numerically an ODE model of a gene circuit. Just from simulations, it appears that once a parameter passes some critical value a stable fixed point splits into three other fixed ...
3
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0answers
96 views

What is this bifurcation?

I have a discrete dynamical System $x_{n+1}=f(x_{n},x_{n-1},x_{n-2},x_{n-3},x_{n-4},\lambda)$ with a paramteter $\lambda>0$, and where all $x_{n}$s are in [0,1]. $f$ is actually a larger ...
2
votes
1answer
198 views

Help with brute force method of producing bifurcation diagrams of discrete-time systems

I have a homework question concerning a brute force method of creating bifurcation diagrams. This seems really abstract for me and would like a clearer description of how the method works. Can someone ...
0
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1answer
375 views

Relationship between Turing bifurcation, saddle-node bifurcation, and Hopf bifurcation?

Quoting from http://jxshix.people.wm.edu/2009-harbin-course/mississippi-bifurcation-2.pdf a Turing bifurcation occurs when for an ODE and related PDE $u' = f(u,v), v' = g(u,v)$ $u_t = d_1 \nabla ...
3
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1answer
928 views

Possible ways to do stability analysis of non-linear, three-dimensional Differential Equations

For example Lorenz system, $$ \frac{d}{dt}\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} -\sigma & \sigma & 0\\ \rho & -1 & -x\\ y & 0 & -\beta ...
4
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2answers
232 views

Chaos without period doubling

I have been studying the Duffing oscillator rather intensively lately, mainly based on the theory in of the book by Guckenheimer and Holmes. From all that I have gathered, it seems that most dynamical ...
1
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0answers
27 views

Divergent limit cycle frequency along a hopf boundary

I am studying the linear stability of continuous system as a function of two parameters (a and b) and I observe that a hopf bifurcation with frequency w happens along the line described by f(a,b). ...
4
votes
1answer
248 views

What is a good text on bifurcation theory?

What is a a good text on bifurcation theory for mathematicians who haven't seen it before? I'm looking to get a feel for the intuition behind the subject, major standard theorems, etc. I do not mind ...
1
vote
1answer
80 views

Is there a bifurcation node in the activation energy of chemical reactions?

I was thinking about the activation energy of chemical reactions (obviously), and I was wondering if there exists a bifurcation node somewhere in the transition state. Let me give you a link to an ...