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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Sketching bifurcation diagrams.

Consider the system: $$\dot{x}=x(\mu - x + y^2), \ \dot{y} = y(1-x+y^2)$$ I've been asked to consider bifurcations of the fixed points $(\mu, 0)$ and $(0,1)$ at $\mu = 1$ and $\mu = -1$ ...
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21 views

Bifurcations of dynamical systems, different parameters [closed]

I've found the following excercise and I've broken my head about it, but I don't know how to answer it. So we have a system $$\frac{dx}{dt} = f(x,a_1,a_2,...,a_n)$$ And say we first take $a_1$ as ...
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16 views

Draw the orbits in the {$\rho,\theta$}-plane and describe the evolution in each cases, classfying subcritical and supercritical if appropriate.

Given $$\frac{dA}{d\tau}=\sigma A-\beta A|A|^2, $$ where $\sigma=\sigma_r+i\sigma_i$, $\beta$ is real and $A(\tau)=\rho(\tau)\exp(i\theta(\tau))$. Draw the orbits in the {$\rho,\theta$}-plane and ...
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28 views

What type of Hopf bifurcation takes place here?

Consider the system: $\dot{x} = \mu x-y-xy^2-x^3$ $\dot{y} = x+\mu y - x^2y-y^3$ I have shown that a Hopf bifurcaiton takes place at the origin $(0,0)$ as a stable spiral becomes an unstable spiral ...
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28 views

How is this an untable limit cycle?

I am investigating the Lorenz equations and in MATLAB I have plotted a case with $\sigma = 10, b = 8/3, r = 21$ and I have this phase portrait: However I am not exactly sure how this is an unstable ...
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52 views

Is this a spontaneous symmetry-breaking?

I have a system of equations: \begin{cases} f\left(x_{1}\right)+f\left(x_{2}\right)+P=0\\ \\ g\left(x_{1}\right)+g\left(x_{2}\right)=0 \end{cases} where $f,g$ are some functions, ...
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26 views

5-dimensional First Order Nonlinear ODE stability analysis

I have a system of 5 first order nonlinear ODEs, with 2 nonzero equilibria. Is it possible to perform a stability/ bifurcation analysis with 10 unknown parameters? Is it possible to fix 8 parameters ...
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2answers
70 views

Need a textbook for math course

The undergrad course is called intro the applied math, and it covers: "The unit introduces some of the principal mathematical techniques such as difference equations, differential equations and ...
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1answer
38 views

How can this function be considered to have a saddle node bifurcation?

Say I have the function $f(x,\mu) = (1 + \mu)x − x^2 − 0.1$. By definition a Saddle Node bifurcation occurs if: $f_{\mu_0}(0) = 0$ $f'_{\mu_0}(0) = 1$ $f''_{\mu_0}(0) \neq 0$ $\frac{\delta ...
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39 views

Find the bifurcation points for the system of ODEs

Frist of all, I found post Find the bifurcation points for the following system of ODEs by Antonio Vargas and comment by Pragabhava very useful, nevertheless could anyone be pleased to be more ...
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7 views

Is there a faster way to determine the stable branches from here?

I'm currently trying to draw the bifurcation diagram of the model $\frac{dx}{dt}=(\mu-x)(\mu-x^3+x^2)$ What I've done so far is to let that be zero to find any possible equilibria. By doing so, I ...
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49 views

Spontaneous or not spontaneous symmetry breaking? That is the question.

I have the following system of ODEs: \begin{cases} \frac{du_{i}}{dt}=F_{i}\left(\boldsymbol{u},\boldsymbol{v}\right)+A, & i=1,\ldots M\\ \\ ...
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1answer
164 views

Bifurcations in the Duffing oscillator

I'm trying to describe all the bifurcations in the two parameter Duffing oscillator: $$\ddot{x} + ax + bx^3 = 0$$ In phase space with $y = \dot{x}$ I've found the origin to be a centre for $a>0$ ...
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34 views

Topology equivalence in dynamical system

my name is Eric. I've got trouble when proofing that system $\dot{x}=\alpha+x^2+O(x^3)$ is topological equivalence with system $\dot{x}=\alpha+x^2$. I don't understand how to build the homeomorphism ...
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1answer
69 views

Simple question on group theory

Suppose we have the following system of differential equations: \begin{cases} \frac{dx_{i}}{dt}=f_{i}\left(\boldsymbol{x},\boldsymbol{y}\right), & i=1,\ldots M\\ \\ ...
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25 views

Find and classify bifurcation point

Find and classify the bifurcatioin point of the following system, if any x'(t)=ax2(t) y'(t)=x(t)y(t) where a is a real number. Clearly the equilibrium is the origin and when I evaluate the ...
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20 views

Discrete Model Finding Stability

For the discrete model $$x_{t+1} = (\lambda +1)x_t +x_t^3$$ Draw a bifurcation diagram (expressing the equilibrium vs $\lambda$ for values of $\lambda$ near zero. I have the bifurcation diagram. It ...
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1answer
88 views

How to make a cobweb diagram

I am struggling making a cobweb diagram for the function $$x_{t+1}=8x_t/{1+2x_t}$$ So I understand when making the cobweb diagram, that I have to draw the line $y=x$ But where I have trouble ...
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1answer
34 views

Finding Bifurcations Where Number of Equilibiria Don't Change

I'm looking at the system of ODEs \begin{align*}x' & = a - x - \frac{4xy}{1 + x^2} \\ y' & = bx(1 - \frac{y}{1 + x^2})\end{align*} and I've been asked to find values of $a$ and $b$ for which ...
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51 views

Stability and Bifurcation Diagram

if $ \frac{dx}{dt}= x(a^2-x^2) $ find and classify the stability of the equilibrium points, and draw the bifurcation diagram. I have found the stability of the function at $x =0,-a,a$ I believe $0$ ...
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1answer
92 views

Finding Bifurcation Point

Consider $X' = AX$ where $A = \begin{pmatrix} a & 1 \\ 2a & 2 \end{pmatrix}$. For which values of $a$ do you find a bifurcation? I attempted to solve this by finding the eigenvalues which I ...
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1answer
41 views

Bifurcation Points Vs. Values

Can someone explain to me what Bifurcation Points and Bifurcation Values are? I'm looking over a problem my professor gave out regarding the Supercritical Pitchfork Bifurcation: $\frac{dx}{dt} = \mu x ...
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How to find the value of a parameter such that the map has a period-doubling bifurcation?

For example: $f(x)=x_{n+1}=\mu+x_n^2$. Is it when $|f'(x^*)|=1$, where $x^*$ is a fixed point of the system? In this case, $\mu=1/4$? Also how to determine whether it is supercritical or ...
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1answer
57 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 ...
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73 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 ...
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1answer
202 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 ...
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1answer
58 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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101 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 ...
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122 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 ...
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77 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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63 views

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

I am trying to write a bifurcation diagram for $f'(x)=r-cosh(x)$ 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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1answer
137 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 ...
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39 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 ...
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76 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 ...
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1answer
53 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 ...
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80 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 ...
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32 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?
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1answer
27 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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1answer
68 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 ...
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45 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 $\delta$ ...
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1answer
55 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 ...
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1answer
67 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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70 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 ...
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2answers
83 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 ...
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1answer
216 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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343 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} ...
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1answer
150 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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157 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 ...
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1k 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 ...
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1answer
89 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?