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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Why do we need to find the multiple roots? (bifurcation curve)

Consider the system $$ \dot{x}=x+ay-y^3,\quad \dot{y}=b-2y+x. $$ The task is to give the bifurcation curve for the equilibria. First of all, equilibria are determined by $$ ...
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48 views

Planar system: Analyse the existence of equilibria and determine their bifurcations

Consider the system $$ \dot{x}=x+ay-y^3,\quad \dot{y}=b-2y+x. $$ Analyse the existence of equilibria and determine their bidurcation. The equilibria can be determined by setting ...
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43 views

Understanding Hopf's theorem

Hopf's Theorem Suppose, we have a family of systems which depend on a parameter $\varepsilon$ and suppose that at $\varepsilon=0$, $(x,y)=(0,0)$ is an equilibrium that undergoes Andronov-Hopf ...
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25 views

Not sure how to get this normal form

Consider $$ \dot{x}=f(x,\alpha), x\in\mathbb{R},\alpha\in\mathbb{R} $$ with smooth $f$ and equilibrium $x=0$ at $\alpha=0$, $\lambda=f_{x}(0,0)=0$ and, moreover, $f_{xx}(0,0)\neq 0, ...
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1answer
15 views

Normal form of subcritical pitchfork bifurcation.

I'm working the a dynamical system $\dot{x} = r x - \frac{x}{1+x^2}$. I have already worked out that it is a subcritical pitchfork bifurcation. At least, that what my bifurcation diagram shows. ...
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39 views

Name of bifurcation that causes eigenvalues to switch sign in a saddle?

What is the name for a bifurcation where the signs of the eigenvalues switch? E.g. Given a 4-dimensional saddle (two positive, two negative real eigenvalues), as I bifurcate a parameter two ...
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40 views

Does the normal form of the Fold bifurcation has something to do with the Dulac-Poincaré normal form?

Maybe this is a silly question but does the normal form $\dot{x}=\mu\pm x^2$ of the fold bifurcation has something to do with the normal form by Dulac and Poincaré or are this completely different ...
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11 views

Examples of state-space models that show strong homeostasis but also substantial change after critical threshold?

The question is, can can anyone provide examples of systems or math models that exhibit patterns of homeostasis but which can be exhibit substantial transitions or bifurcations after some critical ...
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2answers
48 views

Bifurcation diagram and bifurcation value

Determine the bifurcation values of $\dot{x} = x(x-r^2)$, and sketch the bifurcation diagram. My attempt: First, we see that if $f(x_0, r_0) = Df(x_0, r_0) = 0$, then $x_0$ is a non-hyperbolic ...
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18 views

Bifurcation diagram

Consider the logistic map $x_{n+1}=rx_n(1-x_n)$, whose bifurcation diagram is shown below for $2.4 < r < 4.0$: I need to find a particular value of $r$ so that "attracting $2^k$ periodic points ...
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21 views

Find all bifurcation values of a function

I need to find all bifurcation values of the function $x' = u + cos(x) + cos(2x)$. How do I find all bifurcation values of $u$? I know the solution is $u < -2, u = -2, -2 < u < 0$, and that ...
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21 views

compltely integrable vector field-Definition

I have a question about definition. when I have some ode system : $x'=f(x)$ where $x\in \mathbf{R^n}$ and $f$ is some smooth vector field. What does it mean that the system is completely ...
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1answer
45 views

Bifurcation - first-order ODEs

Construct a first-order ODE with one critical point if $\left\lvert \mu \right\rvert \ge 1$ and three critical points if $\left\lvert \mu \right\rvert \lt 1$ and draw a bifurcation diagram. ...
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2answers
98 views

What is the correct approach for studying bifurcations?

Probably a trivial question. Let's say I have the following system of equations: \begin{cases} f\left(x,y,p\right)=0\\ \\ y=g\left(x\right) \end{cases} where $p$ is a parameter, and I want to study ...
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45 views

Finding bifurcation values of one-parameter families of first-order differential equations

Consider the following one-parameter families of first-order differential equations defined on the reals: $$ \dot x = \mu - x - e^{-x} $$ $$ \dot x = x(\mu + e^x) $$ $$ \dot x = x - \frac{\mu ...
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37 views

How to draw bifurcation diagram:$\dot{x}=x^3-C*sin(\frac{\pi x}{2})$

I want to draw the bifurcation diagram but since I can't solve this equation by hand it is difficult. I can graph it by having $f1=x^3$ and $f2 = C*sin(\frac{\pi x}{2})$ and the intersection points ...
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13 views

Find the bifurcation value given the differential equations

Given $\dot{x} =-2x+y$ $\dot{y} =x^2-y+r$ Where r is a real number. Find the bifurcation values. The answer is $r=1$ but I don't see how that can be obtained so if someone can help me further I ...
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25 views

Explanation for the following bifurcation diagrams.

I am asked to plot the bifurcation diagrams of $x'(t)=ax+3$, $x'(t)=x^3-x+a$, and $x'(t)=x^2-ax$ respectively. The solutions (from an instructor) are as follows. Can anyone explain how to do this ...
2
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1answer
61 views

Find and classify the bifurcations that occur as $\mu$ varies for the system

Find and classify the bifurcations that occur as $\mu$ varies for the system \begin{align}\frac{dx}{dt}&= y-2x \\ \frac{dy}{dt}&=\mu +x^2 -y\end{align} What I have so far: The ...
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1answer
55 views

Find the values of $r$ at which bifurcation occurs

Consider the system $$\dot{x}= rx - \frac{x}{1 + x^2}$$ where $r \in \mathbb R$. Find the value(s) at which bifurcations occur and where possible classify those as saddle-node, transcritical or ...
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53 views

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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25 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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21 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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36 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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37 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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57 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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40 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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102 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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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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47 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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9 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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51 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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240 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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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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73 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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28 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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28 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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230 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
38 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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97 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
137 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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55 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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27 views

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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60 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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1answer
138 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 ...
4
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1answer
263 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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60 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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153 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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2answers
146 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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80 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 ...