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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)

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. Reference: Wikipedia.

Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour.

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Continued fraction analog to zeta function - how to properly define it and find its properties?

I do not mean the continued fraction representation of zeta function; I mean the function which has the form: $$f(s)=\cfrac{1}{1^s+\cfrac{1}{2^s+\cfrac{1}{3^s+\cfrac{1}{4^s+\cdots}}}}$$ For some values, we know the function…
Yuriy S
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Why does ${x}^{x^{x^{x^{\,.^{\,.^{\,.}}}}}}$ bifurcate below $\sim0.065$?

When you calculate what ${x}^{x^{x^{x\cdots }}}$ converges to between $0$ and $1$, before approximately $0.065$ the graph bifurcates. Why does this happen and is there a reason for it happens at that number?
user894224
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Convergence of the quadratic map $\left(x-\left(x-\left(x- \dots \right)^2 \right)^2 \right)^2$?

Edit - I changed the title and much of the body to better reflect my full question. The old one I don't really care about, although I appreciate Fabian's answer of course. Here is the plot for the function (or map), defined as…
Yuriy S
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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 $a>0$ and a saddle for $a<0$ and there are two other fixed…
Wooster
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What are the lines on a Bifurcation Diagram?

Here is the Bifurcation Diagram for the logistic map $x_{n+1}=rx_{n}(1-x_{n})$: And here is an enlargement of a particular section What are the faded lines that appear in the dark sections of the diagram, and why are they so pronounced?
Plato
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What causes bifurcation?

Suppose let's say we have a recursive function, $$x_{n+1}=rx_{n}(1-x_{n})$$ From what I understand, the $x_{n+1}$ VS r graph starts to split (bifurcate) after a particular value of r (apparently it's around 3). Does that mean after they split up, we…
basilisk
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How to determine the existence of limit cycle?

I'm student learning about dynamical system. I understood well how to find fixed points and determine stability thanks to eigenvalues of the jacobian matrix, but not how to find limit cycle... I heard about Poincaré–Bendixson theorem, but it remain…
Dadep
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Looking for references about a graphical representation of the set of roots of polynomials depending on a parameter

Answering, some time ago, to this question : Change in eigenvalues on changing one entry of a matrix, I had the idea of a graphical representation of roots of polynomial equations $$P(x,a)=P_{a}(x) \in \mathbb{R}[a,x]=\mathbb{R}[a][x],$$ such…
Jean Marie
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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 some level of rigor and sophistication, although…
Ray Yang
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Why does the bifurcation diagram of the logistic map exhibit chaotic behavior past exactly r=3.57? What about 3.57 causes it to be the breaking point?

The logistic map typically converges to a "carrying capacity", or output value across iterations, at r values (the growth rate) under 3. After 3, the value the system converges to bifurcates multiple times until r reaches 3.57. After 3.57, it…
Juan Pablo
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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 elementary functions. (We need the so-called Bessel…
Fly by Night
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Center Manifold Theorem

Let $f:M \rightarrow M$ be a partially hyperbolic diffeomorphism of $M$ with the usual definition that at each $p$ tangent space splits to $Df$ invariant subspaces: $T_pM = E^s_p \oplus E^c_p \oplus E^u_p$ with the relevant constants on the norm of…
Sina
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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 systems show a period doubling cascade before…
PianoEntropy
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Asymptotics for solutions of a version of Lienard's differential equation

Consider the second order differential equation $ x'' + f(x)x' + g(x) = 0 $ with $$ f(x) = -\lambda + x^2, \quad g(x) = (-1 + x^2)x \, . $$ with $\lambda > 0$. Note: The original post had a misprint in this formula. $g$ should be a cubic, not a…
Hans Engler
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How can I calculate Feigenbaum's constant?

So I am trying to calculate Feigenbaum's constant for the logistic map: $$ x_{n+1} = 4 \lambda x_n (1-x_n) $$ I am writing this through python and the main pieces I have for my code that are relevant are: def logistic(L,x): return 4*L*x*(1-x) n…
Robert
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