Questions tagged [chaos-theory]
For questions in chaos theory.
685 questions
2 votes
0 answers
91 views
Suppose a continuous function $f : [a,b] \to [a, b]$ has no point of period 2. Does each sequence $x, f(x), f(f(x)), \cdots$ converges?
A continuous function $f$ maps $[a,b]$ to itself. For each $x$ in $[a,b]$, $f(f(x)) = x \Rightarrow f(x) = x$. Is it true that each sequence generated by iterating $f$ over $x$ always commerges? ...
- 43
1 vote
0 answers
59 views
Admissibility of chaos in an infinite system of ODEs
Consider a sequence $A=(a_i)_{i=0}^{\infty}$ with $a_0\neq 0$ that determines the evolution of an infinite system of ODEs \begin{align*} \dot{X}_n&=-\epsilon X_n-a_n+\frac{a_0}{X_1}X_{n+1}, \end{...
- 158
1 vote
1 answer
66 views
Family of truncated tent maps
I'm trying to study the behaviour of the family of maps $T_h:[0,1]\to[0,1]$ defined by $T_h(x)=\min{(h,1-2|x-\frac{1}{2}|)}$. I stumbled upon the family at the end of this paper on Sharkovsky's ...
- 183
3 votes
1 answer
78 views
Chaos theory problems and arbitrary precision
Given any system that behaves chaotically (e.g. three-body problem), can the chaotic nature of it be erased by "rescaling" it? In other words, will the system behave in a non-chaotic manner ...
- 163
3 votes
0 answers
90 views
Lebesgue measure of the set of invariant tori in KAM theorem
Fasano & Marmi Analytical Mechanics, states the K.A.M. theorem in section 12.6, page 528 as follows: Theorem 12.12 (KAM) Consider a quasi-integrable Hamiltonian system $$H(J, χ, ε) = H_0(J) + εF(...
- 602
5 votes
1 answer
148 views
Strange phenomenon in iterative operations
Based on the concept of logistic iterative operations, $x\rightarrow kx(1-x)$,where $k\in[0,4]$, I have proposed a similar iterative operation relationship.Specifically, I considered the form $${} x\...
- 53
4 votes
1 answer
216 views
Iterations of orthic triangles
The orthic triangle exists for any given triangle with the following remarks: For acute triangle orthic triangle is inscribed triangle For a rectangular triangle it is degenerated triangle (a segment)...
- 897
1 vote
1 answer
70 views
Stochastic version of Lorentz equation based on the system size
The Lorenz equation is given by $\frac{dx}{dt}=\sigma\left(y-x\right)$ $\frac{dy}{dt}=\rho x - y - xz$ $\frac{dz}{dt}=xy - \beta z$. I want to change the deterministic equation to stochastic version ...
- 53
36 votes
2 answers
955 views
Reflection between Two Parallel Cylinders
I am interested in the following question. Consider two identical cylinders (or in 2D, two circles) of radius $r$, with centers separated by a distance $s$. A point particle is released above with a ...
- 411
0 votes
0 answers
53 views
Literature on $\beta$-transformation composed with tent map
I'm currently studying the following map: $$ T(x)=2\min\{1.5x\bmod1,1-(1.5x\bmod 1)\} $$ If $f_\alpha$ denotes the family of tent maps (with $\alpha\in[0,2]$) and $g_\beta$ denotes the family of beta ...
- 171
5 votes
1 answer
148 views
Invariance of Lyapunov exponents under diffeomorphisms
I have a fundamental question about the behavior of Lyapunov exponents under smooth transformations. Intuitively, I would expect that a chaotic system's Lyapunov exponents will not be preserved if, ...
- 151
1 vote
0 answers
45 views
Determining if a quadratic map is a strange attractor using Lyapunov exponent
I stumbled across an interesting paper from decades ago, about a simple way to find strange attractors of the following quadratic map: $$ x_{n+1}=a_1+a_2x_n+a_3x_n^2+a_4x_ny_n+a_5y_n+a_6y_n^2\\ y_{n+1}...
- 31
1 vote
1 answer
70 views
Behavior of the iterative process $f(x)=x^2-1$: divergence for $x>\phi$ and convergence to a 2-cycle for $0<x< \phi$
Behavior of the iterative process $f(x) = x^2 - 1$: divergence for $x > \phi$ and convergence to a 2-cycle for $0 < x < \phi$ I am analyzing the iterative process defined by $f(x) = x^2 - 1$, ...
- 21
0 votes
1 answer
62 views
A confusion about the chaotic invariant set
Consider $(M,d)$ to be a metric space, and we have a continuous map $f: M\to M$ . Then we say that $A\subset M$ is invariant under $f$ if it has property $f(A)\subset A$ . Also, if $A\subset M$ , we ...
- 2,015
-1 votes
2 answers
120 views
Chaos Theory: how to go from time series to phase space? [closed]
I've been reading "Chaos: Making a New Science" by James Gleick, and I came across the following visualization which relates time series data to phase space. However, I can't quite make ...
