I am reading about Feigenbaum attractor (FA) and am getting very confused with something that is described in some books. It is written that FA is not an attractor because in its neighbourhood however small there are points of unstable periodic orbits (UPOs). But I used to think that, apart from other criteria of invariance and irreducibility, an "attractor" has an open set of points constituting basin of attraction which has non-zero measure. Now, I think FA satisfies this property as well because any open set about it is attracted to it except for a set of points (UPOs) which anyway are countable and hence of zero-measure. Am I missing something?
2026-03-27 08:48:20.1774601300
Feigenbaum attractor is not an attractor?
143 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in DYNAMICAL-SYSTEMS
- Stability of system of parameters $\kappa, \lambda$ when there is a zero eigenvalue
- Stability of stationary point $O(0,0)$ when eigenvalues are zero
- Determine $ \ a_{\max} \ $ and $ \ a_{\min} \ $ so that the above difference equation is well-defined.
- Question on designing a state observer for discrete time system
- How to analyze a dynamical system when $t\to\infty?$
- The system $x' = h(y), \space y' = ay + g(x)$ has no periodic solutions
- Existence of unique limit cycle for $r'=r(μ-r^2), \space θ' = ρ(r^2)$
- Including a time delay term for a differential equation
- Doubts in proof of topologically transitive + dense periodic points = Devaney Chaotic
- Condition for symmetric part of $A$ for $\|x(t)\|$ monotonically decreasing ($\dot{x} = Ax(t)$)
Related Questions in CHAOS-THEORY
- Chaotic behaviour from iteration process.
- Doubts in proof of topologically transitive + dense periodic points = Devaney Chaotic
- Homeomorphism between Smale set and $\{0,2\} ^{\mathbb{Z}}$
- Making something a control parameter or a variable when analysing a dynamical system
- Lorenz attractor as a position-velocity-acceleration problem
- Do chaos and/or limit cycles always require the existence of an unstable fixed point?
- Is the "sensitive dependence on initial conditions" of chaotic systems referring to the seed, control parameters, or both?
- Logistic map and chaos on a Cantor set
- Moser invariant curves in discrete dynamical systems and how they give stability
- Why do we need folding and a finite domain for chaos?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
I do not know about the exact nature of the UPOs in question, but if they are dense, the basin of attraction cannot be an open set: In every ε-ball around a point from the basin of attraction, there is an UPO, which is hence not in the basin of attraction. This is analogous to $ℝ\backslashℚ$ not being an open set.
However, I would disagree that openness of the basin of attraction is a good criterion for an attractor.