chaos

  1. P

    Nonlinear Dynamics and Chaos

    Hi, So my one class is using the book Nonlinear Dynamics and Chaos by Steven H. Strogatz, and I've been struggling a lot to keep up, does anyone here have some experience with that book or the material in it? For example one of our problems: I've been having the most trouble with Chapter 6 -...
  2. D

    Chaos thru period doubling

    Hi, Im on something that associates with period doubling bifurcation and wanted to know: The idea of a system that demonstrates pdb, is that the period-time doubles as a function of some parameter (call it p), until the length of a single period reaches "infinity". Bascially, if I double a...
  3. A

    Fractals & Chaos!

    Hi all, I am in my final year at University doing a horrible module called Fractals & Chaos (really it's called Applied Mathematics - even though I find it to be purely pure mathematics) Now my main problem is the proofs of the module.. turns out it's more proof heavy than I ever imagined and...
  4. S

    non linear systems

    Can any one help me with this: Find the exact values of r, k, and x at the cusp point shown in Figure 3.7.5 r(x) = (2* (x^2))/ (1+(x^2))^2 , k(x) = 2*(x^3) /( x^2) - 1 [/IMG]
  5. A

    Chaos and Dynamical Systems

    The equation P_n_+_1 = rP_n(1-P_n/C) -k may be considered a density - dependent population model with constant harvesting. If the growth rate is r = 3 and the carrying capacity is C = 6000, what is the largest number k that could be harvested each generation so that the population has a stable...
  6. D

    Chaos- Correlation dimension and randomness

    Hello, I know there is a way to distinguish between random and pseudo random data using the correlation dimension, but I don't know how exactly. Any help will be much appreciated.
  7. M

    Logistic dynamics & chaos

    (a) Show that a solution to the discrete logistic dynamics Xn+1 = 4Xn(1-Xn) can have the form of Xn = A(sin^v)(b^n) Determine the A, v, and b. (b) How can you use the result in (a) to illustrate chaos?
  8. C

    Trying to figure out the intial conditions that would lead to chaos.

    So I've been working to achieve chaos from a damped pendulum. The acceleration equation for a damped simple pendulum that I'm using is: a= -cv + bx - dx^3 + F\cos{\omega t} Where -cv is the Restoring Force bx-dx^3 is the Duffing Oscillator F cos(...) is the Driving force I have...
  9. K

    Chaos Theory Problem

    The one-dimensional mapping x(n+1) =ax^2(n)(1 − x(n)) has been used as a toy model for the sunspot cycle. 0 < x(n) < 1 and 0 < a< 6.75; here x(n) is the phase space variable, and a is a parameter. (a) Show algebraically that there are one, two or three fixed points, giving the values of...
  10. G

    a simple chaos problem

    i just forget how it was worked out (Angry) Choose any irrational number x in the interval (0,1). Construct a series {X(i)} as follows: X(0)=x and Xi=(2*X(i-1))mod1 for i=1,2,3... so that the whole series is contained in (0,1). The question is the series thus constructed dense in (0,1)...
  11. H

    Chaos Theory: Logistic Map

    Hi I have this question, What is the least number of points of prime period n that the logistic map can have in each of the cases: (i) n is a prime number, (ii) n is 2^k where k is a natural number. I know that the logistic map is Q_{\mu}=\mu x(1-x) and that x is a prime period n point if...
  12. H

    Chaos theory: bifurcation diagram

    Hi, I have this problem. I need to sketch a bifurcation diagram for the fixed points of the family \mu \rightarrow f_\mu where f_\mu= \mu x^2(1-x). I have that the fixed points are x=0 and x=\frac{1\pm\sqrt{1-\frac{4}{\mu}}}{2} I know that: When \mu < 4 f_\mu drops below the identity...
  13. H

    Chaos Theory: Linear Mappings

    Hi, I have the following question Prove that each linear mapping f: R \rightarrow R (R is the set of real numbers) has only periodic points with prime period 1 or 2. How do I go about this? I know a linear mapping has the form f(x)=ax for some a. Do I need to split it into cases for...
  14. H

    Chaos Theory: period points

    Hi, I have this question Let f: R \rightarrow R (R is the set of real numbers) and let f^2 = f(f(x)) intersect the graph of the identity mapping id in just one point (call it p). How many points of period-1 does f have? In how many points does the graph of f intersect that of id? I know that...
  15. H

    nonlinear dynamics and chaos

    hey guys, i am having a little trouble with my homework assignment. although it's the first of the semester i'm a little confused about some of the concepts. i attached the assignment as a pdf file so i don't have to write the equations in bizarre forms. it's only the first two questions...
  16. T

    Logistic Map & Chaos

    Hi Everyone, I am having some trouble with finding information relating to an assignment i am doing for school. The assignment is like this: We are using the logistic map function of: L(x)=bx(1-x) with an x value of 0.1 This is a pretty standard function used to highlight how chaos can...
  17. JoeyCC

    Chaos Theory

    Lately I have been working with Fractal Art in photoshop and UF. I read into the subject and got interested, but I still have a few questions. For Example, In simple terms, what is Chaos Theory? How is Fractal Art (Geommetry) related to Chaos Theory? What exactly is a Mandlebrot? This is...
  18. S

    question about chaos theory

    i'm working on a project for mathematical modeling on chaos project, so for a system to be classified as chaotic its periodic orbits must be dense. what would be a good way to make sense of that? (i need to explain to the whole class, and im not sure what this means at all) thanks!
  19. S

    chaos question

    a.let g(x)=x^2 be a map acting on the real line. 1 find the lyapunove exponent(s) for all bounded orbits. 2 is a chaotic orbit possible?why? b.possible periods for periodic orbits of quadrtic map y= -2x^2 + 5x (real line again)