Dynamics & Chaos

A dynamical system is a system where the next step only depends on the current position and a map that will decide where to go next.

A forward orbit O+(x0) is the set {x0, x1, x2, …} for which F(xi) = xi+1

An N cycle is a set {x0, x1, … xN-1} where F(xi) = x_{i+1 modN} but no value appears twice.

x: F(x) = x

F(x) = x and

Attracting: |F’(x)| < 1

Repelling: |F’(x)| > 1

It means that there is an 𝜺-range around x’ such that for all x in this interval we find FN(x) -> x’ for N -> infty

It means that there is an 𝜺-range around x’ such that for all x in this interval we find that F^N(x) is not in that interval for some N.

~ : asymptotic to eachother, meaning that f(x)/g(x) -> 1 as x-> x0,

g >> f means that g is much greater than f, ie that f(x) / g(x) -> 0 as x-> x0

This means that we choose different terms in our equation and argue what happens if they are the dominant terms, ie they are >>

A bifurcation is a point where there is a significant change in the dynamic of fixed points as the parameter µ goes through that point

Saddle Node Bifurcation

Fx(0,0) = 1, but Fxx, Fµ ≠ 0

Transcritical Bifurcation

Fx(0,0) = 1, Fµ = 0 but Fxx ≠ 0

Pitchfork Bifurcation

Fx = 1, Fµ = 0, Fxx = 0

Supercritical: if only exists for µ > 0. Else subcritical.

Sensitive Dependence on Initial Conditions.

There exists a 𝛅 > 0 such that for whatever point x you choose and any 𝜺 range around that x, there is a y in that range such that |F^n(y) - F^n(x)| > 𝛅.

For any two points x and y there is a point z that is arbitrarily close to x and will at some point be arbitrarily close to y, i.e. |x-z| < 𝜺 and |F^n(z) - y| < 𝜺 for some n∈ℕ.

This means that for every point x, there is a y in the orbit that gets arbitrarily close to that x.

F is chaotic if

• F is SDIC

• F is TT

• periodic points are dense

There exists a homeomorphism h such that F•h = h•G.

A homeomorphism is a continuous bijection with a continuous inverse.

d(a,b) = ∑𝛄(ai,bi) / 3^i, where 𝛄(x,y) = 1 if x=y and else 0. The sum begins at k=0.

The two extremes are that from, from m onwards, they are all the same or all different.

All different -> Maximum -> 3^-m * ∑1/3^i = 3^-m • 3/2

All same -> Minimum -> 3^-m

The shift map is a map that shifts the sequence a0a1a2… -> a1a2a3…

Since they are all non-empty, we can find a sequence ai such that ai ∈Si for all i. This sequence is bounded as S0 is bounded and hence thanks to Bolzano Weierstraß it has a converging subsequence. This subsequence has a limit and that value is in all of the sets. Hence it is non empty.

Let F: I->ℝ be a continuous map and G: Y->Y also be continuous on a metric space.

Then, if there is a invariant set 𝛬 ⊂ I and a continuous surjection h: 𝛬-> Y with h•F = G•h, we call F semiconjugate to G via h.

The matrix A has entries aij, that are 1 if i can be followed by a j in the sequence space and a 0 if not.

A subshift is like a regular shift but only on a sequence space that only allows certain sequences.

A transition matrix is irreducible if there exists a path from each i to every j.

If there exists some j for which there are two possible points to go to, i.e. the row j has more than one 1.

If you thought about doing this by induction and multiplying the necessary columns and rows, then that is alright.

You can prove this by induction.

The base case is quite simple, since if there is a fixed point, then Aii will be one and hence the trace will count this.

Next, the i-th column holds the information on possiblle paths to node i. Therefore, if in A^n, the i,k entry shows that there are m many ways to get from i to k, meaning we count these m ways if we multiply the ith row of A^n by A.

So we know that Nq is the number of cycles with minimum period q. Hence, they each add q new points to the cycle. Therefore, if we calculate the amount of points in n-cycles, then we have to add all the minimum cyces hat divide n and count them according to how long they are.

is TT

SDIC

This means that J ⊂ F(I).

The transition graph is a directed graph has nodes for each interval and an edge from I ->J if J ⊂ F(I).

If there is a horseshoe for Fn for some n, then F is chaotic.

let x0<x1<x2 be the 3-cycle. Either, we have x0 -> x1 -> x2 ->x0 or x0 -> x2 -> x1 ->x0. In the latter case, look at G(x) = -F(-x), which again leads to the first case.

Let I = [x0,x1] and J = [x1,x2]. We have J ⊂ F([x0,x1]) and I,J ⊂ F([x1,x2]). Therefore, we get the dynamic

I <->J<->J.

THence, F2 has a horseshoe.