Discrete dynamical system

A one-dimensional discrete dynamical system iterates a map $x_{n+1}=F(x_n)$. A fixed point satisfies $F(x^{\ast })=x^{\ast }$, while a point of least period $k$ satisfies $F^k(x)=x$ but no corresponding equation for a smaller positive period.

For a map $F:X\to X$, its iterates are defined by $F^0$ equal to the identity map and $F^{n+1}=F\circ F^n$.

A map on $(0,1)$ can satisfy $0<f(x)<x$ and still have no orbit tending to zero. Partition the domain into $I_j=(1/(j+1),1/j]\cap (0,1)$ and define $f(x)=(x+1/(j+1))/2$ on $I_j$. Every interval is invariant under the iteration of a map, and $f^n(x)=1/(j+1)+2^{-n}(x-1/(j+1))$ has a positive limit. The discontinuities at interval boundaries prevent that limit from being a fixed point. In contrast, continuity of a self-map satisfying $f(x)<x$ permits passage to a positive orbit limit and would force an impossible fixed point.

Topological entropy measures the exponential growth rate of distinguishable orbit segments of a dynamical system.

A dynamical system has positive topological entropy when $h_{\mathrm{top}}>0$; this expresses exponential orbit complexity.

A continuous interval map has positive topological entropy if and only if some positive iterate has a horseshoe for an interval map.

An interval map is a continuous function $F:I\to I$ from a real interval to itself. Its iterates form a one-dimensional discrete dynamical system.

For a real number $\beta >1$, the beta transformation is the interval mapon $[0,1)$. Each interval of continuity is expanding with slope $\beta$.

The full tent map on the unit interval isIts two affine branches expand lengths by two, and it preserves Lebesgue measure.

Given a finite measurable partition of an interval, the itinerary of $x$ records which partition element contains each iterate $T^n(x)$. Prescribing a finite initial word defines an itinerary cylinder. For the full tent map, every length-$N$ binary cylinder has Lebesgue measure $2^{-N}$ up to endpoint conventions, so its itinerary process is a fair i.i.d. Bernoulli process.

An itinerary cylinder is the set of points whose first finitely many itinerary symbols equal a prescribed finite word. Such cylinders generate the symbolic σ-algebra and pull back to finite intersections of inverse images of partition elements.

The logistic map is the one-parameter family $F_{\mu }(x)=\mu x(1-x)$ on the unit interval. As $\mu$ increases it exhibits a period-doubling cascade, periodic windows and chaotic parameter ranges.

A point $x$ is periodic for an interval map $F$ when $F^n(x)=x$ for some positive integer $n$. Its least such $n$ is its period, and the finite setis its periodic orbit.

For closed intervals $J$ and $K$, write $J⟶K$ under $F$ when $F(J)\supseteq K$. The intermediate value theorem implies that there is a closed subinterval $L\subseteq J$ with $F(L)=K$.

Given finitely many intervals $J_1,\dots ,J_r$, their directed covering graph has an directed edge $J_i\to J_j$ whenever $F(J_i)\supseteq J_j$.

Every closed walkin a directed covering graph of an interval map has a point $x\in J_0$ with $F^j(x)\in J_j$ and $F^n(x)=x$. This follows by successively pulling $J_0$ back through the covering relations and applying the fixed-point property of a closed interval. If the itinerary has least period $n$ and avoids shared endpoints, $x$ lies on an $n$-cycle.

If $A$ is the adjacency matrix of a directed covering graph, then $\mathrm{tr}(A^n)$ counts its pointed closed walks of length $n$. For a prime number $p$, subtracting the $\mathrm{tr}(A)$ constant walks and identifying the $p$ cyclic choices of starting point givesprimitive closed itineraries of length $p$.

For prescribed values $F(x_i)$ at ordered points $x_0<\cdots <x_n$, the connect-the-dots interval map is the unique piecewise-linear function obtained by linear interpolation between consecutive data points.

The Jury criterion tests whether every root of a real discrete-time characteristic polynomial lies strictly inside the unit circle. For $z^2+az+b$, this is equivalent to $|b|<1$, $1+a+b>0$, and $1-a+b>0$.

For a period-$k$ orbit $x_0,\dots ,x_{k-1}$ of a differentiable map, the multiplier isThe orbit is locally asymptotically stable when the modulus of this product is less than one.

A period-doubling bifurcation occurs when a fixed-point multiplier crosses $-1$ and a nearby period-two orbit is created. Under the generic nondegeneracy conditions, the orbit amplitude is proportional to the square root of the parameter displacement.

For $F_{\mu }(x)=\mu x+b{x}^2+a{x}^3$ near $\mu =-1$, write $\mu =-1+ϵ$. Then $F_{\mu }^2(x)-x=-2ϵx-2(a+b^2)x^3+O({ϵ}^{2}x,ϵx^2,x^4)$. The small two-cycle is attracting on $\mu <-1$ if $a+b^2>0$ and repelling on $\mu >-1$ if $a+b^2<0$. Vanishing of $a+b^2$ gives a degenerate flip requiring higher terms.

For a $C^4$ real map $f(u)=-u+c_2{u}^2+c_3{u}^3+O(u^4)$, the second iterate isIf $c_3+c_2^2>0$, sufficiently small nonzero $u$ retains its sign under $f^2$ and strictly decreases in absolute value. Its iterates therefore converge to zero, proving local asymptotic stability even though $f^{\prime }(0)=-1$. If $c_3+c_2^2<0$, the second iterate moves small positive points away from zero, proving instability. The zero-coefficient case requires higher-order terms. With strictly stable transverse directions, the centre manifold theorem for a discrete dynamical system applies this scalar test to a higher-dimensional discrete dynamical system.