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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 butterfly effect is the intuitive idea that an arbitrarily small change in an initial condition can eventually change the observed future substantially. In dynamical systems, sensitive dependence makes this precise for a map $F:X\to X$ on a metric space $(X,d)$: there is one separation scale $\delta >0$ such thatThe same $\delta$ works at every point, whereas $y$ and the separation time $n$ may depend on the point and the requested closeness. For a flow map, replace the iteration of a map $F^n$ by ${\Phi }_t$ with some $t\ge 0$. The state space matters: restricting to a single fixed point removes the alternative nearby states needed for sensitivity.

Sensitivity is compatible with deterministic evolution. Given an exact initial condition, there is still one orbit; uncertainty concerns which exact state a measurement represents. It also allows accurate short-term prediction. For a continuous function $F$, each fixed finite iterated function $F^n$ is continuous, so sufficiently close initial states remain close at any specified finite step. What fails is a single initial accuracy that controls every future step.

The logistic map illustrates both sides. At $r=2.8$, nearby orbits in the basin of attraction of its attracting fixed point approach one another. At $r=4$, arbitrarily close states can eventually separate by a macroscopic fraction of $[0,1]$. The definition asserts the existence of separating neighbours, not that every nearby pair must separate: $x$ and $1-x$ have exactly the same next state under the logistic map and hence the same subsequent orbit. Nor must separation increase at every step.

A positive Lyapunov exponent describes average exponential growth of an infinitesimal displacement along an orbit, when the relevant limit exists. It is related to sensitivity but is not a substitute for its uniform definition. For a differentiable interval map, the chain rule givesAt $r=4$, the logistic map has $\lambda =\log 2$ for Lebesgue measure-almost every initial condition, while exceptional orbits can have different exponents. Finite-separation growth is only approximately exponential while the linear approximation remains useful. In a bounded state space, a separation cannot grow exponentially forever.

Sensitivity alone is also weaker than Devaney chaos, which additionally requires topological transitivity and dense periodic points. For example, $F(x)=2x$ on the real line has sensitive dependence and a positive Lyapunov exponent, but its only periodic point is zero, so it is not Devaney-chaotic. This distinguishes growth of uncertainty from the richer organization of recurrent orbits.

Edward N. Lorenz's "Deterministic Nonperiodic Flow" (1963) studied a thermal convection model with three ordinary differential equations. Its numerical orbits exhibited persistent nonperiodic motion and instability to small changes of the initial condition. Lorenz discussed the consequences for very-long-range weather forecasting: a deterministic law need not make a detailed distant future reliably recoverable from approximate observations. The historically relevant lesson of the Lorenz system is this growth of uncertainty under deterministic evolution.^[ref]

A continuous interval map is chaotic in Glendinning's sense when some positive iterate has a horseshoe for an interval map.

An interval map $F:I\to I$ has a horseshoe when there are an open interval $J\subseteq I$ and disjoint open intervals $K_0,K_1\subseteq J$ such thatIterating inverse branches produces full two-symbol itinerary dynamics.

Suppose a interval map $F:A\to A$ maps every nondegenerate subinterval onto $A$ after finitely many iterates. Choose two separated closed intervals inside the interior of $A$ and a common iterate mapping each onto $A$. Between first-passage endpoint levels in each interval, take an open interval that maps exactly onto the interior of $A$. These two disjoint intervals are the branches of a horseshoe for an interval map for that iterate.

// Target: dynamical-systems.bigb

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

An inverse branch of a function $f$ on a set $J$ is an inverse function $g:J\to I$ of a bijective restriction $f{|}_I:I\to J$. Different restrictions can give different branches on the same $J$. Their function compositions describe successive preimages and itinerary cylinders in an interval map.

In Glendinning chaos, a continuous interval map is chaotic when some positive iterate has a horseshoe for an interval map. In the interval version, one may take an open interval $J$ containing two disjoint open intervals $K_0,K_1$ with $F^m(K_0)=F^m(K_1)=J$.

Let a three-cycle be ordered $a<b<c$. There are only two possible cyclic orderings; reflection of the coordinate interchanges them, so assume $F(a)=b$, $F(b)=c$, $F(c)=a$. ThenAlso $F(b)=c$ and $F(c)=a$, so the intermediate value theorem gives $d\in (b,c)$ with $F(d)=b$, whence $F^2(d)=c$. Both $[a,b]$ and $[b,d]$ therefore map across $[a,c]$ under $F^2$.

To obtain exact open-interval images, use the following elementary first-passage observation. A continuous function on $[r,s]$ with endpoint values $a,c$ has a subinterval on which its endpoint values are $a,c$ and all interior values lie strictly between them: take the first hit of the second level, then the last hit of the first level before it. The intermediate value theorem makes its interior image exactly $(a,c)$. Apply this to the two intervals, reversing endpoint order on the first. Their selected interiors $K_0\subset (a,b)$ and $K_1\subset (b,d)$ are disjoint and lie in $J=(a,c)$, withThis proves the three-cycle forces a two-iterate interval horseshoe result and establishes the required horseshoe for $F^2$.

An interval map has a horseshoe for an interval map when two disjoint subintervals are each mapped across a common interval. It is Glendinning-chaotic when one of its positive iterates has such a horseshoe.

Let the points of the three-cycle be $x_1<x_2<x_3$ and put $I_0=[x_1,x_2]$, $I_1=[x_2,x_3]$. There are two possible cyclic orders. If $x_1\mapsto x_2\mapsto x_3\mapsto x_1$, the intermediate value theorem givesfor the interval covering relation. Thus there are two closed covering walks of length two based at $I_1$. In the reverse cyclic order there are instead the arrows $I_0\to I_0$, $I_0\to I_1$ and $I_1\to I_0$, again giving two closed walks of length two. The horseshoe from two closed covering walks therefore shows that $F^2$ has a horseshoe in either case, so $F$ is chaotic.

The Sharkovsky theorem orders the positive integers asand says that a cycle of a given period forces cycles of every period to its right.

Suppose $N=2^{r}q$ with $q>1$ odd. Then $F^{2^r}$ has a $q$-cycle. The Sharkovsky theorem gives that map a six-cycle, and squaring it produces a three-cycle. Hence $F^{2^{r+1}}$ has a three-cycle and an iterate of $F$ has a horseshoe. Thus

A horseshoe contains the symbolic dynamics of the Bernoulli shift. Periodic binary words give cycles of every symbolic period; after translating from an iterate back to $F$, this supplies cycles of many periods that are not powers of two.

For the logistic map, a ten-cycle would imply chaos because $10$ is not a power of two. Hence there is no ten-cycle for $\mu <{\mu }_{\infty }$. A three-cycle forces every period, so a ten-cycle exists for $\mu >1+\sqrt{8}$. The stated facts alone give no conclusion for the intermediate range:

An interval map $F:I\to I$ has a horseshoe when there are an open interval $J\subseteq I$ and two disjoint open intervals $K_0,K_1\subseteq J$ such that

For $0<\mu <1$, all orbits converge to one fixed point, and for $\mu =1$ every orbit is eventually fixed or period two, so no iterate has a horseshoe. For the symmetric tent map with slope magnitude $\mu >1$, the topological entropy is $\log \mu >0$. The interval-map positive-entropy horseshoe theorem says that a continuous interval map has positive topological entropy exactly when some iterate has a horseshoe. HenceFor $\mu \ge \sqrt{2}$ this already follows from part (iii); for $1<\mu <\sqrt{2}$, a higher iterate supplies the horseshoe.

An interval map $F:I\to I$ has a horseshoe if there are two closed subintervals $J_0,J_1\subseteq I$ with disjoint interiors such thatIt is chaotic in Glendinning's sense if some positive iterate $F^n$ has a horseshoe.

Suppose $a<b<c$ are the points of a 3-cycle and put $J_0=[a,b]$, $J_1=[b,c]$. There are two possible cyclic orders. Ifthen the intermediate value theorem givesConsequently both $F^2(J_0)$ and $F^2(J_1)$ contain $J_0\cup J_1$. If insteadthenand again both second images contain $J_0\cup J_1$. Thus in either cyclic order $F^2$ has a horseshoe.

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 an interval map, this is the usual periodic point condition. Its least positive return time is the period.

A periodic orbit of a differentiable interval map is superstable when its periodic-orbit multiplier vanishes. A cycle passing through a critical point has zero multiplier because the derivative product includes $f^{\prime }(c)=0$. For the nonconstant quadratic family $f_{\mu }(x)=1-\mu x^2$, a superstable $2^n$ cycle satisfies $f_{\mu }^{2^n}(0)=0$ and does not have a smaller critical period.

// Target: dynamical-systems.bigb

For a continuous interval map $F$ with a three-cycle $a<b<c$, after reflecting the coordinate if needed its ordering is $F(a)=b$, $F(b)=c$, $F(c)=a$. Then $F^2(a)=c$, $F^2(b)=a$. Also the intermediate value theorem gives $d\in (b,c)$ with $F(d)=b$, hence $F^2(d)=c$. Each of $[a,b]$ and $[b,d]$ therefore maps across $[a,c]$ under $F^2$. Choose first-passage subintervals between endpoint levels to avoid overshoots; their interiors are disjoint and both map exactly onto $(a,c)$. This is a horseshoe for an interval map, and establishes Glendinning chaos.

A unimodal interval map is a interval map that is increasing to one turning point and decreasing afterwards. For a differentiable map with a maximum at zero, a critical point of even order $d$ means $f(x)=f(0)+c{x}^d+o(x^d)$ with $c<0$. The critical order distinguishes universality classes of period-doubling cascades.

// Target: dynamical-systems.bigb