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.

A map is noninvertible when it is not one-to-one and onto on the phase space under consideration. Forward iteration can be well-defined while backward states are absent or nonunique. A zero Jacobian multiplier can provide strong contraction at a fixed point. For the Lozi family, setting $b=0$ sends the whole plane into a line after one iteration, illustrating why a discrete saddle may have a zero stable multiplier without a unique local backward evolution.

An aperiodic orbit does not repeat with a finite period. An orbit that is not eventually periodic never enters a finite cycle after any number of transients. Aperiodicity alone is insufficient for chaos: an irrational circle rotation is aperiodic but has no exponential separation of nearby states. Chaotic interval maps can have both uncountably many aperiodic orbits and dense periodic points.

A piecewise affine planar map whose derivative changes across $x=0$. Away from that line its Jacobian is $\left(\begin{array}{cc}-a\mathrm{sgn}x& 1\\ b& 0\end{array}\right)$, with determinant $-b$. For $b\ne 0$ it is invertible; at $b=0$ an iteration collapses the image onto a line. The switching line requires piecewise analysis rather than applying a smooth-map theorem across it without checking its hypotheses.

The positive fixed point exists when $b<1+a$ and equals $(1,b)/(1+a-b)$; the negative one exists when $b>1-a$ and equals $(1,b)/(1-a-b)$. Equality on either boundary does not give a branch, because its fixed-point equation becomes $0=1$. For a genuine two-cycle and $a^2\ne (1-b{)}^2$, subtracting its scalar equations rules out equal signs of the two first coordinates. Solving with $x_1>0>x_2$ gives the displayed coordinates and second coordinates $y_1=b{x}_2$, $y_2=b{x}_1$. Their signs require exactly $a>|1-b|$. The cycle is unique up to interchanging its two points. Degenerate boundary parameter lines can support additional period-two families and are outside this statement.

An iterate is the result of repeatedly applying a self-map to an initial state. The $n$th iterate of $x$ is $F^n(x)$, where the superscript denotes composition.

A planar diffeomorphism is area-preserving if the phase-space area of every measurable set is unchanged. The change-of-variables formula proves this property when $|\det DF|=1$ everywhere, and conversely gives that determinant condition for a smooth map preserving all sufficiently small areas. Such a map cannot have an asymptotically attracting fixed point with a positive-area local basin: iterates of a small invariant neighborhood would have to enter arbitrarily small disks while retaining its phase-space area. For a map with constant positive determinant $b$, invariance of a bounded region of positive phase-space area instead forces $b=1$.

This triangular planar map has diagonal Jacobian matrix entries one and determinant one, for differentiable $f$. Its inverse subtracts $f(v)$, so it is an area-preserving map. The analogous vertical map $(u,v)\mapsto (u,v+g(u))$ has the same property. Their composition preserves area even when the shifts are nonlinear, as in the logarithmic area-preserving seasonal predator-prey map.

For $F(x,y)=((a+1)x-x^3+y,(b-1)y-(b-1)x^3+y^3)$, the origin multipliers are $a+1$, $b-1$. A nonzero fixed point has $S=x^2>0$, $y=x(S-a)$, and $S[1-(S-a{)}^3]=a(2-b)$. Since the map is odd, a symmetric two-cycle has $F(x,y)=-(x,y)$; it satisfies $y=x(S-a-2)$ and $S[1+(S-a-2{)}^3]=b(a+2)$. Eliminating $y$ from the two map equations gives these identities directly. They locate the side of each pitchfork or flip without relying on a potentially misleading cubic sign in the original coordinates. At $(a,b)=(-1,0)$ the second identity becomes $3S^2-3S^3+S^4=b$, giving the quartic amplitude scaling of a generalized flip bifurcation.

The Hénon map is a two-parameter quadratic planar map, conventionally $(X,Y)\mapsto (1-a{X}^2+Y,b_{H}X)$. The alternative family $(x,y)\mapsto (y,\mu -bx-y^2)$ has constant Jacobian determinant $b$. When $\mu \ne 0$, the change $X=y/\mu$, $Y=-bx/\mu$ gives the conventional parameters $a=\mu$, $b_H=-b$. Fixed points in the alternative normalization satisfy $x=y$ and $y^2+(1+b)y-\mu =0$. Its invertibility requires $b\ne 0$; at $b=1$ it preserves area.

At $b=1$ the Hénon map has unit Jacobian determinant. The upper fixed point is linearly elliptic for $-1<\mu <3$, and its newborn two-cycle is elliptic for $3<\mu <4$. These are neutral multiplier statements, not attraction. The endpoints have double multipliers $+1$ and $-1$. Constant-area scaling also excludes a simple invariant closed curve enclosing positive area when $b>0$ and $b\ne 1$: invariance of its interior would require both $\mathrm{area}(F(D))=b\mathrm{area}(D)$ and $F(D)=D$. Thus the $b=1$ elliptic stability boundary is a conservative degeneracy, not a generic dissipative Neimark-Sacker bifurcation. Arbitrary non-area-preserving perturbations can change this behavior even though generic fold and flip curves persist.

For the Hénon map $(x,y)\mapsto (y,\mu -bx-y^2)$ with $b>-1$, fixed points exist at $y_{\pm }=-(1+b)/2\pm \sqrt{\mu +(1+b{)}^2/4}$. Their multipliers satisfy ${\lambda }^2+2y\lambda +b=0$. The Jury stability criterion gives asymptotic stability only for the upper branch, with $-1<b<1$ and $-(1+b{)}^2/4<\mu <3(1+b{)}^2/4$. The two-cycle has alternating values $y,z$ satisfying $y+z=1+b$ and $(y-z{)}^2/4=\mu -3(1+b{)}^2/4$. Its second-iterate trace is $4[(1+b{)}^2-\mu ]-2b$ and determinant $b^2$, so it is attracting for $-1<b<1$ until $\mu =(5b^2+6b+5)/4$. The fixed-point fold and first flip are nondegenerate away from endpoints and persist under small smooth perturbations.

If $g$ is even and an orbit does not hit zero before its final iterate, induction gives $f^n(x)=(-1{)}^n{g}^n(x){\prod }_{k=0}^{n-1}\mathrm{sgn}(g^k(x))$. The step uses oddness of $f$ on nonzero arguments. A one-sided assigned value of $f(0)$ can invalidate that step after a critical hit; for $g(x)=\mu -x^2$, $\mu >0$, $x=\sqrt{\mu }$, the claimed second-iterate formula would give $+\mu$ while the actual value is $-\mu$.

// Target: dynamical-systems.bigb

For a noncritical least-period-$n$ cycle of $g(x)=\mu -x^2$, the multiplier sign is $(-1{)}^n{\prod }_k\mathrm{sgn}(x_k)$. A positive multiplier gives two symmetry-related period-$n$ cycles of the odd quadratic Lorenz map; a negative multiplier gives one period-$2n$ cycle. Their multipliers are respectively $|(g^n{)}^{\prime }|$ and $|(g^n{)}^{\prime }{|}^2$. Distinct points of a quadratic-map cycle cannot have the same absolute value, since the map is even and acts bijectively on its cycle; this proves the asserted least periods.

// Target: dynamical-systems.bigb

Away from zero this discontinuous map is odd, and its branch derivatives are $2|x|$. With the convention $\mathrm{sgn}(0)=1$, its value at zero is $-\mu$, so it is not odd at zero unless $\mu =0$. A global transition at $\mu =0$ turns two small stable fixed points into one stable symmetric two-cycle. Its unsigned dynamics can be related to the even quadratic map $g_{\mu }(x)=\mu -x^2$, provided critical hits are treated separately.

// Target: dynamical-systems.bigb

The two-cycle of $g(x)=\mu -x^2$ has points $(1\pm \sqrt{4\mu -3})/2$ and multiplier $4(1-\mu )$. Crossing $\mu =1$ changes its sign and passes through the critical cycle $\{0,1\}$. The sign lift therefore changes from two stable two-cycles to one stable four-cycle. The second iterate is continuous at zero at the transition, but retains jumps at the nonzero preimages of zero. Thus the return-map interpretation concerns a neighborhood of zero, not continuity of the whole second iterate on the real line.

// Target: dynamical-systems.bigb

The displayed map has a symmetric periodic orbit $\{\pm \sqrt{\mu +1}\}$ for $\mu >-1$, with multiplier $(-2\mu -3{)}^2>1$. For $\mu >2$, put $s_{\pm }=(\mu \pm \sqrt{{\mu }^2-4})/2$. The two additional period-two orbits are $\{\sqrt{s_{+}},\sqrt{s_{-}}\}$ and their negatives. Their multiplier is $9-2{\mu }^2$, so they are stable for $2<\mu <\sqrt{5}$ and undergo a period-doubling bifurcation at $\sqrt{5}$. At $\mu =2$ these points coincide with fixed points, not genuine two-cycles.

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

For a partial function $f:D\to X$ with $D\subseteq X$, its $n$th iterated function is defined only when each intermediate application is defined. Put $D_0=X$, $f^0={\mathrm{id}}_X$, and use the displayed recursion for the domain of a function. Simplifying an algebraic expression for $f^n$ does not restore points excluded by an intermediate application. For example, $f(x)=1/x$ has $f^2(x)=x$ on $\mathbb{R}\setminus \{0\}$; the simplified expression $x$ has a larger natural domain of a function.

The sine fixed-point iteration converges to zero from every real starting value. After one step, $|x_n|\le 1$, and $|x_{n+1}|=\sin |x_n|$. Since $0<\sin t<t$ for $0<t\le 1$, the absolute values form a bounded monotone sequence. Its limit $a$ satisfies $a=\sin a$ by continuity, forcing $a=0$. This proves convergence even though the sine is not a contraction mapping on any interval containing zero in its interior.

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.

For a two-dimensional hyperbolic toral automorphism, the stable and unstable eigenvalues obey $|{\lambda }_s|<1<|{\lambda }_u|$. In a sufficiently small Bowen ball, lifting to the stable and unstable coordinates gives widths proportional to $ϵ$ and $ϵ|{\lambda }_u{|}^{-(n-1)}$. Its normalized Haar measure is therefore proportional to ${ϵ}^2|{\lambda }_u{|}^{-(n-1)}$, uniformly in the centre. Maximal separated sets cover by radius-$ϵ$ Bowen balls and pack disjoint radius-$ϵ/2$ Bowen balls, so their exponential growth rate is $\log |{\lambda }_u|$. Smallness of $ϵ$ must exclude nonzero lattice jumps between successive lifted iterates; this is essential to the ball-volume argument.

For a continuous map on a compact metric space, the Bowen metric compares the first $n$ points of two forward orbits. Its radius-$ϵ$ ball is $B_n(x,ϵ)=\{y:d_n(x,y)<ϵ\}$. If $s_n(ϵ)$ is the largest cardinality of an $ϵ$-separated set in this metric, then topological entropy is ${\lim }_{ϵ\downarrow 0}{\mathrm{limsup}}_{n\to \infty }{n}^{-1}\log s_n(ϵ)$. The uniform continuity of the identity between two compatible metrics gives the comparison of separated-set counts needed for independence of the original metric.

A radius-$ϵ$ ball for a Bowen metric consists of the points whose first $n$ iterates stay within $ϵ$ of the corresponding iterates of the centre. Balls at a fixed small radius distinguish forward orbit segments at that observational resolution.

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.

The family $x_{n+1}=\mu -x_n^2$ has fixed points $(-1\pm \sqrt{1+4\mu })/2$. Its upper branch attracts linearly for $-1/4<\mu <3/4$ and loses stability in a period-doubling bifurcation. The resulting two-cycle has points $(1\pm \sqrt{4\mu -3})/2$ and multiplier $4(1-\mu )$, attracting for $3/4<\mu <5/4$. At $\mu =2$, the map is conjugate on $[-2,2]$ to the full tent map.

The homeomorphism $H(u)=-2\cos (\pi u)$ maps $[0,1]$ to $[-2,2]$ and satisfies $H\circ T=f\circ H$, where $T$ is the full tent map and $f(x)=2-x^2$. This transfers dense periodic points, topological mixing and sensitivity to the quadratic interval map. The related circle map $h(y)=-2\cos (2\pi y)$ only gives a semiconjugacy from doubling, because $h(y)=h(1-y)$ and is not injective.

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

For a normalized even unimodal interval map with $f(0)=1$ and $a=f(1)\in (-1,0)$, defineThis returns the second iterate near the critical point to the original spatial and height normalization. Then $\mathcal{T}(f)(0)=f(1)/a=1$. A restrictive-interval condition is also needed for the composition to remain unimodal and map $[-1,1]$ into itself; normalization alone is not sufficient.

// Target: dynamical-systems.bigb

Fix $R>1$ and write $h(z)={\sum }_{n\ge 0}{h}_n((z-1)/R{)}^n$ with absolutely summable real coefficients. The coefficient space is a Banach space under $\Vert h\Vert ={\sum }_n|h_n|$. The normalized even maps $p(x)=1-x^{2}h(x^2)$ form an affine copy of that space and are analytic where $|x^2-1|<R$. On any smaller coefficient disk, the tail beyond degree $N$ is bounded by $q^{N+1}{\sum }_{n>N}|h_n|$ for a radius ratio $q<1$. Analytic composition and Cauchy estimates on smaller domains permit certified derivative and truncation bounds. A small ball about an admissible map must additionally preserve the quadratic critical maximum and restrictive-interval conditions; the coefficient norm alone does not impose them.

// Target: banach-space.bigb

A Feigenbaum fixed point is an even analytic unimodal interval map $g$ with $g(0)=1$, a critical maximum of even order $d$, $a=g(1)\in (-1,0)$ andIn the quadratic class $g^{\prime \prime }(0)<0$. Its spatial scaling uses the Feigenbaum constants convention $\alpha =-1/a$; the values depend on the critical order. The central restrictive intervals are successively reduced by $|a|$ under renormalization. This fixed point describes the limiting shape of repeatedly renormalized maps at a period-doubling accumulation.

// Target: dynamical-systems.bigb

An approximate analytic fixed point of the period-doubling renormalization operator can be certified by a frozen Newton correction for a fixed-point equation. A coefficient Banach space, rigorous residual and derivative bounds, and an a posteriori contraction ball give existence and local uniqueness in the full function space. Interval arithmetic controls finite computations; analytic estimates control the infinite coefficient tail.

// Target: dynamical-systems.bigb

Suppose $g(x)=1+c{x}^d+O(x^{d+2})$, with even $d$ and $c<0$. Expanding the period-doubling renormalization operator exactly at zero gives $a^{d-1}{g}^{\prime }(1)=1$. Approximating globally by $1+c{x}^d$ gives $c\approx a-1$ and $g^{\prime }(1)\approx dc$, hence $d(a-1)a^{d-1}\approx 1$. For $d=2$, $a\approx (1-\sqrt{3})/2$, $g(x)\approx 1-(1+\sqrt{3})x^2/2$ and $\alpha \approx 1+\sqrt{3}$. The coefficient relation is a polynomial-truncation approximation, not an exact identity for the full analytic fixed point.

// Target: numerical-analysis.bigb

At the normalized quadratic fixed point $g$, one expanding eigenvalue $\delta >1$ of $D\mathcal{T}(g)$ and a contracting complement give a stable manifold of finite codimension one. A generic one-parameter family crosses that manifold transversely at its cascade accumulation. Its unstable coordinate is initially proportional to parameter distance and multiplies by $\delta$ per renormalization, yielding $s_{\infty }-s_n\sim C{\delta }^{-n}$. Stable components disappear, while the fixed point's orientation-reversing scale $a$ yields signed spatial ratios $a^{-1}=-\alpha$.

// Target: dynamical-systems.bigb

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.

For $T_s(x)=1-s|x|$ with $1<s\le 2$, the largest closed interval mapped onto itself is $A=[1-s,1]=[T_s^2(0),T_s(0)]$. It equals the image of $[-1,1]$. Every point outside $A$ enters $A$ after one iterate, because $T_s([-1,1])=A$. It subsequently remains in $A$.

// Target: dynamical-systems.bigb

If $\sqrt{2}<s\le 2$, every nondegenerate subinterval of $A=[1-s,1]$ eventually maps onto $A$. Away from the critical point, lengths multiply by $s$; one fold loses at most a factor of two. If an interval and its image both contain zero, its second image is $A$. Otherwise every two steps increase length by at least $s^2/2>1$, which cannot continue inside bounded $A$. Applying the intermediate value theorem to an interval that covers itself then gives a periodic point in every subinterval.

// Target: dynamical-systems.bigb

For $1<\mu \le \sqrt{2}$, the interval $J=[1/(\mu +1),\mu /(\mu +1)]$ is invariant under the second iterate of the tent map. The affine coordinate $h(x)=(\mu /(\mu +1)-x)/((\mu -1)/(\mu +1))$ conjugates that restriction to the tent map with parameter ${\mu }^2$. Its two linear pieces give the identity directly. This turns the two-iterate horseshoe threshold into a four-iterate threshold and explains why chaotic invariant sets need not have periodic points near the original nonzero fixed point.

For $1<s\le \sqrt{2}$, set $h_s(x)=-(s-1)x$ and $J_s=[-(s-1),s-1]$. The tent map sends $J_s$ to $[1+s-s^2,1]$, with disjoint interiors, and $T_s^2(J_s)\subseteq J_s$. Directly,Successive squaring eventually brings every $s>1$ into $(\sqrt{2},2]$. A horseshoe for an interval map of an iterate in the renormalized coordinate lifts to a horseshoe of an iterate of the original map.

// Target: dynamical-systems.bigb

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 repeatedly applies a quadratic function:The closed interval $[0,1]$ is a positively invariant set: $0\le F_r(x)\le r/4\le 1$. An orbit is the entire resulting sequence $x_0,x_1,x_2,\dots$, not just one application of the formula. For example, with $r=2$ and $x_0=0.2$, the first states are $0.2,0.32,0.4352,0.49160192,\dots$, approaching the fixed point $1/2$.

As a simple model of population dynamics, $x_n$ is a scaled abundance in generation $n$. When abundance is small, $x_{n+1}/x_n=r(1-x_n)$ is close to $r$; increasing abundance reduces this per-capita multiplier. The falling side of the quadratic function represents sufficiently severe crowding reducing the next generation. The scale $x=1$ is not generally the carrying capacity: the positive fixed point is $1-1/r$. The logistic map is a discrete-generation model; the continuous-time logistic growth equation has a different evolution rule and does not produce its period-doubling cascade.

The practical question is how abundance behaves after the initial adjustment: does it settle, alternate, or continue varying? The same formula permits all these possibilities. The following figure shows three orbits, their cobweb plots, and a bifurcation diagram. In a cobweb plot, moving vertically to $y=F_r(x)$ calculates the next state, and moving horizontally to $y=x$ makes that value the input for the next step.

The fixed points solve $F_r(x)=x$:The second fixed point is in $[0,1]$ for $r\ge 1$ and coincides with zero at $r=1$. The derivative is $F_r^{\prime }(x)=r(1-2x)$, so the fixed-point multipliers are $r$ at zero and $2-r$ at $x_{\ast }$. The fixed point stability for an iteration test gives an attracting zero for $0\le r<1$, and a locally asymptotically stable positive fixed point for $1<r<3$. A multiplier of modulus one needs a nonlinear argument; the strict derivative test alone settles neither $r=1$ nor $r=3$.

This derivative test has a direct interpretation. A small error $e_n=x_n-x_{\ast }$ obeysAt $r=2.8$, the leading multiplier is $-0.8$: the error changes sign, explaining the alternating approach, while its magnitude eventually shrinks by approximately a factor $0.8$ per step. At $r=3.2$, the multiplier is $-1.2$, and the positive fixed point repels nearby states. However, a different outcome has asymptotic stability: the logistic map two-cycle attracts nearby orbits.

For $3<r<1+\sqrt{6}$, the attracting period-two orbit has the two valuesAt $r=3.2$ they are approximately $0.513045$ and $0.799455$, with $F_r(x_{-})=x_{+}$ and $F_r(x_{+})=x_{-}$. Their periodic-orbit multiplier is $4+2r-r^2=0.16$, so a small displacement shrinks over two steps even though the fixed point is unstable. The brief algebraic justification is to factor $F_r^2(x)-x$, remove the fixed point factors, and multiply the two derivatives using the chain rule.

At $r=3$, the positive fixed-point multiplier passes through $-1$ and a period-doubling bifurcation creates the period-two orbit. At $r=1+\sqrt{6}\approx 3.44949$, its periodic-orbit multiplier passes through $-1$ and an attracting period-four periodic orbit emerges. At $r=3.5$, the period-four values are approximately $0.382820,0.826941,0.500884,0.874997$ in that order. Further period-doubling bifurcations accumulate near $r\approx 3.56995$. Beyond that value there are also parameter windows with attracting periodic orbits, including a period-three window near $r=3.83$. Neither a crowded numerical plot nor a parameter beyond the accumulation value proves sensitive dependence on initial conditions on the attracting set.

The endpoint $r=4$ supplies an analytically transparent example. The homeomorphism $h(u)={\sin }^2(\pi u/2)$ satisfieswhere $T$ is the full tent map. Thus these maps have topological conjugacy; stretching and folding in the tent map explain sensitive dependence on initial conditions and Devaney chaos for $F_4$ on $[0,1]$. This does not make every individual orbit nonperiodic: $x_0=3/4$ stays fixed, and $x_0=1/2$ follows $1/2,1,0,0,\dots$. Sensitivity means that these exceptional orbits also have arbitrarily close initial states that eventually depart substantially.

Robert M. May's "Simple mathematical models with very complicated dynamics" (1976), Nature 261, pp. 459–467, reviewed how simple deterministic recurrence relations can pass from stable abundances through period-doubling cascades to irregular behaviour. May presented the logistic map as an especially accessible example and discussed the relevance of such models to population dynamics. The paper explicitly describes itself as a review of results found by several researchers, bringing their findings together for a broad scientific audience.^[ref]

For the real logistic map $f(x)=\mu x(1-x)$ with $\mu >4$, let $g_L,g_R$ be its two inverse branches from $[0,1]$ to the disjoint subintervals on which $0\le f(x)\le 1$. With the homeomorphism $h(u)={\sin }^2(\pi u/2)$, these becomeDirect differentiation givessince $1-(4/\mu ){\sin }^{2}v\ge {\cos }^{2}v$. Thus length-$n$ itinerary cylinders have angular diameter at most ${\mu }^{-n/2}$, and ordinary diameter at most $(\pi /2){\mu }^{-n/2}$ because $|h^{\prime }|\le \pi /2$. The nested interval theorem then gives exactly one point for each infinite itinerary. This proves the coding by the one-sided full shift even when the ordinary-coordinate inverse branches are not contractions.

For the logistic map $F(u)=\lambda u(1-u)$, factor $F(F(u))-u$ and remove its two fixed points. The remaining quadratic equation is ${\lambda }^2{u}^2-\lambda (\lambda +1)u+\lambda +1=0$. Its distinct roots form a periodic orbit of period two when $\lambda >3$. The periodic-orbit multiplier is $F^{\prime }(u_{-})F^{\prime }(u_{+})=4+2\lambda -{\lambda }^2$, so this orbit is asymptotically stable for $3<\lambda <1+\sqrt{6}$. It is born at a period-doubling bifurcation at $\lambda =3$.

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.

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 $F(L)=K$.

For a continuous function from an interval to itself with a three-cycle $a<b<c$, reflection if needed gives $F(a)=b,F(b)=c,F(c)=a$. Set $I=[a,b],J=[b,c]$. The intermediate value theorem gives $F(I)\supseteq J$ and $F(J)\supseteq I\cup J$. Successive closed-interval pullbacks realize every cyclic covering word by a point fixed under the corresponding iterate. The word $I{J}^{n-1}$ has a unique visit to the interior of $I$ per cycle, forcing least period $n$; the only boundary ambiguity is the original three-cycle, which realizes $n=3$ and cannot realize any other such word. A covering of $J$ by itself also gives a fixed point. Thus every positive least period occurs.

The covering words $IJJJJ$ and $IJIJJ$ both occur under the interval coverings in period three implies all periods. Their points cannot be the boundary three-cycle. Each has least period five, because five is prime and each itinerary visits both intervals. Their respective one and two visits to the interior of $I$ distinguish the orbits even after cyclic changes of starting point.

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 cycle points ordered $x_3<x_1<x_0<x_2<x_4$, successive gap vertices have arrows $0\to 3$, $1\to 1,2$, $2\to 0$, $3\to 0,1$. Their transition-matrix traces through length four are $1,3,1,7$, forcing periods one, two and four but not three. The affine interpolation through ordered values $(4,3,1,0,2)$ has no three-cycle: its only length-three closed interval itinerary is $111$, on which the map is $5-2x$ and its third iterate has only its fixed point. Two distinct length-four return words at vertex one give a horseshoe for the fourth iterate.

A five-cycle whose points are in orbit order along the real line gives four interval vertices with arrows $0\to 1\to 2\to 3$ and $3\to 0,1,2,3$. The transition-matrix traces for lengths one through four are $1,3,7,15$. Primitive closed words force at least one fixed point, one two-cycle, two three-cycles and three four-cycles. The second iterate has a two-branch horseshoe for an interval map on vertices two and three. The map itself need not have a one-step horseshoe: the unimodal affine example $F(x)=x+1$ for $x\le 3$, $F(x)=16-4x$ for $x\ge 3$ has precisely this five-cycle and has no two disjoint self-covering branches.

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.

A period-doubling cascade is a succession of period-doubling bifurcations creating attracting periods $1,2,4,8,\dots$, with bifurcation parameters tending to an accumulation value. Superstable periodic orbits inside successive stability intervals give a second parameter sequence with the same limiting parameter-scaling ratio in a quadratic universality class.

// Target: dynamical-systems.bigb

For a generic quadratic period-doubling cascade, the Feigenbaum constants are the limiting parameter ratio $\delta$ and spatial ratio $\alpha$. With $s_n$ the successive superstable period-$2^n$ parameters and $d_n=f_{s_n}^{2^{n-1}}(0)$ the signed central-orbit spacing,The positive convention gives $\delta \simeq 4.669201609$ and $\alpha \simeq 2.502907875$. The orientation-reversing spatial scale at the Feigenbaum renormalization fixed point is $a=-1/\alpha$.

// Target: dynamical-systems.bigb

A generalized flip is a codimension-two period-doubling point with a simple multiplier $-1$, a vanishing cubic coefficient in its scalar centre-manifold map, and a nonzero quintic coefficient. In the odd normal form $z^{\prime }=(-1+\beta )z+c\alpha z^3+d{z}^5+\cdots$, symmetric two-cycles solve $\beta +c\alpha z^2+d{z}^4=0$. The discriminant of this quadratic in $z^2$ produces a fold of two-cycles where its positive roots collide. For $c=3$, $d=-3$, this fold is $\beta =-3{\alpha }^2/4$, $z^2=\alpha /2$, $\alpha >0$. At $\alpha =0$ the amplitude is proportional to $|\beta {|}^{1/4}$.

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.