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In the Known-variance Gaussian mixture EM example, initialize both means at the same value $m$. The mixture responsibilities are then $1/2$ for every observation. Each M-step sets both means to the sample mean, zero here; the next E-step again gives $1/2$. Thus $(0,0)$ is a fixed point with log-likelihood $-14.593631199$, despite the better fit found from $(-1,1)$.

One can check the failure without trusting an iteration plot. Separate the means as $(-a,a)$ and use the hyperbolic cosine identity:The Taylor expansion uses ${\sum }_i{y}_i^2=18.16$. Small nonzero separation therefore improves the log-likelihood, whereas translating both means to $m$ changes it by $-3m^2$. This stationary point has both improving and worsening directions. Identical initialization cannot discover the separation because every observation keeps identical label probabilities. Trying several distinct initial fits and comparing their final log-likelihoods is useful, but does not prove global optimality.

Keeping the variance fixed also removes a different pathology. With known $\sigma =1$, every component probability density function is at most $(2\pi {)}^{-1/2}$, so $\ell \le -3\log (2\pi )$. If separate component variances are allowed to shrink without restriction, one mixture component can concentrate at an observation while another retains positive density at the other observations, making the likelihood function unbounded. The fixed positive variance in this example prevents that collapse; it does not remove initialization traps.

For finite sets $A_1,\dots ,A_n$, the cardinality of their union isAdd the individual cardinalities, subtract pairwise intersection counts, add triple intersection counts, and continue alternating. This is useful whenever one object can satisfy several conditions: a search result can match several filters, or a permutation can have several forbidden fixed points.

Among the integers from 1 to 30, let $A$ contain the multiples of 2 and $B$ the multiples of 3. Then $|A|=15$, $|B|=10$, and $|A\cap B|=5$, because their common multiples are the multiples of 6. The naive sum $25$ counts those five twice; the correct union size is $15+10-5=20$. The regions contain ten in $A$ alone, five in both, five in $B$ alone, and ten in neither.

Add a third set $C$ containing the multiples of 5. Its size is 6; the three pairwise intersections have sizes $5,3,2$; and the triple intersection contains only 30. The inclusion-exclusion principle gives $15+10+6-5-3-2+1=22$ integers divisible by at least one of 2, 3, and 5.

The alternating corrections have a local explanation. An object belonging to exactly $r\ge 1$ of the sets is counted $\left(\genfrac{}{}{0ex}{}{r}{1}\right)-\left(\genfrac{}{}{0ex}{}{r}{2}\right)+\cdots +(-1{)}^{r+1}(\genfrac{}{}{0ex}{}{r}{r})=1$ times, by the binomial theorem applied to $(1-1{)}^r$. An object in none contributes zero. Thus overlapping descriptions are repaired without requiring the sets themselves to be disjoint. For probability, the same calculation applies to event probabilities in place of cardinalities, giving the inclusion-exclusion principle for probabilities.

The inverse function theorem answers a practical question: can nearby outputs be used to recover nearby inputs uniquely and smoothly? Its useful surprise is that a nonlinear function can be tested using its linear derivative. If the first-order changes preserve all input directions, then, on a sufficiently small region, the nonlinear map can also be reversed. This is the basis for treating a nonlinear map as a coordinate transformation.

Precisely, let $f:U\to {\mathbb{R}}^n$ be continuously differentiable on an open subset $U$ of Euclidean space, let $a\in U$, and put $b=f(a)$. If $Df(a)$ is an invertible linear map, there are open neighbourhoods $U_0$ of $a$ and $V_0$ of $b$ such that $f:U_0\to V_0$ is a bijection with a continuously differentiable inverse function $g$. Thus this restriction is a diffeomorphism. In coordinates the test is $\det Df(a)\ne 0$, using the Jacobian determinant. The derivative of a local inverse map is obtained by inverting the Jacobian matrix, at the corresponding input. ^[ref]

For a one-dimensional example, $f(x)=x^2$ has $f^{\prime }(1)=2$. Near $x=1$, the inverse function is $g(y)=\sqrt{y}$, and the inverse function rule gives $g^{\prime }(1)=1/2$. But $x^2$ has no inverse function on the whole real line, since $x$ and $-x$ give the same output. The theorem supplies a local restriction, not global injectivity. At zero, the derivative test fails; the nonnegative branch still has a square root inverse, but that inverse has no finite derivative at zero. ^[ref]

Here is a two-dimensional example with an invertible derivative everywhere and infinitely repeated outputs:The exponential function turns $s$ into a positive radius, while $\theta$ is the angle in polar coordinates. Restricting, for example, to $|s|<0.35$ and $|\theta |<0.6$ produces the reversible patch pictured below. On its image, with Cartesian coordinates $(u,v)$ and $u>0$, the inverse function isThe arctangent selects the angle near zero. Without an angle restriction, $f(s,\theta +2\pi k)=f(s,\theta )$ for every integer $k$, so there is no single global inverse function. This map is a local diffeomorphism but is not injective.

The short proof idea is to correct a guessed input using the inverse linear response. With $A=Df(a)$, solving $f(x)=y$ is equivalent to finding a fixed point ofAt $a$, the derivative $D{T}_y(a)=I-A^{-1}Df(a)$ vanishes. Continuity makes $T_y$ a contraction mapping on a sufficiently small closed ball, which it maps into itself when $y$ is sufficiently close to $b$. The contraction mapping theorem then gives existence and uniqueness there. Establishing the smooth dependence on $y$ completes the inverse theorem. This explains why the nonlinearity can be tolerated: its correction has a small local slope.^[ref]

If a self-map $T$ of a nonempty complete metric space has an iterate $T^n$ that is a contraction mapping, the contraction mapping theorem gives a unique fixed point $a$ of $T^n$. The maps commute, so $T^n(Ta)=T(T^{n}a)=Ta$, and uniqueness forces $Ta=a$. Conversely every fixed point of $T$ is fixed by $T^n$. Thus $a$ is the unique fixed point of $T$.

A nonidentity Möbius transformation with only one fixed point is conjugate to a nonzero translation of the Riemann sphere. Its translation iterates are $z+nb$, tending to infinity in the topology of the Riemann sphere for finite $z$, with infinity itself fixed. Conjugating back proves that every point tends to the unique fixed point. This is convergence on the sphere; if the fixed point is finite, intermediate iterates can still pass through infinity.

A different interpretation can treat cyclic spreadsheet formulas as simultaneous equations and seek a fixed point by repeated numerical updates. This requires an explicit update rule, initial values and stopping condition. It is not automatically the meaning of a spreadsheet formula. In particular, a directed cycle alone tells us neither whether a fixed point exists nor whether a chosen update rule approaches it.

Two original toy examples make the distinction exact. The cyclic pair X1=Y1+1, Y1=X1+1 has no solution over the real numbers: substitution gives $X1=X1+2$. For X1=Y1, Y1=X1, every equal pair is a fixed point, so there is no unique answer. Simultaneously updating both spreadsheet cells from the previous pair gives $(x_{n+1},y_{n+1})=(y_n,x_n)$. Starting at $(0,1)$ produces the periodic sequence $(0,1),(1,0),(0,1),\dots$, which does not approach a fixed point. Updating X1 first and immediately using its new value for Y1 instead gives $(1,1)$ in one sweep. That difference follows from the specified rules, not from a claim about any current spreadsheet product. Numerical iteration can be useful for a deliberately chosen model with a justification such as the contraction mapping theorem; ordinary acyclic spreadsheet dependency evaluation already has its answer from prerequisite order.

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

Let $f$ be order-preserving and inflationary on a directed-complete partial order. Define $xRy$ when every set containing $x$ and closed under $f$ and directed joins contains $y$. This is a partial order refining $\le$. The order-preserving maps $h$ satisfying $xRh(x)$ form a composition-closed family $H$. The values $Hx$ are directed, since $h(k(x))$ bounds $h(x)$ and $k(x)$. Their least upper bound $h_0(x)$ lies in every such closed set containing $x$, so $h_0\in H$. Then $f\circ h_0\in H$ gives $f(h_0(x))\le h_0(x)$, while inflationarity gives the reverse inequality. If $p=f(p)$ and $x\le p$, the lower set $\{y:y\le p\}$ is closed, proving $h_0(x)\le p$. Thus $h_0(x)$ is the least fixed point of $f$ above $x$.

A Life oscillator is a nonconstant periodic orbit of the Conway's Game of Life update function $F$. Its period is the least positive integer $p$ for which $F^p(A)=A$, comparing the entire live-cell set at the same coordinates. A configuration that never changes is instead a fixed point; a Life glider is not a Life oscillator on the infinite board because it returns at displaced coordinates. The Life blinker has period two.^[ref]

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]