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

In a supercritical Hopf bifurcation, a small periodic orbit attracts within the critical centre manifold on the side where the equilibrium point repels within that manifold. Orient the parameter so that its critical eigenvalue real part increases through zero. For the exact Hopf normal form $\dot{z}=(\mu +i\omega )z-|z{|}^{2}z$, with $\omega >0$, the nonzero constant radius solvesFor $0<r<r_{\ast }$ the radius grows; for $r>r_{\ast }$ it shrinks. A radial disturbance $\delta r$ therefore has leading equation $\dot{\delta r}=-2\mu \delta r$, since ${\partial }_r(\mu r-r^3){|}_{r_{\ast }}=-2\mu <0$. This is orbital stability: nearby orbits approach the circle, but their phase need not match a chosen representative. The exact angular frequency is $\omega$, so the period is $2\pi /\omega$ and the frequency is $\omega /(2\pi )$.

The Jacobian matrix at zero is $\left(\begin{array}{cc}\mu & -\omega \\ \omega & \mu \end{array}\right)$, with eigenvalues $\mu \pm i\omega$. Zero attracts for $\mu <0$ and repels for $\mu >0$. Exactly at $\mu =0$, $\dot{r}=-r^3$ givesThus zero still has asymptotic stability in this model, with algebraic decay, and there is no nonzero limit cycle at the threshold itself. As $\mu \downarrow 0$ on the cycle branch, both its radius $\sqrt{\mu }$ and its radial attraction rate $2\mu$ vanish. Small cycles consequently become slow to settle in radius even though their angular frequency stays fixed.

// Target: dynamical-systems.bigb

For $\ddot{q}=-{\omega }^{2}q+g_2{q}^2+g_3{q}^3+(\tau +f_{1}q+f_2{q}^2)\dot{q}+\cdots$, the Hopf bifurcation amplitude equation has cubic coefficient $(f_2+f_1{g}_2/{\omega }^2)/8$ in the leading position amplitude. The quadratic restoring term produces mean shift $g_2{r}^2/(2{\omega }^2)$ and second harmonic $-g_2{r}^2\cos (2\omega t)/(6{\omega }^2)$. Consequently $⟨q{\dot{q}}^2⟩=g_2{r}^4/8$ and $⟨q^2{\dot{q}}^2⟩={\omega }^2{r}^4/8$, giving the coefficient by energy averaging. A negative value gives a supercritical stable limit cycle; a zero value requires higher-order analysis.

A generalized Hopf bifurcation has a vanishing cubic radial coefficient and nonzero quintic coefficient. The two parameters unfold the eigenvalue real part and cubic coefficient. With $R=r^2$, periodic orbits satisfy $\tau +aR+b{R}^2=0$. Their saddle-node bifurcation curve is $\tau =a^2/(4b)$ with $R=-a/(2b)>0$, hence $ab<0$. For $b>0$ the smaller-radius limit cycle is stable and the larger one unstable where both exist; their collision explains why the criticality change of a Hopf bifurcation curve brings a saddle-node bifurcation of periodic orbits.

// Destination: partial-differential-equation.bigb

Two complex Hopf amplitudes of even and odd spatial parity transform under reflection as $(A,B)\mapsto (A,-B)$. In a nonresonant double Hopf bifurcation, the cubic terms are $A|A{|}^2,A|B{|}^2$ and their $B$ counterparts. At equal temporal frequencies, common phase covariance also permits $B^2\overline{A}$ and $A^2\overline{B}$. Distinct frequencies alone do not guarantee nonresonance: a ratio of two allows the quadratic terms $B^2$ and $A\overline{B}$. The frequency assumptions must therefore be included when deriving the normal form of a dynamical system.

A Hopf algebra is commutative when its multiplication is commutative. This is independent of whether its coproduct is cocommutative; function algebras of finite nonabelian groups give commutative but noncocommutative examples.

A Hopf algebra is cocommutative when flipping the two tensor factors leaves its coproduct unchanged. A group algebra has coproduct $g\mapsto g\otimes g$, so it is cocommutative even for a nonabelian group, whose group algebra is not commutative.

Given a Yang-Baxter R-matrix, take generators $t_{ij}$ subject to the displayed matrix relation, with $T_1=T\otimes I$, $T_2=I\otimes T$. The coproduct $\Delta t_{ij}={\sum }_k{t}_{ik}\otimes t_{kj}$ and counit $\epsilon (t_{ij})={\delta }_{ij}$ give a bialgebra. An antipode requires additional inverses or an appropriate determinant quotient; an arbitrary RTT bialgebra is not automatically a Hopf algebra.

Over a field with $q\ne 0$ and $q^2\ne 1$, this Hopf algebra has generators $E,F,K,K^{-1}$ and relations $KE=q^{2}EK$, $KF=q^{-2}FK$ and $EF-FE=(K-K^{-1})/(q-q^{-1})$. One comultiplication convention is $\Delta E=E\otimes K+1\otimes E$, $\Delta F=F\otimes 1+K^{-1}\otimes F$ and $\Delta K=K\otimes K$. Its counit takes $E,F$ to zero and $K$ to one. The antipode takes $E$ to $-E{K}^{-1}$, $F$ to $-KF$ and $K$ to $K^{-1}$.

An algebra over a field acted on by a bialgebra is a module algebra when multiplication and unit are module maps: the displayed product identity holds and $h1=\epsilon (h)1$. This expresses the product rule determined by comultiplication.

An element $x$ of a Hopf algebra $H$ is primitive whenNo grading is required. Applying the counit to either tensor factor shows $\epsilon (x)=0$. If $x,y$ are primitive, the algebra-homomorphism property of comultiplication givesbecause elements in the two separate tensor factors commute. Thus the primitive elements form a Lie algebra under the commutator.

In a universal enveloping algebra over $\mathbb{Q}$, the primitive elements are exactly the original Lie algebra. The Poincaré-Birkhoff-Witt theorem identifies the associated graded algebra with a symmetric algebra; in characteristic zero, a nonzero homogeneous polynomial of degree greater than one cannot be additive under the diagonal. The highest-degree part of a primitive element must therefore have degree one. Positive characteristic of a field changes this conclusion: if $x$ is primitive in characteristic $p$, then $x^p$ is primitive too, since the intermediate binomial coefficients vanish modulo $p$.

A fibration is a map with the homotopy lifting property. A Hurewicz fibration has this property for every source space; a Serre fibration requires it for CW complexes. Replacing a map by a fibration gives a model for its homotopy fiber sequence.

Approximate a continuous sphere-valued map by smooth real coordinate functions, then normalize. A sufficiently close approximation is homotopic to the original map by normalized straight interpolation. Sard theorem implies the smooth map misses a point when its source dimension is smaller than the sphere dimension. The punctured sphere is contractible, so the map is null-homotopic, even if the source is disconnected.

// Target: geometry-and-topology.bigb

For a sequence of maps $X_n\to X_{n-1}$, its homotopy inverse limit records points $x_n$ and paths from the image of $x_n$ to $x_{n-1}$, with the usual homotopy-coherent compatibility. A fibrant replacement tower computes it by an ordinary inverse limit. For a tower whose homotopy groups are eventually constant in each degree, the limit has those stable groups: the Milnor inverse-limit exact sequence has zero derived-limit term. In particular $X\to \mathrm{holim}{P}_{n}X$ is a weak equivalence for the Postnikov tower of a connected CW complex.

In the based homotopy category of CW complexes, the objects are based CW complexes and the morphisms $X\to Y$ are based homotopy classes $[X,Y{]}_{\ast }$. Their function composition is $[g]\circ [f]=[g\circ f]$. It is well-defined because composing a homotopy with a continuous map again gives a homotopy. A based homotopy equivalence becomes an isomorphism in this category.

For example, based continuous maps $S^1\to S^1$ are classified by their topological degree $m\in \mathbb{Z}$. Composing maps of degrees $m,n$ gives degree $mn$, so this particular endomorphism operation is multiplication of integers. A based contractible space of CW complex type becomes isomorphic to a point. Such examples explain how the category retains deformation information while forgetting a choice of representative continuous map.

Within the simply connected homotopy category, rationalization of a topological space instead inverts rational homotopy equivalences, producing a category represented by rational spaces. This qualification matters: the definition via homotopy groups tensored with $\mathbb{Q}$ does not describe arbitrary nonnilpotent fundamental groups. More generally a model category has a homotopy category obtained by formally inverting its weak equivalences.

A kernel records the information lost by a homomorphism. For a group homomorphism it is the preimage of the identity element, giving the kernel of a group homomorphism; for a ring homomorphism or linear map it is the preimage of zero, giving the kernel of a ring homomorphism or kernel of a linear map. More generally, the relation $a\sim b⟺f(a)=f(b)$ records which inputs the homomorphism identifies. In the group case, its equivalence classes are cosets of the group kernel, as explained by fibres of a group homomorphism are cosets of its kernel.

The universal coefficient theorem relates homology or cohomology with different coefficient groups through tensor and Tor, or Hom and Ext, respectively. The universal coefficient theorem for homology is its tensor-Tor sequence; the universal coefficient theorem for cohomology is its Hom-Ext sequence.

For a closed manifold $M$ with an orientation and dimension four, its intersection form is the symmetric bilinear formon $H^2(M;\mathbb{Z})$ modulo its torsion subgroup. By Poincare duality it is the signed-intersection pairing on the free part of $H_2(M;\mathbb{Z})$, and it is a unimodular intersection pairing. Its real extension has positive and negative indices whose difference is its signature.

This is the four-dimensional case of the general intersection form. On a compact oriented four-dimensional manifold with boundary, the perfect intersection pairing uses one absolute and one relative homology class. An absolute self-pairing alone can be degenerate, so it should not be identified with the closed-manifold unimodular form without a boundary qualification.

A chain cycle is null-homologous with the chosen coefficients when its homology class is zero. Equivalently it is a chain boundary: there is a chain $b$ one degree higher such that $\partial b=c$.

A geometric bounding argument over the integers needs compatible orientations. If a compact oriented submanifold $W$ has oriented boundary $C$, a chain representing its relative fundamental class has boundary representing $C$, so $[C]=0$ in the ambient integral homology. With mod-two coefficients, orientation signs disappear and an unoriented bounding submanifold suffices.

For example, the boundary circle of a Möbius band bounds that band as an unoriented surface, but represents twice the core generator in $H_1(\text{Mo¨bius band};\mathbb{Z})\cong \mathbb{Z}$. It is not integrally null-homologous inside the band. Its mod-two homology class is zero. This distinguishes bounding a geometric subset from bounding a chain with the required coefficients.

A generalized homology theory obeys the homotopy, exactness, suspension and wedge axioms, without imposing the ordinary dimension axiom. A represented homology theory is obtained by taking stable homotopy groups after a smash product of spectra.