8 · bifurcation

structural instability, invariant sets and poincare maps for families

A family of vector fields $v_\mu$ (or maps $f_\mu$) depending smoothly on a parameter $\mu\in\mathbb{R}^p$ is structurally unstable at $\mu_0$ when every neighborhood of $v_{\mu_0}$ contains vector fields that are not topologically equivalent to it. The organizing centers of such instability are the non-hyperbolic invariant sets: equilibria whose linearization possesses eigenvalues on the imaginary axis, or periodic orbits whose Floquet multipliers lie on the unit circle.

For a periodic orbit the Poincare map reduces the problem to the bifurcation of fixed points of a family of diffeomorphisms. The multipliers of the orbit become the eigenvalues of the derivative of the Poincare map, and the classic codimension-one bifurcations (saddle-node, period-doubling, Neimark-Sacker) appear as the corresponding bifurcations of fixed points. The same reduction converts the study of bifurcations of limit cycles into the far simpler study of bifurcations of maps.

definition (parametrized family). a $C^r$ family of vector fields is a $C^r$ map $(\mu,x)\mapsto v_\mu(x)$ with $\mu\in\mathbb{R}^p$ and $x\in M$. likewise for families of maps $f_\mu:M\to M$.

definition (bifurcation value). a parameter value $\mu_0$ is a bifurcation value if $v_{\mu_0}$ is not structurally stable in the $C^1$ topology of vector fields (or, more weakly, if the topological equivalence class of the phase portrait changes for $\mu$ near $\mu_0$ in every neighborhood of $\mu_0$).

theorem (hyperbolicity implies local structural stability of equilibria). if $v_{\mu_0}(x_0)=0$ and $Dv_{\mu_0}(x_0)$ is hyperbolic, then there exist neighborhoods of $x_0$ and of $\mu_0$ and a unique $C^r$ equilibrium branch $x(\mu)$ with $x(\mu_0)=x_0$ such that $x(\mu)$ remains hyperbolic and the local phase portrait near $x(\mu)$ is topologically equivalent to that of the linearization, for all $\mu$ near $\mu_0$.

proof. consider $F(\mu,x)=v_\mu(x)$. then $F(\mu_0,x_0)=0$ and $D_x F(\mu_0,x_0)$ is invertible by hyperbolicity. the implicit function theorem yields a unique $C^r$ branch $x(\mu)$ of zeros. eigenvalues depend continuously on the matrix, so hyperbolicity persists for $\mu$ near $\mu_0$. the Hartman-Grobman theorem (chapter 2) supplies local topological conjugacy to the linearization at each such hyperbolic equilibrium, and continuous dependence of the stable/unstable dimensions gives topological equivalence of nearby phase portraits. $\square$

remark (reduction to maps). by chapter 5, a hyperbolic periodic orbit produces a hyperbolic fixed point of a Poincare map. bifurcations of the orbit are therefore equivalent to bifurcations of that fixed point in a family of maps of one lower dimension.

discrete symmetries and constrained bifurcations

When the system commutes with a discrete group action, most commonly a $\mathbb{Z}_2$-action given by a linear involution, the determining equations for bifurcations are forced to be odd (or even). Consequently the generic eigenvalue-crossing conditions of the unrestricted case become non-generic. In the presence of $\mathbb{Z}_2$-equivariance a simple eigenvalue may be constrained to remain zero, or a complex conjugate pair may be forced to cross the imaginary axis simultaneously with a real eigenvalue. The resulting bifurcations of equilibria and of cycles are still of codimension one, but their normal forms contain only even or only odd terms, and the associated branching diagrams exhibit the characteristic pitchfork or symmetry-breaking structure.

definition ($\mathbb{Z}_2$-equivariant family). let $R$ be a linear involution $R^2=I$. a family $v_\mu$ is $R$-equivariant if $v_\mu(Rx)=R v_\mu(x)$ for all $x,\mu$.

theorem (odd normal form for a simple zero under $\mathbb{Z}_2$). let $v_\mu:\mathbb{R}\to\mathbb{R}$ be a $C^3$ family of scalar fields with $v_\mu(-x)=-v_\mu(x)$ for all $\mu$, and suppose $v_0(0)=0$, $v_0'(0)=0$, and $\partial_\mu v_0'(0)\neq 0$. then after a smooth parameter and coordinate change the family is equivalent, near $(\mu,x)=(0,0)$, to $$ \dot x=\mu x\pm x^3 $$ (supercritical or subcritical pitchfork according to the sign).

proof. equivariance forces $v_\mu(x)=x\,a(\mu,x^2)$ for a smooth $a$. write $a(\mu,u)=\alpha(\mu)+\beta(\mu)u+r(\mu,u)$ with $r=O(u^2)$. conditions $v_0'(0)=0$ and $\partial_\mu v_0'(0)\neq 0$ become $\alpha(0)=0$ and $\alpha'(0)\neq 0$. reparametrize $\mu$ so that $\alpha(\mu)=\mu$. if $\beta(0)\neq 0$ set $\eta=\mathrm{sign}(\beta(0))$ and rescale $x$ by $|\beta(0)|^{1/2}$ to normalize $|\beta(0)|=1$, obtaining $\dot x=\mu x+\eta x^3+O(x^5)$. a further near-identity equivariant change of coordinates (standard, from invertibility of the homological operator on odd quintic remainders once the cubic coefficient is nonzero) removes terms of order $\ge 5$ at the level of $C^3$ topological normal forms, leaving $\dot x=\mu x\pm x^3$. $\square$

remark. without the $\mathbb{Z}_2$ constraint the generic bifurcation of a simple zero eigenvalue is the saddle-node $\dot x=\mu\pm x^2$, which is forbidden for odd fields.

two-parameter bifurcations and normal forms

When two parameters are free to vary, new resonances and degeneracies become generic. The classification of these codimension-two bifurcations proceeds by computing a normal form: a polynomial vector field, unique up to higher-order terms and smooth coordinate changes, that captures the qualitative dynamics in a neighborhood of the critical point. The computation relies on the Fredholm alternative in the space of homogeneous polynomials and on the action of the group of invertible linear changes of coordinates that preserve the linear part. Once the normal form is known, a standard unfolding, containing the minimal number of parameters, is derived by adding the lowest-order terms that restore transversality with respect to the original family.

definition (normal form relative to a linear part $A$). write a vector field $v(x)=Ax+v_2(x)+v_3(x)+\cdots$ in homogeneous components. a polynomial change of coordinates $x=y+h_k(y)$ with $h_k$ homogeneous of degree $k$ transforms the degree-$k$ term by the homological operator $$ L_A h=[A,h]:=Dh\cdot A\,y-A\,h(y). $$ a complement to $\mathrm{im}\,L_A$ in the space of homogeneous fields of degree $k$ is a space of normal-form terms of that degree.

theorem (homological solvability by fredholm). on the finite-dimensional space $H_k$ of homogeneous polynomial vector fields of degree $k$ on $\mathbb{R}^n$, the operator $L_A:H_k\to H_k$ satisfies $$ H_k=\mathrm{im}\,L_A\oplus C_k $$ for any algebraic complement $C_k$ of the image. a term $v_k\in H_k$ can be removed by a degree-$k$ change of coordinates if and only if its projection onto $C_k$ vanishes (equivalently, if $v_k$ is orthogonal to $\ker L_A^*$ with respect to a chosen inner product on $H_k$).

proof. $H_k$ is finite-dimensional, so $L_A$ is a linear endomorphism and $\mathrm{im}\,L_A$ admits a complementary subspace $C_k$. by the finite-dimensional Fredholm alternative (chapter 4), $v_k\in\mathrm{im}\,L_A$ if and only if $\langle v_k,w\rangle=0$ for all $w\in\ker L_A^*$. choosing $C_k=(\ker L_A^*)^\perp$ makes the condition equivalent to vanishing of the $C_k$-component of $v_k$. solving $L_A h_k=v_k^{\mathrm{im}}$ removes that image part and leaves only the normal-form component in $C_k$. $\square$

definition (versal unfolding). a family $v_\mu$ through $v_0$ is a versal unfolding of a singularity if every other perturbation of $v_0$ is fiberwise equivalent, after smooth reparametrization of parameters, to a family induced from $v_\mu$. the minimal number of parameters is the codimension of the singularity.

the five critical codimension-two bifurcations

Five bifurcations organize the overwhelming majority of two-parameter phenomena observed in applications.

definition (cusp). a saddle-node bifurcation in which the quadratic coefficient also vanishes. the normal form is $\dot x=r+sx\pm x^3$. its unfolding produces a hysteresis curve and a pair of saddle-node lines meeting tangentially at the cusp point.

definition (bautin / generalized hopf). a Hopf bifurcation in which the first Lyapunov coefficient vanishes. the normal form contains a fifth-order term; the unfolding produces a curve of neutral saddles and a locus of limit-point-of-cycles bifurcations that bound a region of bistability between a stable equilibrium and a stable periodic orbit.

definition (bogdanov-takens / double-zero). a double-zero eigenvalue with a single Jordan block. the normal form is $$ \dot x=y,\qquad \dot y=\beta_1+\beta_2 x+ax^2+bxy. $$ its unfolding contains a saddle-node curve, a Hopf curve, and a homoclinic curve that meet at the origin; it is the principal organizing center for the transition between equilibria and large-amplitude periodic orbits in planar systems.

definition (fold-hopf / zero-hopf). a zero eigenvalue and a purely imaginary pair. the normal form lives on a three-dimensional center manifold and produces, among other phenomena, invariant tori and Shilnikov-type homoclinic orbits.

definition (hopf-hopf). two distinct pairs of purely imaginary eigenvalues. the normal form on a four-dimensional center manifold yields a rich web of Neimark-Sacker bifurcations, resonant tori, and heteroclinic cycles.

theorem (saddle-node normal form, codimension one). let $v_\mu$ be a $C^2$ family of scalar fields with $v_0(0)=0$, $v_0'(0)=0$, $v_0''(0)\neq 0$, and $\partial_\mu v_0(0)\neq 0$. then there is a $C^1$ change of coordinates and parameter reducing the family to $$ \dot x=\mu\pm x^2 $$ near the origin.

proof. write $v_\mu(x)=a(\mu)+b(\mu)x+c(\mu)x^2+o(x^2)$ with $a(0)=b(0)=0$, $c(0)\neq 0$, and $a'(0)\neq 0$. reparametrize $\mu$ so that $a(\mu)=\mu$. a near-identity shift $x=y+\alpha(\mu)$ with $\alpha(0)=0$ chosen to kill the linear term (possible by invertibility of $c(0)$ against $b(\mu)$ for small $\mu$) leaves $\dot y=\mu+\tilde c(\mu)y^2+o(y^2)$ with $\tilde c(0)=c(0)\neq 0$. rescale $y$ and the time (or absorb constants into $\mu$) to normalize $|\tilde c|=1$, obtaining $\dot y=\mu\pm y^2$ at leading order, which is $C^1$-equivalent to the quadratic normal form near the bifurcation. $\square$

bogdanov-takens as organizing center

In planar systems the Bogdanov-Takens point is the most important codimension-two singularity. By varying the two unfolding parameters one successively encounters a saddle-node bifurcation that creates a pair of equilibria, a Hopf bifurcation that creates a small limit cycle, and a homoclinic bifurcation in which the cycle collides with a saddle and disappears. All of the local and global transitions that connect equilibria to large-amplitude oscillations are therefore already present in the two-parameter unfolding of this single normal form.

theorem (equilibria of the BT unfolding). for the truncated normal form $$ \dot x=y,\qquad \dot y=\beta_1+\beta_2 x+x^2+xy $$ (a fixed choice of nonzero coefficients $a,b$ after scaling), the equilibria occur where $y=0$ and $\beta_1+\beta_2 x+x^2=0$. the discriminant $\Delta=\beta_2^2-4\beta_1$ bounds a region of two equilibria (when $\Delta>0$) from a region of none (when $\Delta<0$), separated by a saddle-node curve $\Delta=0$ with $\beta_2$ free (except at the origin).

proof. equilibria satisfy $y=0$ and $\beta_1+\beta_2 x+x^2=0$. the quadratic $x^2+\beta_2 x+\beta_1=0$ has discriminant $\Delta=\beta_2^2-4\beta_1$ and opens upwards. two real roots exist precisely when $\Delta>0$; a double root when $\Delta=0$; none when $\Delta<0$. the set $\Delta=0$ is the parabola $\beta_1=\beta_2^2/4$ in the $(\beta_1,\beta_2)$-plane, which is the saddle-node locus in the unfolding. $\square$

remark (hopf and homoclinic loci). linearization about the nontrivial equilibria yields a Hopf curve for a range of $(\beta_1,\beta_2)$ where a pure imaginary pair occurs with nonzero first Lyapunov coefficient. matching of a homoclinic orbit of the saddle can be proved by a Melnikov or blow-up analysis in a rescaled neighborhood of the BT point; the three curves meet at the origin with standard asymptotic tangencies. the geometric program of the chapter is complete once this skeleton is in place: codimension-two BT organizes the entire planar equilibrium-cycle-homoclinic scenario.

numerical continuation

Theoretical normal forms are complemented by rigorous numerical continuation. Isolated equilibria are located by Newton or Broyden iteration applied to the vector field. Periodic orbits are continued by solving a boundary-value problem on a fixed interval with an integral phase condition that eliminates the translational invariance in time (chapters 4 and 5). Period-doubling cascades are tracked by appending the period-doubling condition (a multiplier equal to $-1$) and continuing in two parameters. Arnold tongues, regions of phase-locking in periodically forced systems, are delimited by continuing the saddle-node bifurcations of periodic orbits that bound each resonance zone. Modern continuation packages implement these algorithms with adaptive step-size control, automatic detection of bifurcations via test functions, and branch-switching at detected critical points.

definition (test function for a bifurcation). a smooth function $\psi(\mu,x)$ on the solution manifold of the extended equilibrium or periodic problem that vanishes precisely when a prescribed degeneracy occurs (e.g. $\det Dv_\mu(x)=0$ for a fold, or $\det(DP+I)=0$ for period-doubling of a Poincare map).

remark. Newton convergence, bordering for phase constraints, and Fredholm solvability of variational equations (chapter 4) are exactly the linear-algebraic ingredients of these continuum algorithms; bifurcation theory supplies the test functions and branch-switching logic.

singular symplectic reduction as geometric counterpart of symmetry-breaking

When a continuous symmetry group acts on a symplectic manifold, the ordinary Marsden-Weinstein reduced space is smooth only at regular values of the moment map. At singular values the reduced space becomes a stratified Poisson space whose strata are the symplectic manifolds corresponding to the different orbit types. The passage from a smooth reduced space to a stratified space as a parameter (for example a momentum value) crosses a critical level is the precise geometric analog of a symmetry-breaking bifurcation.

Cylinder-valued momentum maps and optimal reduction techniques extend the construction to actions that do not admit an ordinary $\mathfrak{g}^*$-valued moment map; the resulting reduced spaces are again stratified, and their symplectic leaves change dimension precisely when the isotropy type jumps. Thus the algebraic geometry of singular reduction supplies the global, coordinate-free counterpart of the local normal-form analysis of bifurcations that break continuous symmetries.

definition (marsden-weinstein reduced space, regular case). if $G$ acts freely and properly by symplectomorphisms on $(M,\omega)$ with equivariant moment map $\mu:M\to\mathfrak{g}^*$, and $\nu$ is a regular value, then $$ M_\nu=\mu^{-1}(\nu)/G_\nu $$ is a smooth symplectic manifold (with form uniquely characterized by pullback of $\omega$ to $\mu^{-1}(\nu)$).

theorem (singular values force orbit-type strata). if $\nu$ is a critical value of $\mu$, then $\mu^{-1}(\nu)$ is not a smooth manifold in general, and the quotient $\mu^{-1}(\nu)/G_\nu$ decomposes into a union of smooth symplectic strata labeled by conjugacy classes of stabilisers (orbit types). the dimension of a stratum drops precisely when the stabiliser enlarges.

proof (dimension count). at $x\in\mu^{-1}(\nu)$ the tangent to the level set (where smooth) is $\ker D\mu(x)$, while $T_x(G\cdot x)$ is spanned by fundamental fields $\xi_M(x)$. the identity $\ker D\mu(x)=(T_x(G\cdot x))^\omega$ (chapter 6 Noether geometry) implies $$ \dim\mu^{-1}(\nu) =\dim M-\dim\mathfrak{g}+\dim G_x $$ when $\nu$ is regular enough for the level to be a manifold near $x$. quotienting by the residual $G_\nu$-action of dimension $\dim G-\dim G_x$ (for free action of the residual group on the orbit type) yields stratum dimension $\dim M-2\dim G+2\dim G_x$. thus as the conjugacy class of $G_x$ jumps to a larger stabiliser, the stratum dimension decreases. different conjugacy classes give smooth manifolds of constant stabiliser type (tube theorem / slice theorem for proper actions), and symplectic forms descend on each such piece by the same Marsden-Weinstein formula restricted to the orbit type. this is the stratified Poisson structure of singular reduction. $\square$

remark. crossing a critical value of $\mu$ as a control parameter therefore changes the stratified topology of the reduced phase space, mirroring a bifurcation of relative equilibria and relative periodic orbits that break continuous symmetry.

summary

Bifurcation theory combines the local analytic technique of normal forms with the global geometric notions of structural stability, equivariance, and symplectic reduction. The five codimension-two singularities, together with their numerical continuation, organize the transition scenarios observed in generic two-parameter families, while singular reduction realizes the same transitions as changes in the stratified structure of the reduced phase space.

exercises

exercise 1 (saddle-node sketch). for $\dot x=\mu-x^2$ locate the equilibria as functions of $\mu$, determine their stability, and sketch the bifurcation diagram in the $(\mu,x)$-plane.

exercise 2 (pitchfork). for $\dot x=\mu x-x^3$ show that $x=0$ changes stability at $\mu=0$ and that two new stable branches exist for $\mu>0$. which $\mathbb{Z}_2$ symmetry forces the normal form to be odd?

exercise 3 (hopf linear data). let $$ A(\mu)=\begin{pmatrix}\mu&-1\\1&\mu\end{pmatrix}. $$ compute the eigenvalues and verify that a complex conjugate pair crosses the imaginary axis transversally at $\mu=0$ (a Hopf condition for a planar linearization).

exercise 4 (BT equilibria). for $\dot x=y$, $\dot y=\beta_1+\beta_2 x+x^2$, find the saddle-node curve in the $(\beta_1,\beta_2)$-plane and determine for which parameters there are two equilibria.

exercise 5 (saddle-node normal form). for $\dot x=\mu+x^2$ on the line, compute the equilibria as functions of $\mu$, determine their stability, and draw the bifurcation diagram in the $(\mu,x)$ plane. show that $\mu=0$ is the only value where structural stability fails in this one-parameter family.