Codimension-two bifurcation

A codimension-two bifurcation requires two independent parameter conditions to hold simultaneously. Simultaneous stationary and Hopf bifurcation thresholds require both kinds of critical mode in the reduced amplitude equation. A crossing of modes at different wavenumbers is different from a zero-frequency double-zero degeneration of one fixed-wave-number cubic.

A cusp is a two-parameter degeneracy at which two saddle-node bifurcation curves meet and the reduced steady equation has a triple root. In the displayed scalar normal form, simultaneous vanishing of the polynomial and its derivative gives ${\beta }_2=3x^2$, ${\beta }_1=-2x^3$. Thus the folds satisfy $4{\beta }_2^3=27{\beta }_1^2$, with ${\beta }_2\ge 0$. Between the folds there are three real equilibria, and outside them one. A transverse stable direction may be present in a higher-dimensional realization.

A fold-Hopf bifurcation combines a simple zero eigenvalue and a nonzero imaginary pair at an equilibrium point. In a three-dimensional centre manifold, a normal form of a dynamical system separates a real fold amplitude from a complex oscillatory amplitude. A rotationally symmetric normal form permits reduction to two real amplitude equations, while the oscillation phase continues to evolve. Secondary bifurcations of the amplitude flow can produce invariant tori in the full dynamical system.

The amplitude equations $u^{\prime }=-2uv$, $v^{\prime }={\lambda }_2+v^2+u^2$ satisfy $u^{\prime }=F_v$, $v^{\prime }=-F_u$, hence preserve the displayed first integral. For ${\lambda }_2=-a^2<0$ and $u\ge 0$, the centre equilibrium is $(a,0)$ with $F=2a^3/3$, while the two boundary saddle equilibria are $(0,\pm a)$ with $F=0$. The level $F=0$ contains the boundary connection and the arc $u^2=3(a^2-v^2)$ joining the two saddle equilibria. Levels $0<F<2a^3/3$ form closed periodic orbits. Adding an independent rotating phase turns these closed amplitude trajectories into invariant tori.

A saddle-node separatrix-loop point occurs when a saddle-node equilibrium simultaneously has a global returning separatrix at its strong stable manifold. In the local form $\dot{x}=x^2-\mu$, $\dot{y}=-\lambda y$, a global return to $y=h$ at $x=\nu$ makes the codimension-two point $\mu =\nu =0$. For $\mu >0$ a saddle loop occurs on $\nu =\sqrt{\mu }$. The saddle-node boundary $\mu =0$ has an invariant-circle segment on the $\nu <0$ side, while for $\nu >0$ a nearby periodic orbit avoids the bottleneck and survives the local equilibrium fold. The return geometry, rather than the local saddle-node normal form alone, determines the global orbit.

A Bogdanov–Takens bifurcation has a double zero eigenvalue with a nontrivial Jordan block and generic two-parameter unfolding. Its local diagram includes saddle-node bifurcations, Hopf bifurcations, and a curve of small homoclinic orbits. A weakly perturbed Hamiltonian scaling can locate the homoclinic curve through energy balance.

For $\lambda >0$, the equilibrium points are the origin and $(s,s^2)$ with $s=(1\pm \sqrt{1+4\mu })/2$. The nonzero branches have Jacobian trace $s-\lambda$ and determinant $\lambda s(2s-1)$. They have a saddle-node bifurcation at $\mu =-1/4$, $s=1/2$; the origin exchanges stability with the lower branch at the transcritical bifurcation $\mu =0$. A Hopf bifurcation occurs on the upper branch at $\mu =\lambda (\lambda -1)$ for $\lambda >1/2$. The point $(\mu ,\lambda )=(-1/4,1/2)$ has a double zero eigenvalue and a nontrivial Jordan block. Its generic unfolding includes a saddle-loop curve and the intervening stable periodic orbit region.

For $\dot{u}=u(\mu +u-v)$, $\dot{v}=\lambda u^2$ in $u>0$, the time change $d\tau /dt=u$ gives the affine linear system $u_{\tau }=\mu +u-v$, $v_{\tau }=\lambda u$. Shifting $w=v-\mu$ removes the forcing. Its eigenvalues are $(1\pm \sqrt{1-4\lambda })/2$. For $\lambda >1/4$, putting $\omega =\sqrt{4\lambda -1}/2$ givesFor fixed $\mu >0$ and initial $h,v_0$ tending to zero, the first return to small positive $u$ occurs after approximately $\pi /\omega =2\pi /\sqrt{4\lambda -1}$. This is a half-turn, not a full turn. At $\lambda =1/4$, $u=e^{\tau /2}[h+(\mu +h/2-v_0)\tau ]$ does not return and grows without bound. The loss of return diagnoses escape to infinity rather than a finite-amplitude homoclinic orbit. Physical time obeys $dt=d\tau /u$ and need not share the divergent desingularized travel time.

For $\lambda >0$, the equilibrium points are $x=\pm \sqrt{\lambda }$, $y=0$. The positive branch is a saddle; the negative branch has a subcritical Hopf bifurcation at $\mu =\sqrt{\lambda }$. A homoclinic orbit occurs at $\mu =(5/7)\sqrt{\lambda }+O(\lambda )$ near the double-zero point. Between this homoclinic curve and the Hopf curve, an unstable periodic orbit surrounds a stable equilibrium point. The factor $5/7$ follows from a Melnikov energy-balance method, using the integrals of $v^2$ and $u{v}^2$ along the leading conservative saddle loop.

// Destination: dynamical-systems.bigb

The sign $\sigma =\pm 1$ distinguishes an inverted quartic potential from a double-well potential. The origin has determinant $\lambda$ and trace $\kappa$. Nonzero equilibria obey $q^2=\lambda /\sigma$, with determinant $-2\lambda$ and trace $\kappa -2\lambda$. Its energy $p^2/2+\lambda q^2/2-\sigma q^4/4$ has derivative $(\kappa -2\sigma q^2)p^2$. The origin's Hopf bifurcation is supercritical for $\sigma =1$ and subcritical for $\sigma =-1$. In the latter case the nonzero equilibria have supercritical Hopf bifurcations at $\kappa =2\lambda <0$.

// Target: dynamical-systems.bigb

For a reflection-symmetric vector field $\dot{u}=v+f_{30}{u}^3+f_{21}{u}^{2}v+\cdots$, $\dot{v}=g_{30}{u}^3+g_{21}{u}^{2}v+\cdots$, introduce $x=u$, $y=\dot{u}$. Differentiating gives $\dot{x}=y$ and a cubic acceleration with $x^{2}y$ coefficient $b=g_{21}+3f_{30}$. The contribution $3f_{30}$ comes from differentiating the first equation's cubic term. Further cubic near-identity transformations can remove nonresonant terms but cannot turn $b=0$ into a nonzero coefficient by a nonsingular scaling. This is a useful check before assigning a generic reflection-symmetric Bogdanov–Takens bifurcation to a calculated Taylor expansion.

For a vector field with linear part $\dot{u}=v$, $\dot{v}=0$ and cubic terms $a_i{u}^3+b_i{u}^{2}v+c_{i}u{v}^2+d_i{v}^3$, write $(u,v)=(x,y)+(h_1,h_2)$ with homogeneous cubic $h_i$. The transformed cubic field is $f_3+Ah-DhA(x,y{)}^T$, where $A(x,y{)}^T=(y,0{)}^T$. Taking the coefficients of $h_2$ to be $(-a_1,c_2/2,d_2,0)$ and those of $h_1$ to be $((b_1+c_2/2)/3,(c_1+d_2)/2,d_1,0)$ eliminates every cubic term in $\dot{x}$ and the $x{y}^2,y^3$ terms in $\dot{y}$. The remaining normal form is $\dot{x}=y$, $\dot{y}=P{x}^3+Q{x}^{2}y$ through cubic order. The invariant damping coefficient contains the contribution $3a_1$ from differentiating the first equation.

Near a nilpotent origin with $\sigma =1$, $\mu =0$, the weighted scaling $u_{\mathrm{old}}={\epsilon }^{2}u$, $v_{\mathrm{old}}=\epsilon v$, $\tau =\epsilon t$, ${\mu }_{\mathrm{old}}={\epsilon }^2\mu$, $\sigma =1+{\epsilon }^{2}s$ makes the leading cubic confinement flow $u^{\prime }=-sv+v^3/2$, $v^{\prime }=2u$. Direct differentiation proves conservation of $H=u^2+s{v}^2/2-v^4/8$. For $s>0$ the origin is a centre and $(0,\pm \sqrt{2s})$ are saddles. Their heteroclinic connections have $H=s^2/2$ and $u=\pm (2s-v^2)/(2\sqrt{2})$ for $|v|\le \sqrt{2s}$. Positive lower energy levels surround the centre; higher levels escape the potential well.

For a chosen orientation and phase-reduced steady amplitude $a$ and nonresonant oscillatory amplitude $z$, a schematic normal form is $\dot{a}=r_{s}a-g_s{a}^3-h_{s}a|z{|}^2$, $\dot{z}=(r_o+i{\omega }_0)z-g_o|z{|}^{2}z-h_o{a}^{2}z$. For $r_s,g_s>0$ the pure steady branch exists and is radially attracting; its oscillatory perturbation is damped when $r_o-h_{o,r}{r}_s/g_s<0$, where $h_{o,r}=\mathrm{Re}(h_o)$. For $r_o,g_{o,r}>0$, $g_{o,r}=\mathrm{Re}(g_o)$, the pure oscillatory branch exists and is radially attracting; its steady perturbation is damped when $r_s-h_s{r}_o/g_{o,r}<0$. The transverse inequalities are not existence or full stability tests by themselves. Mixed intensities $u=a^2>0$, $v=|z{|}^2>0$ solve $g_{s}u+h_{s}v=r_s$, $h_{o,r}u+g_{o,r}v=r_o$. Their intensity Jacobian matrix has trace $-2(g_{s}u+g_{o,r}v)$ and determinant $4uv(g_s{g}_{o,r}-h_s{h}_{o,r})$, giving attraction in intensities when the trace is negative and determinant positive. Resonant phases, oppositely travelling modes or zero-frequency limits need additional retained amplitudes; this schematic pair is not a universal complete normal form.