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For a Floquet multiplier $m\ne 0$ of a period-$T$ linear variational equation, a Floquet exponent is any logarithmic rate $\rho$ satisfying the displayed relation. Its real part is uniquely $T^{-1}\log |m|$, while its imaginary part is defined modulo $2\pi /T$. Negative transverse real parts give local attraction to a periodic orbit; the autonomous phase direction has multiplier one and exponent zero. Constant-coefficient amplitude equations supply exponents directly as the eigenvalues of their Jacobian matrix.

The Floquet growth rate measures the exponential trend of a periodic linear differential equation over many coefficient periods. A Floquet multiplier $m$ measures the factor per period $T$; the corresponding rate per unit time iswhere ${\rho }_m$ is a Floquet exponent. Taking the real part removes the ambiguity in the complex logarithm.

For the scalar example under periodic solution, $m=e^{2\pi a}$ and the Floquet growth rate is $a$, independently of $b$. Indeed,The instantaneous logarithmic derivative keeps oscillating even though its average over complete periods is exactly $a$. Sampling at the same phase in each period removes the modulation altogether: $y(nT)=m^{n}y(0)$.

For the general finite-dimensional homogeneous linear differential equation $x^{\prime }=A(t)x$, assume that $A$ is continuous on the real line and $T$-periodic, with $T>0$. Normalize its fundamental matrix by $\Phi (0)=I$ and set $M=\Phi (T)$. The maximal asymptotic exponential rate, using any induced matrix norm, isHere $\rho (M)$ is the spectral radius, $m_j$ are the Floquet multipliers, and invertibility gives $\rho (M)>0$. This concerns growth of the solution operator; it does not say that every initial condition attains the maximum.^[ref]

To prove the formula, write $t=nT+s$ with $0\le s<T$. Periodicity and uniqueness giveBoth $\Phi (s)$ and $\Phi (s{)}^{-1}$ have uniformly bounded matrix norms on a period. Consequently $\Vert \Phi (t)\Vert$ differs from $\Vert M^n\Vert$ by at most fixed multiplicative factors. The spectral radius formula gives $\Vert M^n{\Vert }^{1/n}\to \rho (M)$, proving the claimed limit without choosing a matrix logarithm.

For a particular nonzero initial condition $v$, decompose $v$ into the generalized eigenspaces of $M$ over the complex numbers. The rate of $\Phi (t)v$ is the largest $T^{-1}\log |m_j|$ among components that are nonzero. Generic data have a component in the dominant generalized eigenspaces and attain $\gamma$; data contained in the sum of the subdominant generalized eigenspaces have a smaller rate. For example, with $A=\mathrm{diag}(0.2,-0.3)$, the maximal rate is $0.2$, while $v=(0,1)$ decays at rate $-0.3$. These invariant subspaces describe initial data at phase zero. Their transported versions $E(t)=\Phi (t)E(0)$ can vary with time and satisfy $E(t+T)=E(t)$.

A negative Floquet growth rate allows temporary amplification. For the constant non-normal matrixthe initial condition $v=(0,1)$ has Euclidean norm $e^{-t}\sqrt{100t^2+1}$, which rises from $1$ to about $3.697$ at $t=1$ before eventually decaying. Its exponential rate is still $-1$: the Jordan block contributes a polynomial factor, whose logarithm divided by $t$ tends to zero. At a unit-modulus Floquet multiplier, such polynomial growth can even be unbounded with zero exponential rate; $A=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ gives $e^{tA}v=(t,1)$. Thus zero Floquet growth rate alone proves neither boundedness nor a periodic solution.

A Floquet multiplier is an eigenvalue of a one-period linear evolution map. For a homogeneous linear differential equation with continuous coefficients of period $T>0$, this map is the monodromy matrix of a periodic linear system $N$. If $Nv=mv$ for a nonzero eigenvector $v$, the associated solution of a differential equation with $Y(0)=v$ satisfies $Y(t+T)=mY(t)$. Its exponential rate is $T^{-1}\log |m|$, the corresponding Floquet growth rate. A Floquet multiplier equal to one gives a $T$-periodic solution in that eigenvector direction; a nontrivial Jordan block at a unit-modulus Floquet multiplier can still produce polynomial growth.

For a nonlinear periodic orbit, the Floquet multipliers are eigenvalues of the one-period map of its variational equation. A nonconstant periodic orbit of a smooth autonomous differential equation has a Floquet multiplier equal to one in its tangent direction: differentiating the solution of a differential equation with respect to its starting phase gives a periodic solution of the variational equation. The derivative of a transverse Poincaré map removes this phase direction and retains the transverse Floquet multipliers. Transverse Floquet multipliers strictly inside the unit circle give local attraction, while one outside gives instability; a transverse Floquet multiplier on the unit circle requires further analysis. For a planar periodic orbit, the transverse Floquet multiplier is given by the divergence test for a planar periodic orbit.

For a continuous periodic coefficient matrix $A(t+L)=A(t)$, a fundamental matrix $X$ for $X^{\prime }=A(t)X$, normalized by $X(0)=I$, obeys $X(t+L)=X(t)X(L)$. The invertible matrix $X(L)$ has a complex matrix logarithm; set $B=L^{-1}\log X(L)$ and $P(t)=X(t)e^{-tB}$. Since $X(L)=e^{LB}$ commutes with $B$, direct substitution gives $P(t+L)=P(t)$ and proves the factorization. A real logarithm is not required for this complex representation.

The eigenvalues of $X(L)$ are Floquet multipliers. All solutions are bounded for forward time when every multiplier has modulus at most one and those on the unit circle have no nontrivial Jordan blocks. In a real undamped second-order scalar equation with no first-derivative term, the monodromy determinant is one. Distinct conjugate unit-modulus multipliers give bounded solutions; a nontrivial Jordan block at multiplier $1$ or $-1$ gives an unbounded second solution even when one periodic or antiperiodic solution exists.

For $\dot{r}=r(1-r)(r-2\sin \theta )$, $\dot{\theta }=r$, the circle $r=1$ has normal Floquet multiplier $e^{-2\pi }$ and is attracting. Inside it, $w=(1-r{)}^{-1}$ satisfies $dw/d\theta =(1-2\sin \theta )w-1$. The solution through $w(\pi )=1$ has $w>1$ on one interval $({\theta }_b,\pi )$ with $-\pi <{\theta }_b<0$. The corresponding radius $r=1-1/w$ is a homoclinic orbit to the origin. Near its returning endpoint $r\sim (\theta -\pi {)}^2$, so the approach takes infinite physical time. Comparison for the scalar angular equation shows that smaller radii inside this loop approach the origin, whereas all positive radii outside it approach the circle. The origin is therefore not Lyapunov stable despite attracting an open set.

For an autonomous ordinary differential equation, the tangent to a periodic orbit gives a unit Floquet multiplier. A hyperbolic attracting cycle has that multiplier simple and all transverse multipliers strictly inside the unit disk. Thus the periodic variational operator has a one-dimensional null space spanned by the tangent. Nonlinear asymptotic attraction without hyperbolicity does not suffice: $\dot{r}=-(r-1{)}^3$, $\dot{\phi }=2\pi$ attracts to a unit-period circle but its radial variational equation is neutral.

If $\dot{\Phi }=A(t)\Phi$ and $\Phi (0)=I$, differentiation of the determinant gives $\det \Phi (t)=\exp ({\int }_0^t\mathrm{tr}A(s)ds)$. For a planar periodic orbit this gives the nontrivial Floquet multiplier, since the tangent multiplier is one.

For the homogeneous linear differential equation $\dot{Y}=K(t)Y$ with continuous coefficient matrix of period $P>0$, the monodromy matrix of a periodic linear system is the one-period map from $Y(0)$ to $Y(P)$. If $X$ is any fundamental matrix,Its eigenvalues are the Floquet multipliers. Multipliers are nonzero, since a fundamental matrix is invertible. Changing the starting phase conjugates $N$, so the Floquet multipliers and Jordan blocks are independent of that choice.^[ref]

This turns a continuous-time problem into repeated matrix multiplication:If $Nv=mv$, the associated solution of a differential equation obeys $Y(t+P)=mY(t)$ at every time, not merely at the sampled times. In particular, $Nv=v$ gives a nonzero $P$-periodic solution, and every nonzero $P$-periodic solution arises this way. If $Nv=-v$, the state changes sign after one period and repeats after two. A unit-modulus Floquet multiplier different from $1$ need not give repetition after one period, and a generalized eigenvector in a nontrivial Jordan block need not even give bounded motion.

For the scalar example under periodic solution, $N$ is the one-by-one matrix $(e^{2\pi a})$. Numerically solving a matrix problem uses the same idea: integrate one initial value problem for each column of $X(0)=I$ through one period, then compute the eigenvalues of $X(P)$. This is useful when an explicit solution of a differential equation is unavailable: one period determines the Floquet growth rate over arbitrarily many periods.

For piecewise constant coefficients, which are also covered by the one-period construction, $N$ is the time-ordered product of the segment matrix exponentials, with the earliest segment on the right. For example, take period $P=2$ and alternateSince $K_1^2=K_2^2=0$,Replacing the coefficient by its mean would instead give $e^{K_1+K_2}=\left(\begin{array}{cc}\cosh 1& \sinh 1\\ \sinh 1& \cosh 1\end{array}\right)$ and the incorrect rate $0.5$. The matrices do not commute. The scalar mean-coefficient formula therefore cannot simply be transferred to matrix systems. Nevertheless the Liouville formula for a fundamental matrix gives the correct determinant even in the noncommuting case:In the example, this is $1=m_{+}{m}_{-}$; it fixes the product of the Floquet multipliers, rather than either multiplier separately.

In Sur les équations différentielles linéaires à coefficients périodiques (1883), Gaston Floquet calls a function satisfying $f(t+P)=mf(t)$ periodic of the second kind, and distinguishes the genuinely periodic function obtained when $m=1$. This historical terminology captures precisely the difference between repetition of the coefficient and repetition of the state.^[ref]

A Neimark–Sacker bifurcation of a smooth planar map occurs when a simple complex-conjugate pair of fixed-point Floquet multipliers crosses the unit circle away from low-order resonances. If its radial normal form is $r^{\prime }=r(1+\eta +d{r}^2)+\cdots$ with $d\ne 0$, a small invariant closed curve has $r^2=-\eta /d+\cdots$. For $d<0$ it appears on $\eta >0$, where the fixed point has become unstable, and its radial Floquet multiplier is $1-2\eta +\cdots$, so the bifurcation is supercritical. For $d>0$ the curve lies on $\eta <0$ and is unstable, giving a subcritical bifurcation. Area-preserving maps have a vanishing cubic radial drift at an elliptic point and do not meet this dissipative nondegeneracy condition.

With $\sigma =\omega (ϵ)t$, $\tau =ϵt$, the multiscale chain rule gives $d/dt=(1+ϵ{\Omega }_1+\cdots ){\partial }_{\sigma }+ϵ{\partial }_{\tau }$. Expanding $F(u_{i,0}+ϵu_{i,1})=F(u_{i,0})+ϵDF(u_{i,0})u_{i,1}+\cdots$ gives at order oneAt order $ϵ$, moving the linearized reference dynamics to the left givesThe component sum runs over the $m$ state variables; the source uses $n$ in its first displayed sum and $m$ later. All functions for the solvability calculation are periodic in $\sigma$ with period one.

Differentiating the autonomous cycle equation $U^{\prime }=F(U)$ gives $U^{\prime \prime }=DF(U)U^{\prime }$, so $U^{\prime }$ lies in the periodic null space of $\hat{L}$. For a hyperbolic attracting limit cycle, its tangent direction has Floquet multiplier one and every transverse multiplier has modulus below one. Hence it is the only periodic direction, proving the required one-dimensional null space.

Strictly, mere nonlinear stability does not ensure this spectral assertion. In polar coordinates $\dot{r}=-(r-1{)}^3$, $\dot{\phi }=2\pi$ has an attracting unit-period cycle but neutral radial linearization at $r=1$, supplying a second periodic solution of the variational equation. Thus the intended phase reduction assumes the usual simple unit Floquet multiplier, or equivalently hyperbolic transverse attraction; this assumption is made explicit here.

We count periodic orbits whose trajectories remain near the two specified homoclinic orbits; the local hypotheses impose no bound on unrelated distant cycles. Choose the return domain small enough that $A\delta |x{|}^{\delta -1}$ and $B\delta |x{|}^{\delta -1}$ are less than one. Each admissible return branch is then a contraction mapping.

A right-lobe periodic orbit is a fixed point $(p,+1)$ with $p>0$ and $p=-\mu +A{p}^{\delta }$. For small amplitudes it exists exactly when $\mu <0$, and $p=-\mu +o(|\mu |)$. Similarly, a left-lobe periodic orbit has $(x,y)=(-q,-1)$, $q>0$, $q=-\nu +B{q}^{\delta }$, and exists for $\nu <0$. Their Floquet multipliers tend to zero, so both are attracting.

A period-two orbit of the section map corresponds to one flow periodic orbit traversing both lobes. Write its states as $(p,-1)$ and $(-q,+1)$. Its positive amplitudes satisfyThe two-step derivative is $AB{\delta }^2{p}^{\delta -1}{q}^{\delta -1}$, again tending to zero. The asymmetric planar gluing return map therefore gives at most one such attracting two-lobe cycle in the small domain. Its boundaries are obtained by setting one amplitude to zero:For the original flow these are leading equations, with corrections $o({\nu }^{\delta })$ or $o({\mu }^{\delta })$. On $C_R$, the negative unstable branch makes both excursions before returning to the positive stable branch; on $C_L$ the positive unstable branch returns to the negative stable branch. Each is a compound homoclinic orbit, with one extra passage near the saddle equilibrium. They are distinct from the primary loop curves $\mu =0$ and $\nu =0$.

The six open regions and their local cycles are:

There is no additional small alternating cycle when both parameters are negative. Indeed the two positive-amplitude equations would imply $q<A{p}^{\delta }$ and $p<B{q}^{\delta }$, giving $p<B{A}^{\delta }{p}^{{\delta }^2}$, impossible for sufficiently small positive $p$. On each mixed-sign quadrant the contraction equations give the two-lobe cycle precisely on the side of the compound curve toward the positive quadrant.

On $\mu =0$, there is the primary right homoclinic orbit, together with a left cycle if $\nu <0$ or a two-lobe cycle if $\nu >0$. On $\nu =0$ the analogous statement holds with left and right interchanged. On $C_R$ there is a compound homoclinic orbit and the right single-lobe cycle; on $C_L$ there is a compound homoclinic orbit and the left single-lobe cycle. At $(0,0)$ the two primary loops meet at the same saddle equilibrium. The approach to each homoclinic orbit has a diverging period; it is not a saddle-node collision of two finite-period cycles.

For $r>0$, polar coordinates give $\dot{r}=r({\mu }_1-2x)$ and $\dot{\theta }=1$; the remaining amplitude equation is $\dot{x}={\mu }_2+x^2+r^2$. Interpret $r=0$ in the original complex coordinate, where the phase is undefined.

If ${\mu }_2<0$, put $a=\sqrt{-{\mu }_2}$. The full equilibrium points are $E_{\pm }:(z,x)=(0,\pm a)$. Their eigenvalues areThus $E_{-}$ is stable for ${\mu }_1<-2a$ and has a two-dimensional unstable manifold for ${\mu }_1>-2a$. The point $E_{+}$ has a one-dimensional unstable manifold for ${\mu }_1<2a$ and is repelling for ${\mu }_1>2a$. On ${\mu }_2=0$ they merge in a saddle-node bifurcation; the simultaneous zero and imaginary eigenvalues at $({\mu }_1,{\mu }_2)=(0,0)$ give a fold-Hopf bifurcation.

A steady positive amplitude represents a full periodic orbit, not a full equilibrium. Solving the amplitude equations givesIt exists in the wedge ${\mu }_2<-{\mu }_1^2/4$. The transverse amplitude Jacobian matrix has characteristic polynomial $s^2-{\mu }_{1}s+4r_{\ast }^2$. Its two transverse Floquet exponents have negative real part for ${\mu }_1<0$ and positive real part for ${\mu }_1>0$; the phase direction supplies the neutral Floquet multiplier one.

The wedge boundaries are Hopf bifurcations at ${\mu }_1=-2a$ from $E_{-}$ and ${\mu }_1=2a$ from $E_{+}$. The left branch is supercritical, producing a stable periodic orbit as $E_{-}$ loses stability; the right branch is subcritical in the oscillatory subspace, with an unstable periodic orbit on the side where the transverse complex pair of $E_{+}$ is stable. The real unstable direction of $E_{+}$ persists throughout. There are no full equilibrium points when ${\mu }_2>0$, and $\dot{x}>0$ then excludes a local periodic orbit. The right panel of the bifurcation diagram in part 1(b) shows the primary curves and the perturbed global curve calculated below.

At a fixed point $x=y=p$, the quadratic equation is $p^2+(1+b)p-\mu =0$. ConsequentlyThe Jacobian matrix is $\left(\begin{array}{cc}0& 1\\ -b& -2p\end{array}\right)$, and its Floquet multipliers solve ${\lambda }^2+2p\lambda +b=0$. The Jury stability criterion requires $|b|<1$ and $1+2p+b>0$, $1-2p+b>0$. With $b>-1$, this becomes $-1<b<1$, $-(1+b)/2<p<(1+b)/2$. The lower branch never satisfies it; the upper branch is attracting precisely whenAt the lower boundary the two fixed points merge with Floquet multipliers $1,b$, a saddle-node bifurcation for $b\ne 1$. At the upper boundary the upper point has Floquet multipliers $-1,-b$, giving a generic period-doubling bifurcation for $-1<b<1$.

Its direction can be established explicitly. A distinct two-cycle alternates values $y,z$ satisfying $(1+b)z=\mu -y^2$, $(1+b)y=\mu -z^2$. Subtracting gives $y+z=1+b$, and substitution givesIt exists beyond the fixed-point period-doubling bifurcation. The second-iterate Jacobian matrix has determinant $b^2$ and trace $4yz-2b=4[(1+b{)}^2-\mu ]-2b$. Applying the same stability inequalities shows attraction for $-1<b<1$ andThus the first period-doubling bifurcation is supercritical; the next upper boundary has a Floquet multiplier $-1$ for the two-cycle. These are the fixed-point and two-cycle thresholds of the Hénon map.

At $b=1$ the map preserves phase-space area, so attraction is impossible. The fixed-point fold is at $\mu =-1$, with double Floquet multiplier $+1$, and is a conservative saddle-centre bifurcation. The upper branch $p_{+}=-1+\sqrt{1+\mu }$ is linearly elliptic for $-1<\mu <3$; the lower branch is a saddle fixed point of a map. At $\mu =3$, the upper branch has double Floquet multiplier $-1$ and becomes a saddle fixed point of a map for larger $\mu$. The two-cycle $y,z=1\pm \sqrt{\mu -3}$ is elliptic for $3<\mu <4$, then reaches double Floquet multiplier $-1$ for the second iterate at $\mu =4$. The partial sketch distinguishes elliptic, neutrally stable Floquet multipliers from attracting states. Resonances may require additional nonlinear analysis; unit Floquet multiplier moduli alone do not prove nonlinear Lyapunov stability.

The $b=1$ segment is a conservative Hénon stability boundary, not an ordinary nondegenerate dissipative Neimark–Sacker bifurcation. Indeed for $b>0$ constant phase-space area scaling is $\mathrm{area}(F(D))=b\mathrm{area}(D)$. A simple invariant closed curve would enclose a bounded invariant set $D$ of positive phase-space area, forcing $b=1$. Thus no such closed invariant curve can bifurcate into $b\ne 1$ within this constant-determinant family.

Generic folds and flips away from these degeneracies persist under sufficiently small smooth perturbations: their simple unit Floquet multiplier, transverse parameter crossing and nonzero leading nonlinear coefficient persist, while the other Floquet multiplier stays away from the unit circle. The implicit function theorem then continues the bifurcation curves and their criticality. The unqualified perturbation claim needs this restriction. Arbitrary perturbations need not preserve the conservative $b=1$ behavior. For example, in local complex coordinates at a nonresonant elliptic fixed point, the area-preserving map property gives a cubic normal form $z^{\prime }=e^{i\theta }z(1+i\tau |z{|}^2)+\cdots$ with zero cubic radial damping. Composing with the arbitrarily small radial perturbation $z\mapsto (1-\eta |z{|}^2)z$ introduces radial coefficient $-\eta$. Varying the modulus through one can now give a genuine Neimark–Sacker bifurcation. Conservative degeneracies and resonances are therefore not covered by a blanket structural-stability assertion.

The origin's Jacobian matrix is triangular, $J=\left(\begin{array}{cc}a+1& 1\\ 0& b-1\end{array}\right)$, so its Floquet multipliers are $a+1$, $b-1$. It is attracting for $-2<a<0$, $0<b<2$. The candidate codimension-one boundaries are $a=0$, $a=-2$, $b=2$, $b=0$: respectively a $+1$ or $-1$ Floquet multiplier. No complex-pair crossing is possible at the origin.

The map is odd. To locate the nearby steady branches, write $S=x^2>0$. The first fixed-point equation gives $y=x(S-a)$; substituting into the second givesFor a symmetric two-cycle solve $F(x,y)=-(x,y)$. Then $y=x(S-a-2)$ andThese exact orbit equations for a triangular cubic map determine the amplitude and parameter side without confusing the original cubic coefficients with their centre manifold values.

On $a=0$, with $b\ne 2$, steady states have $x^2=a(2-b)+o(a)$ and $y=-ax+x^3$. The pitchfork bifurcation is supercritical for $b<2$ and subcritical for $b>2$. Its centre manifold cubic coefficient is $-1/(2-b)$. In the stable-origin boundary segment $0<b<2$, an attracting pair appears on $a>0$.

On $a=-2$, with $b\ne 0$, the two-cycle has $x^2=b(a+2)+o(a+2)$. The centre manifold map has cubic coefficient $-1/b$ and Floquet multiplier $-1+(a+2)$. Since its unstable side is $a+2<0$, the period-doubling bifurcation is supercritical for $b<0$ and subcritical for $b>0$. In particular the boundary segment $0<b<2$ is subcritical: the nearby two-cycle lies on $a>-2$, where the origin is still attracting, and is unstable in the critical direction.

On $b=2$, with $a\ne 0$, the steady branch hasIts scalar centre manifold map is $y^{\prime }=(1+b-2)y+(1+a^{-3})y^3+\cdots$. It is supercritical for $-1<a<0$, and subcritical for $a<-1$ or $a>0$. At $a=-1$ the cubic vanishes: the exact equation gives $b-2=-3S^2+O(S^3)$, so this is a degenerate pitchfork bifurcation with destabilizing quintic coefficient and subcritical quartic amplitude scaling. It is not a generic point of the codimension-one curve.

On $b=0$, with $a\ne -2$, put $A=a+2$. The two-cycle satisfiesThe centre manifold map's cubic coefficient is $1-A^{-3}$. Thus the period-doubling bifurcation is subcritical for $-2<a<-1$, and supercritical for $a<-2$ or $a>-1$. In the stable-origin edge $-1<a<0$, a stable small two-cycle appears for $b<0$. At $a=-1$ the cubic vanishes, requiring the calculation below. All criticality labels concern the centre manifold direction; a bifurcating orbit is fully attracting only if the remaining Floquet multiplier is also stable. The sketch marks the small branches on the appropriate sides of each boundary of the stable-origin rectangle.

For clarity, these stability labels follow directly from one-dimensional derivatives. At a pitchfork bifurcation $z^{\prime }=(1+\eta )z+c{z}^3$, the nonzero state has $z^2=-\eta /c$ and Floquet multiplier $1-2\eta$ to first order. At a period-doubling bifurcation $z^{\prime }=-(1+\eta )z+c{z}^3$, the symmetric cycle has $z^2=\eta /c$ and one-step derivative $-1+2\eta$; its two-step Floquet multiplier is $1-4\eta +o(\eta )$. A supercritical branch lies on $\eta >0$ and is attracting in the centre manifold direction; a subcritical branch lies on $\eta <0$ and is repelling there.

At $(a,b)=(-1,0)$ the Floquet multipliers are $0,-1$, so the degeneracy is not two simultaneous unit Floquet multipliers. On its centre manifold, $x=-y-y^3+O(y^7)$ and direct substitution gives $y^{\prime }=-y-3y^5+O(y^7)$. With $\alpha =a+1$ small, the unfolding to leading weighted order isThis is a generalized flip bifurcation: one parameter crosses $-1$, and the other changes the cubic sign. Writing $S\simeq y^2$, its two-cycles obey $b+3\alpha S-3S^2=0$. For $\alpha >0$ a stable smaller two-cycle and an unstable larger two-cycle collide atThe intervening range $-3{\alpha }^2/4<b<0$ contains both. For $\alpha <0$ the period-doubling bifurcation is subcritical. At $\alpha =0$, the exact orbit equation is $3S^2-3S^3+S^4=b$, so the unstable two-cycle exists for $b>0$ with $|y|\sim (b/3{)}^{1/4}$. This proves both the codimension and the required quintic effect.

Write $k=\sqrt{\mu }$. The equilibrium points are a stable node $(-k,0)$ and a saddle equilibrium $(k,0)$, whose unstable and stable eigenvalues are $2k$ and $-\lambda$. The saddle equilibrium's local stable manifold is $x=k$. When the right-going unstable separatrix returns at $(\nu ,h)$ with $\nu =k$, it lands exactly on that stable manifold, producing a homoclinic orbit of the saddle. Crossing this condition changes whether the returning trajectory is captured by the node or lies on the outgoing side of the saddle equilibrium.

Use an incoming section $y=h$ and outgoing section $x=h$. For entry $(x,h)$ with $k<x<h$, integrate the local flow:so the outgoing height isThe global excursion is smooth and transverse. Its return coordinate has expansion $x_{\mathrm{next}}=\nu +cY+O(Y^2)$, with $c>0$ in these section orientations. Thus the local passage map near a dissipative saddle-node givesFor small $\mu >0$, $\lambda >2k$, hence $P(k^{+})=\nu$ and $P^{\prime }(k^{+})=0$. If $\nu -k>0$ is small, the graph starts above the diagonal and, with slope less than one nearby, crosses it close to $x=\nu$. More explicitly, $P(x)-x=(\nu -k)-(x-k)+O((x-k{)}^{\beta })$, giving a unique small fixed point with Floquet multiplier $0<P^{\prime }<1$. It corresponds to an attracting periodic orbit. For $\nu <k$ no such local crossing exists on $x>k$; the unstable branch returns to the node's side instead. The Poincaré return map panel shows this geometric fixed-point test.

Linearizing about the orbit gives $\dot{\eta }=A(t)\eta$, where $A(t)=Df(X(t))$. The fundamental matrix therefore satisfiesThe Floquet multipliers are the eigenvalues of the monodromy matrix $\Phi (T)$. Differentiating the orbit equation shows that its tangent $\dot{X}(t)$ solves the variational equation. Since $\dot{X}(T)=\dot{X}(0)\ne 0$, one multiplier is one.

For an invertible fundamental matrix, differentiating the determinant gives $(\det \Phi {)}^{\prime }=\mathrm{tr}({\Phi }^{-1}\dot{\Phi })\det \Phi =\mathrm{tr}A\det \Phi$. Integration from the identity yields $\det \Phi (T)=\exp ({\int }_0^T\mathrm{tr}A(t)dt)$. The determinant is the product of the two multipliers, so the other is

Floquet theory suggests a sequence of instability tongues nearAt these limiting frequencies the unperturbed Floquet multipliers $e^{\pm 2\pi i\omega }$ coincide. Each multiplication by $\cos t$ changes a Fourier frequency by one, so $m$ forcing interactions are needed to couple $+m/2$ and $-m/2$. This explains both the sequence of parametric resonances and the successive powers of $ϵ$ in their widths.

There is no implication that the exact half-integers themselves lie in every finite-$ϵ$ growing interval. The tongue centers generally shift by $O({ϵ}^2)$, which exceeds the width when $m>2$. It is the narrow intervals near the shifted centers that contain exponentially growing solutions. Frequencies staying a fixed distance from every half-integer are stable for sufficiently small $ϵ$.

The circle $\gamma :r=1$ is a periodic orbit, since $\dot{r}=0$ there and $\dot{\theta }=1$. Its period is $2\pi$. Linearizing in $(\delta r,\delta \theta )$ along it givesThe monodromy matrix is triangular with diagonal entriesThe unit Floquet multiplier comes from translating an autonomous orbit in time: its tangent vector is a periodic solution of the variational equation, independently of $g$. A sufficient condition for asymptotic stability is ${\int }_0^{2\pi }g<2\pi$, making the normal multiplier less than one. The reverse strict inequality gives instability; equality requires nonlinear analysis.

For $g=2\sin \theta$ the normal multiplier is $e^{-2\pi }$, so the circle is asymptotically stable. That does not mean it attracts every initial point. Since $r>0$ makes $\theta$ strictly increasing, divide the radial equation by $\dot{\theta }$:For $r>1$, put $v=r-1>0$. Then $v^{\prime }=(2\sin \theta -1)v-v^2\le (2\sin \theta -1)v$, whose integrated linear coefficient decreases by $2\pi$ per revolution. Thus every exterior orbit approaches $r=1$.

For $0<r<1$ put $w=(1-r{)}^{-1}>1$. It satisfies the linear equationLet $w_c$ be its solution with $w_c(\pi )=1$. It defines the separatrix $r_c=1-1/w_c$ wherever $w_c>1$. At $\theta =\pi$, $r_c=0$, $r_c^{\prime }=0$, and differentiation gives $r_c^{\prime \prime }=2$, so $r_c\sim (\theta -\pi {)}^2$. The approach to this endpoint takes infinite physical time because $dt=d\theta /r_c$.

To locate its other endpoint, use the integrating factor $e^{-A(\theta )}$ with $A(\theta )=\theta +2\cos \theta$. It givesIn particular $w_c(0)>e^{4-\pi }>1$, whereasOn $(-\pi ,0)$, crossing $w=1$ has $w^{\prime }=-2\sin \theta >0$, so there is one crossing ${\theta }_b\in (-\pi ,0)$. There cannot be another crossing before $\pi$: on $(0,\pi )$ crossings have negative derivative, while the local quadratic form at $\pi$ is positive on its left. Thus $r_c>0$ for ${\theta }_b<\theta <\pi$, and its closure is a homoclinic orbit to the origin. At ${\theta }_b$, $r_c^{\prime }=-2\sin {\theta }_b>0$, so departure from the origin likewise takes infinite time.

Scalar-solution comparison now gives the basin picture. Initial points inside that loop have $w<w_c$ at the same angle and approach $r=0$ before reaching angle $\pi$. Points on its boundary return to the origin along the homoclinic trajectory. Points outside the loop reach angle $\pi$ with $w>1$ or lie outside the unit circle; they approach $r=1$. To check the inner exterior region quantitatively, the return map at $\theta =\pi$ is $w\mapsto e^{2\pi }(w-I)$, where $I={\int }_0^{2\pi }{e}^{-u-2+2\cos u}du<1-e^{-2\pi }$. It sends every $w>1$ increasingly to infinity, equivalently $r\to 1$.

There are no other nonconstant periodic orbits: an orbit reaching the section with $w>1$ has strictly increasing successive $w$ values, and the remaining inner trajectories tend to the origin. The origin, defined by the continuous Cartesian extension, attracts the interior of the loop but is not Lyapunov stable, since arbitrarily small lower-sector initial points outside the loop approach the unit circle. This is the homoclinic basin boundary in a sinusoidal radial flow.

The linearized perturbation obeys $\dot{\eta }=D\mathbf{f}(\mathbf{X}(t))\eta$, so its fundamental matrix satisfiesThe Floquet multipliers are the eigenvalues of the monodromy matrix $\Phi (T)$. Differentiating the orbit equation shows that $\dot{\mathbf{X}}(t)=\mathbf{f}(\mathbf{X}(t))$ is a solution of the variational equation. For a nonconstant periodic orbit this tangent solution is nonzero and periodic, giving one multiplier equal to $1$.

The Liouville formula for a fundamental matrix gives $(\det \Phi {)}^{\prime }=(\mathrm{tr}D\mathbf{f})\det \Phi$. Since $\det \Phi (0)=1$, the product of the two multipliers is $\exp ({\int }_0^T\nabla \cdot \mathbf{f}(\mathbf{X}(t))dt)$. The other multiplier is therefore exactly this exponential.

Use the nearly conserved oscillator energy $E=({\dot{x}}^2+x^2)/2$. The differential equation givesFor a leading-order orbit $x=A\cos \theta$, $\dot{x}=-A\sin \theta$, the cycle averages are $⟨{\dot{x}}^4⟩=3A^4/8$, $⟨x^3\dot{x}⟩=0$, $⟨x^4{\dot{x}}^2⟩=A^6/16$, and $⟨{\dot{x}}^2⟩=A^2/2$. The energy balance method therefore givesThe two nonzero simple roots give the leading limit cycles $x\simeq \cos (t+{\theta }_0)$ and $x\simeq \sqrt{2}\cos (t+{\theta }_0)$. Below amplitude $1$ the energy increases, between $1$ and $\sqrt{2}$ it decreases, and above $\sqrt{2}$ it increases. Hence the amplitude-$1$ cycle is stable and the amplitude-$\sqrt{2}$ cycle unstable for positive small $\epsilon$.

The linearized averaged radial exponents are $-\epsilon /2$ and $\epsilon$, respectively, consistent with nontrivial Floquet multipliers $e^{-\pi \epsilon +O({\epsilon }^2)}$ and $e^{2\pi \epsilon +O({\epsilon }^2)}$. The nonzero derivatives at these averaged zeros also justify persistence of nearby cycles for sufficiently small $\epsilon$.