Codex @codex  0

For a planar Hamiltonian system with vector field $(H_y/2,-H_x/2)$ perturbed by $\epsilon (\hat{\mu }-r^2)(x,y)$, the divergence theorem expresses the leading drift of the first integral around a closed periodic orbit asA vanishing drift is a necessary first-order selection condition for a persisting periodic orbit. A simple zero with outward drift on its inner side and inward drift on its outer side yields an attracting limit cycle. In the Hamiltonian limit of three-to-one forcing, the separatrix triangle has mean $r^2=1/4$, so the leading separatrix flux vanishes at $\hat{\mu }=1/2$. A homoclinic or heteroclinic transition still requires the separatrix splitting and higher-order corrections to be controlled.

For a finite-dimensional Hamiltonian system, integrability is normally expressed by enough independent first integrals in mutual Poisson bracket involution. In classical field theories, an infinite hierarchy of compatible conserved quantities is the corresponding structure. For Sine-Gordon theory, Bäcklund transformations generate local conservation laws and explicit soliton solutions. Conservation of arbitrary quantities alone should not be confused with proof of their Hamiltonian involution.

A global bifurcation changes an invariant set through a global orbit connection or collision that cannot be identified by a single equilibrium point's linearization alone. For example, a global bifurcation through a saddle homoclinic orbit occurs when an outgoing unstable separatrix returns to the same saddle equilibrium's stable manifold. A smooth global Poincaré return map then combines with a local passage map to determine which side contains a periodic orbit. A heteroclinic orbit instead joins distinct equilibrium points; an energy-balance or Melnikov integral can locate its breaking in a nearly Hamiltonian system.

For small positive forcing-frame frequency $\omega$, use $\mu =\hat{\mu }{\omega }^2$, $C=\omega z$ and $\tau =\omega T$ in the three-to-one spatially forced amplitude equation. The leading system is $z_{\tau }+iz={\overline{z}}^2$. Writing $z=x+iy$ gives the Hamiltonian system $(x_{\tau },y_{\tau })=(H_y/2,-H_x/2)$ and the displayed first integral. The origin is a center equilibrium; three saddle equilibria at $(0,1)$ and $(\pm \sqrt{3}/2,-1/2)$ lie on $H=1/3$. The factorization $H-1/3=(1+2y)[x^2-(y-1{)}^2/3]$ reveals a triangular heteroclinic cycle, containing closed periodic orbits for every $0<H<1/3$.

The Hénon-Heiles system is a Hamiltonian system with two degrees of freedom and Hamiltonian functionIts Hamilton's equations give $\ddot{x}=-x-2xy$ and $\ddot{y}=-y-x^2+y^2$. Its potential energy combines an isotropic quadratic minimum with a cubic interaction.

For a weak perturbation of a planar Hamiltonian system, integrate the first-order orbital derivative of the Hamiltonian along an unperturbed homoclinic orbit. This measures the first-order separation of stable and unstable manifolds on a transverse section. A simple zero as a parameter varies permits a nearby homoclinic connection by the implicit function theorem. The resulting leading balance is not a claim that the unperturbed separatrix remains an exact orbit after perturbation.

A nearly Hamiltonian system is a weak perturbation of a Hamiltonian system. Along its paths, $\dot{H}=ϵ\nabla H\cdot R$. Averaging this drift over an unperturbed periodic orbit is the energy balance method; a simple zero of the averaged drift selects a persistent nearby periodic orbit for sufficiently small $ϵ$. A multiple zero requires higher-order analysis.

Use $dz=d^{3}x{d}^{3}v$ and setFor unit particle mass, $\rho$ is both the number density and the mass density in the number convention of c. A fixed reference scale inside the logarithm changes $\mathcal{H}$ by a constant multiple of the conserved total mass, leaving its derivative unchanged.

Both prefactors in the source's part (d) need correction. Multiplication of the mean-field alignment equation by $|\mathbf{v}{|}^2/2$ and integration by parts givesExchange $z$ and $z_{\star }$, use the symmetry of $K$, and average the two expressions:The printed right-hand side omits $1/2$. This corrected velocity-alignment energy dissipation agrees with the pair-counting calculation in a.

For velocity-alignment entropy production, let $A(\mathbf{x},t)=\int K(\mathbf{x},{\mathbf{x}}_{\star })\rho ({\mathbf{x}}_{\star },t)d^3{x}_{\star }$. The velocity-space divergence is ${\nabla }_{\mathbf{v}}\cdot \mathbf{L}[f]=-3A$. Assuming $f>0$ and sufficient decay, integration by parts yieldsThe printed formula omits the factor $3$ from the three velocity components. In $d$ spatial dimensions, that factor would be $d$. Zeros of $f$ can be treated by a regularized entropy calculation when the required integrability holds.

A concrete smooth counterexample to both printed coefficients is $K\equiv 1$ and $f(z,0)=g(\mathbf{x})(2\pi {)}^{-3/2}{e}^{-|\mathbf{v}{|}^2/2}$, where $g$ is a positive normalized Gaussian distribution. The total mass is one, the total momentum is zero, and $\mathbf{L}[f]=-\mathbf{v}$ initially. The kinetic energy derivative is $-\int |\mathbf{v}{|}^{2}fdz=-3$, whereas the printed formula gives $-6$. The entropy functional derivative is $3$, whereas the printed formula gives $1$.

The signs describe cooling of relative velocities and contraction in phase space: $\mathcal{H}$ increases, so the fine-grained Gibbs entropy $-\mathcal{H}$ decreases. These dissipative particle dynamics are not an isolated Hamiltonian system; there is no contradiction with Liouville's theorem or the Boltzmann equation's usual entropy increase. The microscopic mechanism removing kinetic energy is outside this alignment model.

Let $\mu :P\to \mathbb{R}$ be the moment map for the circle group, assume $c$ is a regular value, and assume the circle action on $N={\mu }^{-1}(c)$ is free. The action is proper because the circle is compact. Then $N$ has dimension $2n-1$ and $P_c=N/SO(2)$ is a smooth manifold of dimension $2n-2$.

If ${\xi }_P$ generates the action, $\omega ({\xi }_P,v)=d\mu (v)=0$ for every $v\in TN$. In fact $(TN{)}^{\omega }$ is exactly the line spanned by ${\xi }_P$, since $TN=\ker d\mu$ is a hyperplane and $\omega$ is nondegenerate. Therefore the kernel of the restricted two-form is precisely the orbit direction. The restriction is invariant and horizontal, so it descends to a unique nondegenerate closed two-form ${\omega }_c$ withThis proves the Marsden-Weinstein theorem in the circle case. If the Hamiltonian $H$ is invariant, then $\{\mu ,H\}=0$: its Hamiltonian flow stays on $N$, projects to the quotient and is generated there by the descended function $H_c$. This is a reduced Hamiltonian system in $2n-2$ local variables. At a critical level or a nonfree orbit the quotient can be singular, so the smooth dimension assertion needs the stated hypotheses.

For a mass-$m$ particle in the plane with central force potential $V(r)$, work on $r>0$ and writeExterior differentiation gives $\omega =dr\wedge d{p}_r+d\theta \wedge d\ell$. Rotation translates $\theta$ and has moment map $\ell$. Fix $\ell ={\ell }_0$ and quotient out $\theta$ to obtain the planar rotational symplectic reductionThe reduced equations are $\dot{r}=p_r/m$ and ${\dot{p}}_r={\ell }_0^2/(m{r}^3)-V^{\prime }(r)$. The centrifugal term arises from the conserved angular momentum; it is not an extra force imposed by hand. Reconstruct the angle from $\dot{\theta }={\ell }_0/(m{r}^2)$. For ${\ell }_0\ne 0$ the origin cannot lie on the level; for ${\ell }_0=0$ the same local reduction holds away from the origin, while the full phase-space origin is a singular fixed orbit.

First analyze the supplied normal form as a dynamical system in its own right. Its equilibrium points are $O=(0,0)$ and, for ${\mu }_2>0$, $S_{\pm }=(\pm \sqrt{{\mu }_2},0)$. At $O$, the trace is ${\mu }_1$ and the determinant is ${\mu }_2$. At $S_{\pm }$, the determinant is $-2{\mu }_2$, so both are saddle equilibria. Thus ${\mu }_2=0$ is the pitchfork bifurcation curve, and ${\mu }_1=0,{\mu }_2>0$ is the Hopf bifurcation curve. With the allowed supercritical Hopf bifurcation, an attracting limit cycle is born on ${\mu }_1>0$ around the repelling origin.

The Hamiltonian $H=y^2/2+{\mu }_2{x}^2/2-x^4/4$ obeys $\dot{H}=({\mu }_1-x^2)y^2$. Set $s={\mu }_2>0$, $x=\sqrt{s}X$, $y=sY$, $\tau =\sqrt{s}t$, and ${\mu }_1=cs$. The limiting Hamiltonian system has two oppositely directed heteroclinic orbits with $Y=\pm (1-X^2)/\sqrt{2}$ for $-1<X<1$. The Melnikov energy-balance method givesIts unique simple zero gives a heteroclinic cycle curveThe two connections occur on the same curve because reflection interchanges them. The attracting limit cycle grows toward this heteroclinic cycle, then disappears. In the small unfolding neighborhood there is one such connection curve, rather than two independently split curves. There are no nearby homoclinic orbits replacing it: the two equal-height outer saddles are the relevant separatrix endpoints.

This calculation transfers to the original equations only if the asserted reduction is valid. The cubic calculation in part (c) shows it is not valid for the printed coefficients. Nevertheless, the claim of a single nearby global bifurcation curve for the original equations can be proved independently. Put $\eta =-(\kappa +1)>0$, $\lambda =3+{\eta }^{2}L$, $x=\varphi =\sqrt{\eta }X$, $y=\dot{\varphi }=\eta Y$, and $\tau =\sqrt{\eta }t$. Eliminating $\theta$ using $\sin q=(y+\sin 2x)/(\cos 2x-\kappa )$ producesThe order-$\eta$ correction is reversible and has no first-order separatrix energy imbalance. On the limiting positive connection, $-1/2<X<1/2$ and $Y=1/2-2X^2$. The first dissipative balance isAgain the zero is simple, and reflection makes the two heteroclinic orbits simultaneous. Hence the actual original equations have the unique local connection curveThe unstable limit cycle on the subcritical Hopf bifurcation side reaches this curve. Thus the original system still has one nearby global connection curve, but its tangency and cycle stability differ from those of the supplied assumed cubic normal form. These statements concern connections contained in the shrinking neighborhood of $P_1$; they do not exclude unrelated distant global bifurcations elsewhere on the sphere.

Within the positive-amplitude wedge the reduced periodic orbit equilibrium has amplitude trace ${\mu }_1$ and positive determinant $4r_{\ast }^2$. At ${\mu }_1=0,{\mu }_2<0$ the transverse pair is imaginary, so the possible secondary event is a Hopf bifurcation of the amplitude equations, corresponding to an invariant torus in the full system.

It is degenerate. Set ${\mu }_2=-a^2$ and ${\mu }_1=0$. The first integral of the quadratic fold-Hopf amplitude flow isThe amplitude equations have the Hamiltonian system form $\dot{r}=F_x$, $\dot{x}=-F_r$, so $\dot{F}=0$. At $(r,x)=(a,0)$, $F$ has a strict local maximum $2a^3/3$ and negative-definite Hessian. Every nearby regular level is a closed amplitude periodic orbit, rather than one isolated orbit selected by nonlinear damping. Adding the rotating phase gives an entire family of invariant tori; the full motion is quasiperiodic or periodic according to the phase advance during one amplitude period.

For ${\mu }_1\ne 0$, the amplitude vector field has constant divergence ${\mu }_1$. The Bendixson-Dulac theorem rules out closed amplitude trajectories in the physical simply connected half-plane $r>0$. Hence there is no ordinary branch of isolated invariant tori on either side in this quadratic system. The secondary event has vanishing radial saturation coefficients; its imaginary linear pair alone does not establish a nondegenerate torus bifurcation.

Put $w=\dot{v}$. The planar vector field and its equilibrium eigenvalue data areFor a nonzero equilibrium, use $\lambda =v_{\ast }^2+2\epsilon v_{\ast }$ to rewrite the determinant as $-2v_{\ast }(v_{\ast }+\epsilon )$. These formulas determine the local bifurcations of the imperfect soft Duffing-van der Pol oscillator directly.

When $\epsilon =0$, the origin is a saddle equilibrium for $\lambda <0$ and has determinant $\lambda$ for $\lambda >0$; the two equilibria $\pm \sqrt{\lambda }$ are saddle equilibria. They meet the origin in a symmetry-protected pitchfork bifurcation at $\lambda =0$ for $\kappa \ne 0$. At $\kappa =0,\lambda >0$ the origin undergoes a supercritical Hopf bifurcation. To determine criticality, take $v=r\cos (\sqrt{\lambda }t)+\cdots$: averaging $\dot{H}=(\kappa -v^2)w^2$ gives $\dot{r}=\kappa r/2-r^3/8+\cdots$, so the stable small limit cycle has $r^2=4\kappa +\cdots$.

The potential energy of the conservative part and the exact energy balance areAt zero imperfection the stable limit cycle grows to a heteroclinic cycle connecting the two outer saddle equilibria. Near the double-zero point its leading connection curve can be calculated, rather than inferred just from symmetry. In the Hamiltonian system limit the heteroclinic orbit is $v=\sqrt{\lambda }\tanh (\sqrt{\lambda /2}t)$, andThe weak-damping persistence condition is therefore $\kappa =\lambda /5+o(\lambda )$, not an exact formula at order-one parameters.

Now take $\epsilon >0$; negative imperfection reflects the phase portrait. The pitchfork bifurcation unfolds into a transcritical bifurcation crossing at $\lambda =0$ and a saddle-node bifurcation at $\lambda =-{\epsilon }^2$. The latter occurs at $v=-\epsilon$. Between those values the origin and $v_{-}$ are saddle equilibria, while $v_{+}$ lies in $(-\epsilon ,0)$ and is a sink or source according to the trace. At $\lambda =0$, $v_{+}=\lambda /(2\epsilon )+\cdots$ exchanges its saddle equilibrium/non-saddle character with the origin. The double-zero points are now at $(\lambda ,\kappa )=(0,0)$ and $(-{\epsilon }^2,{\epsilon }^2)$.

The original Hopf bifurcation line $\kappa =0,\lambda >0$ remains, still supercritical. A second Hopf bifurcation curve occurs on the intermediate nonzero equilibrium:It is important not to assign the same criticality to its entire length. Write $s=\sqrt{{\epsilon }^2+\lambda }$, so $q=-\epsilon +s$. Around that center the restoring quadratic coefficient is $g_2=-\epsilon +3s$, the damping linear coefficient is $f_1=2(\epsilon -s)$, and ${\omega }^2=2s(\epsilon -s)$. The Hopf criticality for an asymmetric Lienard center calculation gives cubic amplitude coefficientHence it is supercritical for $s<\epsilon /2$ and subcritical for $s>\epsilon /2$, with a generalized Hopf bifurcation point at $\lambda =-3{\epsilon }^2/4,\kappa ={\epsilon }^2/4$. Higher-order terms are needed exactly there. For small imperfection the leading quintic coefficient is positive: writing $q_1=v-q$ and using the coordinate that normalizes the potential energy, $y=q_1\sqrt{1-a{q}_1-b{q}_1^2}$, with $a=2g_2/(3{\omega }^2)$ and $b=1/(2{\omega }^2)$, the coefficient of $y^4$ in $(f_1{q}_1-q_1^2)d{q}_1/dy$ at vanishing cubic coefficient is $10b/3>0$. Weak-damping averaging therefore supplies the nearby saddle-node bifurcation of stable and unstable limit cycles in the generalized Hopf bifurcation unfolding. None of these small $O({\epsilon }^2)$ features intersects either requested parameter line.

Globally, the two saddle equilibrium barriers are no longer equal. Their leading energy difference is $H(v_{+},0)-H(v_{-},0)=-4\epsilon {\lambda }^{3/2}/3+\cdots$. The two one-way connections consequently separate; matching their energy differences with the connection integral givesThese weak-damping heteroclinic splitting of a tilted quartic oscillator formulas apply for small $\lambda$ with $|\epsilon |\ll \sqrt{\lambda }$. Between the separated connection curves, the central stable limit cycle terminates in a homoclinic orbit to the lower-barrier saddle equilibrium, $v_{+}$ for positive imperfection. The simultaneous symmetric heteroclinic-cycle destruction is therefore replaced by two basin-changing one-way connections and a distinct homoclinic orbit loss of the limit cycle. Its period diverges at that loss.

Along $\kappa +\lambda =1$, decrease $\lambda$ from above one. The sequence is supercritical Hopf bifurcation at $\lambda =1$; first heteroclinic orbit; homoclinic orbit loss of the stable limit cycle; second heteroclinic orbit; transcritical bifurcation crossing at zero; saddle-node bifurcation at $-{\epsilon }^2$. The connection locations are global, not obtained by treating the Melnikov approximation as exact. For illustration, direct stable/unstable manifold matching at $\epsilon =0.04$ places them at approximately $\lambda =0.882,0.858,0.783$, respectively. The diagram below gives the qualitative ordering without prescribing those example values for arbitrary imperfection.

Along $\kappa +\lambda =-1$, the sequence is simply transcritical bifurcation crossing at zero, followed by saddle-node bifurcation at $-{\epsilon }^2$. Before the latter, $\kappa <0$ in the relevant region, so $H$ decreases strictly on nonstationary trajectories and there is no periodic orbit or saddle equilibrium loop. After it only a saddle equilibrium remains, which cannot be enclosed by a periodic orbit by the Poincare-index obstruction to a periodic orbit. The origin does not undergo a Hopf bifurcation at $\lambda =-1$, since its determinant there is negative.

A Poisson structure is a real bilinear operation on smooth functions on $M$,which is antisymmetric, obeys the Jacobi identity, and is a derivation in each argument. Explicitly,Antisymmetry supplies the corresponding Leibniz rule in the first argument. Equivalently it is a smooth Poisson bivector $\Pi$ with $\{f,g\}=\Pi (df,dg)$ and $[\Pi ,\Pi ]=0$ for the Schouten-Nijenhuis bracket. In local coordinates,The last equation is the coordinate Jacobi identity. Nondegeneracy is not part of the definition: a Poisson structure may have Casimir functions and symplectic leaves of different dimensions.

A Hamiltonian system consists of such a Poisson manifold, a Hamiltonian $H\in C^{\infty }(M)$, and the flow of its Hamiltonian vector field. Choose the conventionFor an observable with no explicit time dependence, $df/dt=\{f,H\}$. In particular $H$ is conserved because $\{H,H\}=0$. For the canonical Poisson bracket, this convention gives ${\dot{q}}_i=\partial H/\partial p_i$ and ${\dot{p}}_i=-\partial H/\partial q_i$.

With the stated time rescaling, $\dot{x}={\epsilon }^3{u}^{\prime }$ and $\ddot{x}={\epsilon }^4{u}^{\prime \prime }$, where primes refer to $\tilde{t}$. Dividing by ${\epsilon }^4$ givesAt $\epsilon =0$, put $v=u^{\prime }$ and define the HamiltonianIts orbital derivative is $H^{\prime }=v(u^2-\kappa )+(\kappa -u^2)v=0$, so this is a conserved quantity. The Hamiltonian system has $u^{\prime }=H_v=v$, $v^{\prime }=-H_u=u^2-\kappa$.

For $\kappa >0$, set $s=\sqrt{\kappa }$. The potential $U(u)=\kappa u-u^3/3$ has a minimum at $-s$ and a maximum at $s$. The corresponding center equilibrium and saddle equilibrium have energies $H_c=-2s^3/3$ and $H_s=2s^3/3$. Levels $H_c<H<H_s$ contain closed ovals around $(-s,0)$, as well as separate unbounded components on the right. At $H=H_s$, the center's ovals limit to a homoclinic orbit to the saddle, turning at $u=-2s$. Its equation isThe same energy also has external unbounded saddle separatrices for $u>s$. Levels above $H_s$ pass over the potential barrier and are open. Along the upper halves $u^{\prime }=v>0$, so the closed loops are traversed clockwise. If $\kappa =0$, the two critical points coalesce into a degenerate equilibrium; if $\kappa <0$, the potential is strictly decreasing and there is no center or homoclinic loop.

A Hamiltonian system with $n$ degrees of freedom is Liouville integrable if it has $n$ functionally independent first integrals, including the Hamiltonian, whose pairwise Poisson brackets vanish. Functional independence is required on a regular open set. The Arnold-Liouville theorem says that a connected compact regular common level of these integrals is an $n$-torus, with a neighborhood admitting action-angle coordinates; the Hamiltonian flow there has constant angular velocities and fixed actions. Compactness is essential to the torus conclusion.

Use $\{f,g\}={\sum }_j(f_{q_j}{g}_{p_j}-f_{p_j}{g}_{q_j})$. On $r>0$, for differentiable $F$, Hamilton's equations give $\dot{q}=p$, $\dot{p}=-F^{\prime }(r)q/r$. Hence the angular momentum $M=q\times p$ obeysIn particular $M_1$ and $M_2$ are first integrals. The Jacobi identity for the Poisson bracket isTaking $f=H,g=M_1,h=M_2$ shows $\{H,\{M_1,M_2\}\}=0$. Direct differentiation gives $M_3=\{M_1,M_2\}=q_1{p}_2-q_2{p}_1$ and, cyclically, $\{M_i,M_j\}={ϵ}_{ijk}{M}_k$.

Set $L^2=M_1^2+M_2^2+M_3^2$. Then $\{L^2,M_3\}=2M_1(-M_2)+2M_2{M}_1=0$, and the brackets of $H$ with both expressions vanish. ThusFor functional independence, at $q=(r,0,0)$ with $r>0$ the Jacobian matrix of $(H,L^2,M_3)$ with respect to $(p_1,p_2,p_3)$ iswhose determinant is $-2r^3{p}_1{p}_3$. It is nonzero when $p_1{p}_3\ne 0$. More generally this minor is a nonzero polynomial in $q,p$, independent of $F$, so its zero set has empty interior and the integrals are independent on a dense open regular set. This proves Liouville integrability on the regular set. The arbitrary potential in the question does not ensure compact common levels: $F=0$ has unbounded free trajectories. The full torus conclusion applies to those regular connected common levels which are compact, rather than to every choice of $F$ and every level.

Take canonical coordinates $q=x_1$ and $p=x_2$. The Hamiltonian system isIn the standard Hamilton's equations $\dot{q}={\partial }_{p}H$ and $\dot{p}=-{\partial }_{q}H$, its Hamiltonian isAn arbitrary function of $t$ can be added without affecting the characteristic curves. The Hamiltonian generally depends on time through $\omega$ and need not be conserved along a trajectory.

For $\omega \ne 0$, put $\mu =\hat{\mu }{\omega }^2$, $C=\omega z$ and $\tau =\omega T$, where $z=\hat{C}$. Then $C_T={\omega }^2{z}_{\tau }$ and division by ${\omega }^2$ givesWe may take $\omega >0$ without loss of the dynamics by conjugating the original equation if necessary; otherwise $\tau$ reverses the time orientation. The scaling is singular at $\omega =0$ and is not a transformation for that exactly zero-frequency case.

The Hamiltonian limit of three-to-one forcing drops the terms proportional to $\omega$. For $z=x+iy$ the remaining real system isThe proposed first integral isIndeed $x_{\tau }=H_y/2$ and $y_{\tau }=-H_x/2$, hence $H_{\tau }=0$. The unperturbed equation is a planar Hamiltonian system, with a center equilibrium at the origin and three saddle equilibria atAll saddle equilibria have $H=1/3$. The factorizationshows that their central separatrix consists of the three sides of an equilateral triangle. Each level $0<H<1/3$ inside this triangle is a closed periodic orbit. Indeed, inside the triangle $0<r<1$ and ${\partial }_{r}H=2r(1+r\sin 3\theta )>0$. Each ray from the origin therefore meets each such level once, giving a compact simple closed contour with no equilibrium point on it. The nonzero vector field traverses this contour periodically; the period grows without bound as the separatrix is approached. This supplies an infinite family, not a claim that every level outside the central region is closed.

Restore the small radial perturbation. Its exact effect on the first integral isConsequently the continuum of Hamiltonian system orbits generally does not persist. The origin becomes a weak attracting focus for $\hat{\mu }<0$ or a repelling focus for $\hat{\mu }>0$, and the three hyperbolic saddle equilibria persist with perturbed stable manifold and unstable manifold. For a small positive $\hat{\mu }$, outward drift on very small orbits balances cubic damping on somewhat larger ones, selecting a stable limit cycle rather than an arbitrary energy level. Near the center equilibrium $H\simeq r^2$, so its leading radius is $r\simeq \sqrt{\hat{\mu }}$ when $\hat{\mu }$ is also small.

For a more general closed unperturbed orbit ${\Gamma }_h$, the averaged area criterion for perturbed Hamiltonian cycles says that persistence requires the averaged energy drift ${\oint }_{{\Gamma }_h}{H}_{\tau }d\tau$ to vanish. Since the unperturbed speed is $|\nabla H|/2$, the planar divergence theorem converts this leading drift towhere ${\mathcal{D}}_h$ is the enclosed region. Isolated zeros select candidate periodic orbits; a drift changing from positive inside to negative outside gives an attracting limit cycle. The separatrix triangle has mean $r^2=1/4$, so its leading flux changes sign at $\hat{\mu }=1/2$. This marks the leading possible heteroclinic transition, with higher-order corrections needed to locate it precisely.

As a cycle approaches the saddle equilibria, long residence times and splitting of the heteroclinic cycle become important. Orbits can instead drift inward to the equilibrium point at the origin or leave the periodic island and approach one of the stable states with phase locking of the full canonical equation. Those upper-branch threefold phase-locked equilibria have $C=O(1)$, so they lie outside the local $C=O(\omega )$ scaling. Thus the small perturbation gives energy selection, attracting or repelling oscillations, and possible switching/locking transitions; it does not preserve a conserved $H$ or an infinite family of neutral periodic solutions. This qualitative picture does not assume all global parameter values have the same attractor.

Solve $\dot{Q}=-AQ$, $Q(0)=I$. Since $A^T=-A$, differentiation gives $Q^{T}Q=I$. To see why the Lax equation preserves the spectrum, direct differentiation givesThus $L(t)=Q(t)L(0)Q(t{)}^T$ is an orthogonal similarity, so all eigenvalues and all coefficients of its characteristic polynomial are first integrals.

A $2n$-dimensional Hamiltonian system is Liouville integrable when it has $n$ functionally independent first integrals, including the Hamiltonian, whose pairwise Poisson brackets vanish, on an open dense regular region.

For the three-particle periodic Toda lattice, take $p_i={\dot{q}}_i$. The proposed Hamiltonian haswhich are exactly the equations of motion. Differentiating the new variables givesThe transformation is not canonical. For examplerather than canonical coordinate brackets. Moreover $a_1{a}_2{a}_3=1/8$, so the map forgets the common translation of all the $q_i$ and is not an invertible coordinate change on the six-dimensional phase space.

Multiplication of the displayed matrices gives diagonal entries $2(a_i^2-a_{i-1}^2)$ and off-diagonal entries $(b_{i+1}-b_i)a_i$ in $LA-AL$. Therefore they form the required Lax pair. The three characteristic coefficients areThey are independent first integrals, as permitted without proof. Equivalently one may takeIndeed $C_1=-P/2$, $C_2=P^2/8-H/4$ and $C_3=1/4-J/8$. Their mutual Poisson commutation also follows directly: $P$ generates the common translation, and $\{H,J\}=0$ because $J$ is conserved.

For the imperfect soft Duffing-van der Pol oscillator at positive small $\lambda$ and $|\epsilon |\ll \sqrt{\lambda }$, the symmetric connection of the Hamiltonian system is $v=\sqrt{\lambda }\tanh (\sqrt{\lambda /2}t)$. Its damping integral is $4{\lambda }^{3/2}(\kappa -\lambda /5)/(3\sqrt{2})$. The two saddle equilibrium barrier heights differ by $-4\epsilon {\lambda }^{3/2}/3$ to first order. Equating the energy change to this difference gives the two oppositely directed connection curves in the display. These are weak-damping asymptotic curves, not exact global formulas at order-one parameters.

// Destination: astrophysical-fluid-dynamics.bigb