Dynamical system

A dynamical system consists of a state space together with a rule that determines how its state evolves in time.

A semiconjugacy from $f:X\to X$ to $g:Y\to Y$ is a map $h:X\to Y$ satisfying $h\circ f=g\circ h$. It transports forward orbits but need not be invertible, A bijection instead gives conjugate functions.

Evolution is deterministic when a specified initial condition and evolution rule select a unique future orbit. For a discrete dynamical system, applying the same map to the same state gives the same next state. For an ordinary differential equation, uniqueness of the initial value problem is required; merely writing a differential equation does not guarantee it.

Determinism permits both asymptotic stability and sensitive dependence on initial conditions. It does not by itself imply reliable prediction from approximate measurements, and sensitivity does not introduce a random choice into the evolution rule.

The Lorenz system is a three-variable autonomous differential equation modeling an idealized thermal convection flow. In Edward N. Lorenz's "Deterministic Nonperiodic Flow" (1963), published in the Journal of the Atmospheric Sciences 20, pp. 130–141, numerical solutions of the model illustrated nonrepeating motion and instability to changes in the initial condition. Lorenz used this example to examine obstacles to very-long-range weather forecasting.^[ref]

The smooth quadratic-critical fixed point of period-doubling renormalization is normalized by $g(0)=1$ and $\beta =g(1)<0$. It has $g^{\prime }(0)=0$ and $g^{\prime \prime }(0)\ne 0$. Differentiating its displayed functional equation twice at zero gives $g^{\prime }(1)=1/\beta$. Its restrictive intervals generate a Cantor attractor described by Feigenbaum geometric potential weights.

The geometric symbolic potential for the Feigenbaum period-doubling map has uniformly negative bounded values and exponentially decreasing variations. Potential sums describe logarithmic lengths of odd-address period-doubling intervals up to a uniform additive error. This is a geometric scaling potential, distinct from an unbounded derivative potential at a critical point. Its pressure is continuous, strictly decreasing and convex by the almost-additive partition-function limit and finite-volume inequalities.

Odd-address intervals lie in the first image of the central restrictive interval, away from the critical point. The derivative of the Feigenbaum map is bounded above and away from zero there, so the next even interval has comparable length by the mean value theorem. For the wraparound image, $g^{2^n}({\beta }^{n}x)={\beta }^{n}g(x)$ makes its length a fixed fraction $(1-\beta )/2$ of the central interval. Thus the odd and full partition length-power sums differ only by constant factors for each fixed real exponent, and have the same exponential growth rate.

The Feigenbaum attractor is the compact Cantor set obtained from the nested restrictive-interval partitions of the Feigenbaum map. Every level interval has two descendant intervals. Their lengths vary with symbolic address, so a geometric topological pressure is useful for estimating its Hausdorff dimension.

A piecewise smooth interval map is uniformly expanding when every smooth branch has derivative magnitude at least $\lambda >1$. A finite branch partition gives a symbolic itinerary and cylinders with diameters at most a constant times ${\lambda }^{-n}$. Expansion alone does not make every symbolic Gibbs measure absolutely continuous; the geometric potential and suitable branch regularity are essential.

For a sufficiently regular uniformly expanding full-branch interval map, the geometric potential assigns the inverse Jacobian determinant weight. Its transfer operator is $\mathcal{L}h(x)={\sum }_{Ty=x}h(y)/|T^{\prime }(y)|$. Under the standard smooth full-branch hypotheses, a positive Lipschitz fixed density $h$ generates the corresponding invariant symbolic Gibbs measure. A different potential generally produces another measure, possibly singular with respect to Lebesgue measure.

If cylinder weights are $\nu (C_w)={\int }_{{\Delta }_w}hdx$, $h\ge h_{\ast }>0$, and $h$ is Lipschitz with constant $L$, then $g_n$ is the logarithm of a cylinder average. For $x\in {\Delta }_w$, $|g_n-\log h(x)|\le L\mathrm{diam}({\Delta }_w)/h_{\ast }$. Thus uniformly shrinking expanding-map cylinders give the geometric convergence estimate. Conversely, if $g_n$ converge uniformly at that rate to $g$ and the symbolic law is shift invariant, $e^{g(E(x))}dx$ has exactly the same cylinder masses: subdivide a fixed cylinder into level-$n$ cylinders, compare integrals within factors $e^{\pm C{\lambda }^{-n}}$, and let $n$ increase. This proves absolute continuity and invariance.

Thermodynamic formalism studies weighted orbit sums, invariant measures and geometric scaling through analogues of partition functions. A symbolic potential assigns a weight to each orbit segment; its asymptotic logarithmic normalization is topological pressure. The resulting Gibbs measure depends on the potential, so choosing an arbitrary Gibbs law does not automatically select geometric volume.

A Gibbs measure is a probability measure whose finite-block weights or conditional laws are determined by exponential potential sums and their normalization. One must specify whether the convention uses boundary-conditioned finite-volume laws, a normalized one-sided transition kernel, or symbolic Gibbs bounds. For the latter, cylinder probabilities are comparable to $\exp (S_n\varphi -nP(\varphi ))$, where $P$ is topological pressure. These conventions have compatibility requirements; an arbitrary unnormalized one-sided potential need not define consistent finite-horizon conditional probabilities.

For the doubling map, level-$n$ cylinders have Lebesgue measure $2^{-n}$. A Bernoulli symbolic law with probabilities $p,1-p$ is a Gibbs measure for a locally constant potential. On a cylinder with $n$ identical symbols, its density ratio is $(2p{)}^n$. If $p\ne 1/2$, these ratios cannot converge uniformly to a finite positive density. For $p=2/3$, the nonboundary alternating sequence has even-level log ratio $(n/2)\log (8/9)$, which tends to $-\infty$. The geometric Gibbs law is the special case $p=1/2$.

For a finite full shift and a potential with summable variations, the displayed symbolic pressure exists and does not depend on the reference tail $\eta$. The almost-additive partition-function limit proves existence. The function $\gamma \mapsto P(\gamma U)$ is convex; finite-volume second derivatives are energy variances divided by block length. If $-B\le U\le -b<0$, its increments for $\delta >0$ lie between $-B\delta$ and $-b\delta$, so it is Lipschitz continuous, strictly decreasing, and has one positive zero.

The sequences $a_n+D$ and $a_n-D$ are respectively subadditive and superadditive. Apply Fekete's lemma to the first and its negative to the second; their normalized difference is $2D/n$, so their limits coincide. For full-shift weighted orbit sums, summable variations of the potential bound the energy change at a concatenation boundary, giving this estimate for the logarithm of the partition function. A fixed initial symbol or fixed remote tail changes only a bounded normalization and not the limit.

A symbolic potential assigns a real weight to an infinite symbolic sequence. Its orbit sum gives the energy in a weighted partition function. Agreement on the first $r$ symbols defines the variation ${\mathrm{var}}_{r}U$; summable variations control sensitivity to a remote history. Geometric potentials encode interval contraction, while other potentials can assign different statistical weights to the same symbolic dynamics.

An integer matrix with determinant $\pm 1$ induces an invertible transformation of the real torus, preserving normalized Haar measure. It is a mixing measure-preserving transformation exactly when none of its eigenvalues is a root of unity. Indeed Fourier frequencies evolve by $A^T$: a nonzero integer frequency cannot revisit a fixed frequency unless some power has eigenvalue one. Conversely a root of unity gives a nonzero integer vector fixed by a power of $A^T$, and its nonconstant character obstructs mixing.

The equilibria are $0$ and $-\epsilon \pm \sqrt{{\epsilon }^2+\lambda }$. Their linear trace is $\kappa -v_{\ast }^2$ and determinant $\lambda -4\epsilon v_{\ast }-3v_{\ast }^2$. For nonzero $\epsilon$ the symmetry-protected pitchfork bifurcation splits into a transcritical bifurcation at $\lambda =0$ and a saddle-node bifurcation at $\lambda =-{\epsilon }^2$. The energy $H={\dot{v}}^2/2+\lambda v^2/2-2\epsilon v^3/3-v^4/4$ obeys $\dot{H}=(\kappa -v^2){\dot{v}}^2$. For positive $\lambda$, the two outer saddle equilibria have unequal barrier heights, so the symmetric heteroclinic cycle unfolds into separated one-way connections and a homoclinic orbit termination of the central stable limit cycle.

Pattern formation is the development of spatial structure from an initially uniform state, often through loss of stability to perturbations with a preferred nonzero wavenumber. If a translation-invariant linearized equation has Fourier growth rate $\lambda (k)$, an unstable maximum near $k=k_c\ne 0$ initially amplifies spatial modes near that scale. Nonlinear interactions then determine whether the pattern saturates, drifts, oscillates or changes scale. The Swift–Hohenberg equation supplies a simple model with $\lambda (k)=r-(1-k^2{)}^2$, selecting $k\simeq 1$ near onset.

A planform is the horizontal spatial structure of a pattern, often described by the Fourier modes active at a bifurcation. For example, one pair of opposite wavevectors gives parallel convection rolls, while two equal-amplitude perpendicular pairs give squares. A planform includes the relative amplitudes and phases needed to distinguish spatial symmetry types.

A phase modulation changes the local position of a periodic pattern. Its local wavenumber is shifted by derivatives of the phase. In a method of multiple scales, a correction profile depending on the translated coordinate $\xi =x+\varphi (Y)$ must be differentiated with ${\mathcal{D}}_Y={\partial }_Y+{\varphi }_Y{\partial }_{\xi }$. Omitting this chain rule can change the nonlinear coefficient of a reduced amplitude equation even when its linear coefficients are unchanged.

Let $p={\varphi }_Y$, $r={\varphi }_{YY}$ and $w_2=C_1(\xi )r+C_2(\xi )p^2$. Two applications of the chain rule give ${\mathcal{D}}_Y^2{w}_2=C_1{\varphi }_{YYYY}+2(C_1^{\prime }+C_2)p{\varphi }_{YYY}+(C_1^{\prime }+2C_2)r^2+(C_1^{\prime \prime }+5C_2^{\prime })p^{2}r+C_2^{\prime \prime }{p}^4$. In a Fredholm solvability condition, the fourth term generally contributes to the coefficient of ${\varphi }_Y^2{\varphi }_{YY}$. It cannot be omitted merely because the fast-coordinate correction functions have been denoted by $C_j(x)$.

A zigzag instability is a long-wave transverse phase modulation of parallel rolls. In ${\varphi }_T=\lambda {\varphi }_{YY}-\gamma {\varphi }_{YYYY}-\alpha {\varphi }_Y^2{\varphi }_{YY}$, with $\gamma >0$, the straight roll loses transverse linear stability analysis when $\lambda <0$. Differentiating gives the conserved slope equation $p_T={\partial }_Y^2(\lambda p-\gamma p_{YY}-\alpha p^3/3)$. When $\lambda ,\alpha <0$, the nonzero preferred slopes satisfy $p^2=3\lambda /\alpha$.

At a transverse phase-diffusion threshold the coefficient of the second spatial derivative is of order ${\epsilon }^2$. Its product with two long-wave derivatives therefore has order ${\epsilon }^4$, the same order as the stabilizing fourth derivative. This balance determines the slow time $T={\epsilon }^{4}t$ and retains both terms in the leading phase modulation equation.

The Swift–Hohenberg equation is a pattern-forming dissipative PDE with linear operator $r-(q_0^2+{\partial }_x^2{)}^2$. A spatial Fourier mode of wavenumber $k$ has growth rate $r-(q_0^2-k^2{)}^2$, selecting modes near $k=q_0$. A cubic-quintic version $w_t=[r-(1+{\partial }_x^2{)}^2]w+s{w}^3-w^5$ has destabilizing cubic feedback for $s>0$ and quintic saturation. Its weakly nonlinear envelope obeys a quintic real Ginzburg-Landau equation.

In the mildly subcritical Swift–Hohenberg equation, set $r={\epsilon }^4\mu$, $s={\epsilon }^2\hat{s}$ and $w=\epsilon [A(X,T)e^{ix}+\overline{A}{e}^{-ix}]+\cdots$. The nonlinear and growth terms first balance at order ${\epsilon }^5$, so $T={\epsilon }^{4}t$. Quadratic detuning of the critical Fourier growth rate gives $X={\epsilon }^{2}x$. At order ${\epsilon }^5$, projection onto the kernel of the self-adjoint fast operator $(1+{\partial }_x^2{)}^2$ requires the resonant $e^{\pm ix}$ forcing to vanish. The cubic and quintic resonances have coefficients $3A|A{|}^2$ and $10A|A{|}^4$, while the operator expansion gives $4A_{XX}$. Thus $A_T=\mu A+3\hat{s}A|A{|}^2-10A|A{|}^4+4A_{XX}$.

A reflection-symmetric planar cubic model can be expressed in polar variables $S=u^2+v^2$, $u=\sqrt{S}\cos \theta$, $v=\sqrt{S}\sin \theta$ as $\dot{S}=2S(\mu +\sin 2\theta -S/2)$ and $\dot{\theta }=\sigma +\cos 2\theta -S/2$. A nonzero equilibrium therefore has $\sin 2\theta =S/2-\mu$, $\cos 2\theta =S/2-\sigma$. Adding their squares gives $S^2/2-(\mu +\sigma )S+{\mu }^2+{\sigma }^2-1=0$. Each positive root $S=\mu +\sigma \pm \sqrt{2-(\mu -\sigma {)}^2}$ determines an antipodal equilibrium pair. The determinant of the polar linearization there is $2S(S-\mu -\sigma )$ and its trace is $2(\mu -S)$. These formulas distinguish folds of nonzero equilibria from symmetry-forced bifurcations at $S=0$.

A Lyapunov exponent measures the asymptotic exponential rate of an infinitesimal tangent displacement under a dynamical system, when the displayed limit exists for a nonzero tangent vector $v$. A positive exponent represents exponential separation along that direction. In a smooth velocity field, the tangent displacement obeys $\dot{\mathbf{l}}=(\nabla \mathbf{u})\mathbf{l}$. Finite-separation growth eventually leaves this linear regime; moments of random separation need not grow at the same rate as typical separation.

Coexistence of multiple stable long-time states at the same parameter values. Bistability is the case of two states; distinct basins of attraction allow a system to remember its initial state or a sufficiently large perturbation.

// Target: mathematical-biology.bigb

Coexistence of two stable long-time states at fixed parameters. Initial conditions and sufficiently strong perturbations can select different basins of attraction, allowing a dynamical system to retain a state-dependent memory.

A biochemical regulatory module with two stable states over a specified range of conditions. History-dependent thresholds distinguish hysteresis from a single steep response.

A homoclinic orbit is a nonconstant trajectory that approaches the same equilibrium point as time tends to both positive and negative infinity. Its closure includes that equilibrium. Such an orbit may form a basin boundary, although a homoclinic orbit alone does not imply a chaotic invariant set.

For a hyperbolic equilibrium point with one unstable eigenvalue ${\lambda }_{+}>0$ and weakest stable decay rate ${\lambda }_{-}<0$, the saddle index is $\delta =-{\lambda }_{-}/{\lambda }_{+}$. For a stable complex pair, use its real part. An incoming distance $x$ from the stable manifold takes time $T={\lambda }_{+}^{-1}\log (h/|x|)$ to reach an outgoing section at unstable coordinate magnitude $h$. A stable coordinate is multiplied by $e^{{\lambda }_{-}T}=(|x|/h{)}^{\delta }$, giving a power-law local passage. A smooth nondegenerate reinjection consequently has derivative tending to zero when $\delta >1$ and an unbounded leading derivative when $\delta <1$. This determines local contraction or expansion near a homoclinic orbit; existence of a particular attractor also depends on the global return geometry.

At a planar saddle equilibrium with $m_{-}<0<m_{+}$, positive saddle quantity $m_{-}+m_{+}>0$ is equivalent to saddle index $\delta <1$. The local passage followed by the regular return has leading transverse map $s\mapsto C{s}^{\delta }$, $C>0$. Its derivative diverges as $s\downarrow 0$, so a nearby periodic orbit created by the saddle-loop splitting is repelling. This local mechanism agrees with the growth of the unstable periodic orbit born at a subcritical Hopf bifurcation in the quadratic Bogdanov–Takens unfolding.

Near two homoclinic orbits of a planar saddle equilibrium with contracting ratio $\delta >1$, the signed Poincaré return map has separate offsets $\mu ,\nu$. Two single-lobe periodic orbits exist for $\mu <0$ and $\nu <0$, respectively. A two-lobe periodic orbit is represented by positive $p,q$ solving the displayed equations. Its boundary consists of the compound homoclinic orbit curves $\mu =-A{\nu }^{\delta }$ for $\nu >0$ and $\nu =-B{\mu }^{\delta }$ for $\mu >0$, with $A,B>0$. In a sufficiently small return domain the derivatives tend to zero, so each admissible symbolic branch has at most one attracting periodic orbit. Restricting to $\mu =\nu$, $A=B$ gives the usual symmetric homoclinic gluing bifurcation.

Two symmetry-related periodic orbits can meet the two loops of a symmetric saddle's figure-eight separatrix. On the other side a single periodic orbit can traverse both lobes. The saddle value and nonlinear return-map coefficients determine stability. In a damped double-well oscillator the symmetry-related stable cycles born at the two well Hopf bifurcations provide a natural setting for this merger.

For a symmetric real saddle with one unstable direction and a strongly contracting transverse direction, composing the power-law local passage with a smooth global return gives this leading Poincaré return map, with $A>0$ and saddle index $\delta$. It is undefined at the stable-manifold crossing $x=0$. For $\delta >1$, small fixed points on each branch are attracting for $\mu <0$; for $\mu >0$ an attracting alternating-sign period-two orbit satisfies $s=\mu -A{s}^{\delta }$, giving gluing of two one-lobe flow cycles into one two-lobe cycle. For $\delta <1$ and small $\mu >0$, on $I=[-c\mu ,c\mu ]$, $0<c<1$, the inverse branchesmap into $I$ and contract. Infinite branch choices define an invariant Cantor set with full two-symbol dynamics. In the two-dimensional return, transverse contraction turns this mechanism into a Smale horseshoe. An attracting Lorenz set additionally requires appropriate trapping and global-return properties.

For $x^{\prime }=-\mu \mathrm{sgn}(x)+A\mathrm{sgn}(y)|x{|}^{\delta }$, $y^{\prime }=\mathrm{sgn}(x)$, the invariant sign sheet has $y=\pm 1$ after one iterate. There $z=-xy$ obeys the displayed one-dimensional map. A negative fixed $z$ lifts to two symmetry-related fixed points, while a positive fixed $z$ lifts to one period-two orbit. The origin of the return map represents the separatrix, where the flow return time diverges, and is not an ordinary finite-period orbit of a group action.

The fixed point relation is $\mu =z-A\mathrm{sgn}(z)|z{|}^{\delta }$. Setting the fixed-point multiplier $A\delta |z{|}^{\delta -1}$ equal to one gives the two saddle-node bifurcations at $z=\pm r$. For fixed $0<A<1$, as $\delta \uparrow 1$ from below their parameter width tends to zero as $(1-\delta )A^{1/(1-\delta )}/e$, forming an exponentially narrow gluing-map cusp. Three scalar fixed points coexist inside this region: two stable outer branches and one unstable inner branch. For $\delta >1$ near one the formal saddle-node bifurcations recede to infinity, outside the local gluing-map domain.

For fixed $0<A<1$ and $\delta <1$, the two folds of the signed gluing-map reduction enclose three scalar fixed-point branches. Writing $\delta =1-\eta$ gives width $C=\eta (1-\eta {)}^{-1}[A(1-\eta ){]}^{1/\eta }$. Since $(1-\eta {)}^{1/\eta }\to e^{-1}$, the displayed essential asymptotic follows. The fold curves meet with exponentially small width; this is not the ordinary algebraic cusp normal form of a smooth cubic vector field. A negative fixed point lifts to two symmetry-related flow cycles, so the cusp describes distinct return-map branches rather than counting every physical cycle separately.

A Shilnikov bifurcation is a homoclinic connection to a saddle-focus in three dimensions, with one real unstable eigenvalue ${\lambda }_{+}>0$ and a stable complex pair ${\lambda }_{-}\pm i\omega$. The eigenvalue ratio $\delta =-{\lambda }_{-}/{\lambda }_{+}$ distinguishes a contracting return for $\delta >1$ from the oscillatory expanding return associated with positive saddle value when $\delta <1$. Generic local and global maps yield infinitely many saddle periodic orbits at a connection in the latter case.

A passage near a saddle-focus contracts a transverse radius by $z^{\delta }$ while rotating it by an angle proportional to $\log z$. A smooth global reinjection converts these transverse coordinates into a sinusoidal return in $z$. The map is defined only for positive passage coordinates that return on the chosen branch. The full two-dimensional map has determinant of order $z^{2\delta -1}$; for $\delta >1/2$ its thin transverse geometry supports a leading one-dimensional approximation. A one-passage orbit has period $T=T_g+{\lambda }_{+}^{-1}\log (z_0/z)$.

At a saddle-focus homoclinic orbit, the leading Shilnikov return map is $f(x)=A{x}^{\delta }\cos (q\log x+\Phi )$, $x>0$, with $q>0$. If $1/2<\delta <1$, its fixed points solve $\cos (q\log x+\Phi )=x^{1-\delta }/A$. Infinitely many solutions accumulate at zero near the zeros of the cosine, with successive size ratios tending to $e^{-\pi /q}$. The derivative is $A{x}^{\delta -1}[\delta \cos (q\log x+\Phi )-q\sin (q\log x+\Phi )]$, so their expanding multiplier is unbounded. Narrow monotone strips near successive oscillations map across a common interval; inverse branches contract, supplying arbitrarily long symbolic itineraries and infinitely many saddle periodic orbits. Varying the splitting parameter shifts the graph vertically, producing accumulating fold and period-doubling thresholds and stable windows. The condition $\delta >1/2$ makes the local flow volume-contracting, whereas $\delta <1$ makes its saddle value positive.

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.

An Anosov diffeomorphism is a continuously differentiable diffeomorphism of a compact smooth manifold with an invariant splitting of its tangent bundle into stable and unstable vector subbundles satisfying the displayed uniform exponential estimates for $n\ge 0$, with $C,\lambda >0$. Hyperbolic toral automorphisms give examples. An Anosov flow has a further one-dimensional invariant bundle along the flow direction; its definition is a distinct continuous-time version of uniform hyperbolicity.

An integer invertible matrix with no eigenvalue on the unit circle induces a hyperbolic toral automorphism. Its stable and unstable tangent subspaces are the corresponding generalized eigenspaces. The matrix $\left(\begin{array}{cc}2& 1\\ 1& 1\end{array}\right)$ has eigenvalues $(3\pm \sqrt{5})/2$, yielding one contracting and one expanding direction.

A smooth flow is a smooth one-parameter family of diffeomorphisms satisfying ${\varphi }_0=\mathrm{id}$ and ${\varphi }_{t+s}={\varphi }_t\circ {\varphi }_s$. Its generating vector field is $F(x)={\partial }_t{\varphi }_t(x){|}_{t=0}$. On a closed manifold, every smooth vector field has a flow defined for all real times.

A smooth function $h$ is a flow coboundary if $h=Fu$ for some smooth function $u$, where $F$ is the flow generator. Equivalently $u({\varphi }_{t}x)-u(x)={\int }_0^{t}h({\varphi }_{s}x)ds$. Thus its integral over every periodic orbit is zero. For a transitive Anosov flow, the Livsic theorem supplies the converse and smooth regularity.

For a volume form $\Omega$, define divergence by ${\mathcal{L}}_F\Omega =({\mathrm{div}}_{\Omega }F)\Omega$. The product rule for the Lie derivative gives ${\mathrm{div}}_{e^h\Omega }F={\mathrm{div}}_{\Omega }F+Fh$. Consequently the flow preserves a smooth positive volume precisely when its divergence in any reference volume is a smooth flow coboundary. The displayed density has the minus sign dictated by this product rule.

A suspension flow moves vertically in the mapping torus of a diffeomorphism, applying that diffeomorphism whenever an orbit crosses the identifying section. A hyperbolic toral automorphism with constant roof gives a three-dimensional Anosov flow: after $k$ crossings its transverse derivative is $A^k$, while the remaining fractional time contributes a uniformly bounded factor.

Write $\rho =1/f$ and $t={\int }_0^s\rho ({\varphi }_{r}x)dr$. For an Anosov stable bundle, the new stable directions are $v+{\lambda }_s(x,v)F(x)$, where ${\lambda }_s=\rho (x{)}^{-1}{\int }_0^{\infty }d{\rho }_{{\varphi }_{r}x}(D{\varphi }_{r}v)dr$. Exponential contraction makes this integral uniformly convergent. Differentiating the clock equation shows that its flow-direction coefficient after time $t$ is $\rho ({\varphi }_{s}x{)}^{-1}{\int }_s^{\infty }d\rho (D{\varphi }_{r}v)dr$, which decays exponentially and has the same graph form at the new point. Negative-time integrals give the unstable graph. Compactness and positivity of $f$ compare old and new times uniformly, preserving the Anosov flow property.

A nonsingular smooth flow on a compact smooth manifold is Anosov if there is a continuous invariant splitting $TN=E^s\oplus \mathbb{R}F\oplus E^u$ and constants $C,\lambda >0$ with $\Vert D{\varphi }_t{v}_s\Vert \le C{e}^{-\lambda t}\Vert v_s\Vert$ and $\Vert D{\varphi }_{-t}{v}_u\Vert \le C{e}^{-\lambda t}\Vert v_u\Vert$ for $t\ge 0$. Uniform contraction and expansion are transverse to the neutral flow direction, which distinguishes the splitting from that of an Anosov diffeomorphism.

Let $F$ generate a nonsingular smooth flow on a closed manifold. Suppose a continuous nondegenerate indefinite quadratic form on the transverse vector bundle $TN/\mathbb{R}F$ has constant signature, is differentiable along the linearized flow, and has continuous positive-definite derivative along that flow. Then the flow is an Anosov flow. Compactness makes the positivity uniform; the positive and negative cones give the unstable and stable transverse directions. The flow direction must be quotiented out: its derivative cannot satisfy strict positivity.

Invariance gives $dh(v_s)=dh(D{\varphi }_t{v}_s)$ for a stable direction. Boundedness of $dh$ and exponential contraction make this zero; negative time does the same for unstable directions. Also $dh(F)=0$. The Anosov splitting spans the tangent bundle, so $dh=0$ and the function is constant on each connected component. This proof does not need an ergodicity hypothesis.

For a topologically transitive Anosov flow, a Hölder function has a Hölder flow coboundary precisely when its integral around every periodic orbit vanishes. The coboundary identity is $u({\varphi }_{t}x)-u(x)={\int }_0^{t}h({\varphi }_{s}x)ds$. For smooth flow and smooth $h$, the smooth regularity theorem gives smooth $u$. Necessity follows by setting $t$ equal to a period. The solution is unique up to a constant by density of an orbit.

The unstable bundle consists of transverse directions exponentially contracted in negative time. Together with the stable bundle and the flow direction it gives the invariant splitting of an Anosov flow. Under a positive time change its graph coefficient is obtained from the convergent past integral $-\rho (x{)}^{-1}{\int }_{-\infty }^{0}d\rho (D{\varphi }_{r}v)dr$.

The stable bundle consists of the transverse directions exponentially contracted in positive time. The unstable bundle has the corresponding negative-time property. A positive time change of a smooth flow can tilt these bundles in the flow direction, even though it preserves the underlying orbits.

The weak stable bundle adjoins the flow direction to the strong stable bundle. It is tangent to the weak stable foliation. For a geodesic flow on a hyperbolic surface, the canonical frame gives $E^{ws}=\mathrm{span}\{X,H-V\}$.

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.

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.

For $\dot{q}=u_0(q)+ϵu_1(q,t)$ with planar Hamiltonian flow $u_0=(-{\partial }_y{\psi }_0,{\partial }_x{\psi }_0)$, let $q_0$ be a heteroclinic orbit between two hyperbolic fixed points. Its first-order displacement satisfies ${\dot{q}}_1=D{u}_0{q}_1+u_1$. The covector $w=\nabla {\psi }_0(q_0)$ satisfies $\dot{w}=-(D{u}_0{)}^{T}w$, obtained by differentiating $\nabla {\psi }_0\cdot u_0=0$. Thus $(w\cdot q_1{)}^{\prime }=w\cdot u_1$. Integrating the stable correction from the future and the unstable correction from the past gives the displayed function. At a regular section the first-order normal separation is $ϵM/|\nabla {\psi }_0|$. A simple zero yields a transverse intersection of the perturbed manifolds, by the implicit function theorem.

Perturb the first integral of the quadratic fold-Hopf amplitude flow by $u^{\prime }=-2uv+\epsilon u({\lambda }_1+v^2)$, keeping $v^{\prime }={\lambda }_2+v^2+u^2$. Then $F^{\prime }=-\epsilon u({\lambda }_1+v^2)({\lambda }_2+u^2+v^2)$. On the interior heteroclinic orbit for ${\lambda }_2=-a^2$, $u=\sqrt{3(a^2-v^2)}$ and $v^{\prime }=2(a^2-v^2)$, so the first-order change is $-\epsilon \sqrt{3}{\int }_{-a}^a({\lambda }_1+v^2)\sqrt{a^2-v^2}dv=-\epsilon \pi \sqrt{3}{a}^2({\lambda }_1+a^2/4)/2$. The boundary connection contributes zero. The simple zero gives the displayed leading persistence curve, rather than an exact all-orders parameter equality.

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

Perturb the Hamiltonian blow-up of a reflection-symmetric double-zero point by $u^{\prime }=-sv+v^3/2+\epsilon (\mu -v^2/2)u+O({\epsilon }^2)$ and $v^{\prime }=2u+\epsilon (\mu -v^2/2)v+O({\epsilon }^2)$. Then $H^{\prime }=\epsilon (\mu -v^2/2)(2u^2+s{v}^2-v^4/2)+O({\epsilon }^2)$. On the positive heteroclinic branch $dt=dv/[\sqrt{2}(s-v^2/2)]$, so its first energy change is $\epsilon M/\sqrt{2}$, where $M={\int }_{-\sqrt{2s}}^{\sqrt{2s}}(\mu -v^2/2)(s+v^2/2)dv$. A simple zero gives cancellation of the leading stable/unstable manifold splitting and continues to a nearby heteroclinic bifurcation. Since $\partial M/\partial \mu >0$, there is a unique first-order balance for fixed $s>0$.

For $u^{\prime \prime }=u^2-\kappa -\epsilon (\lambda +u)u^{\prime }$ with $\kappa >0$, the Hamiltonian homoclinic orbit has $H=2{\kappa }^{3/2}/3$. Its energy balance is $\lambda \int v^{2}dt+\int u{v}^{2}dt=0$. Along this orbit the ratio of the second integral to the first is $-5\sqrt{\kappa }/7$, giving the displayed simple zero and hence the leading homoclinic parameter curve.

The conservative equation $u^{\prime \prime }=u^2-a^2$ has a saddle loop with $v_{+}(u)=\sqrt{2/3}(a-u)\sqrt{u+2a}$ for $-2a\le u\le a$. Its homoclinic orbit integrals are $I_0=2{\int }_{-2a}^a{v}_{+}(u)du$ and $I_1=2{\int }_{-2a}^{a}u{v}_{+}(u)du$. Substitution $s=u+2a$ evaluates them as displayed. For a weak perturbation $\epsilon (\beta +u)u^{\prime }$, integrating the energy derivative gives $\beta I_0+I_1=0$, hence $\beta =5a/7$. The nonzero derivative $I_0$ makes this a simple first-order splitting zero.

Stability theory studies the response of a dynamical system to small changes of its state. Linear stability of a planar equilibrium describes the response near equilibria using the linearized evolution.

A model retaining an oscillation's phase while neglecting amplitude dynamics. Interactions can synchronize relative phases through phase locking; noise can cause phase slips. The Adler phase equation is a simple example for a coupled relative phase.

An event changing the unwrapped relative phase by one cycle, usually $2\pi$, compared with a phase-locked state. Thermally activated phase slips correspond to crossing neighboring barriers of a tilted washboard potential. Successive slips can produce long-term drift even while small fluctuations appear confined on shorter times.

A noise-driven barrier crossing between adjacent phase-locked wells. The Kramers escape rate is proportional to $e^{-\Delta V/T_{\mathrm{eff}}}$ when a mobility-one Langevin equation has white-noise covariance $2T_{\mathrm{eff}}\delta (t-t^{\prime })$. Both the barrier and local relaxation time determine whether an intrawell phase autocorrelation approximation is appropriate.

In a mobility-one noisy Adler phase equation, matching Kramers escape rate prefactors and a barrier difference $2\pi \omega$ give $p_{+}/p_{-}=e^{2\pi \omega /T_{\mathrm{eff}}}$. Positive mismatch favors forward winding. The ratio is a low-noise barrier-crossing result, not a claim that the particle remains in one well at arbitrarily long times.

The relative-phase drift $\dot{\theta }=\omega -ϵ\sin \theta$ describes competing frequency mismatch and phase coupling. For $0<\omega <ϵ$, the equilibrium points are $\arcsin (\omega /ϵ)$ and $\pi -\arcsin (\omega /ϵ)$, stable and unstable respectively. They merge in a saddle-node bifurcation at $\omega =ϵ$; greater mismatch gives running phase dynamics. Adding Gaussian white noise gives overdamped Langevin dynamics in a tilted washboard potential.

For $0<\omega <ϵ$, set $\alpha =\arcsin (\omega /ϵ)$ and $\kappa =\sqrt{{ϵ}^2-{\omega }^2}$. The neighboring saddles of the tilted washboard potential have barriers $\Delta V_{+}=2\kappa -\omega (\pi -2\alpha )$ and $\Delta V_{-}=2\kappa +\omega (\pi +2\alpha )$ above the minimum at $\alpha$. Their difference is $2\pi \omega$; both have curvature $-\kappa$, so forward and backward Kramers escape rates have equal leading prefactors.

A phase that continues winding rather than tending to a locked value. In the noise-free Adler phase equation, $\omega >ϵ>0$ makes the drift positive at every phase, so there are no equilibrium points. The associated tilted washboard potential decreases monotonically; wells vanish at the locking threshold.

A skew product evolves a base coordinate by a dynamical system $R$ and a fibre coordinate by a map $S_x$ depending on the current base coordinate: $F(x,y)=(R(x),S_x(y))$. The base evolution is independent of the fibre, while the fibre may depend on the base. Continuity or measurability is imposed according to the setting. An irrational skew shift is a useful affine example on a torus.

For a minimal dynamical system on a compact metric space, the continuous circle extension $(x,y)\mapsto (Tx,y+\rho (x))$ is minimal precisely when the displayed equation has no continuous circle-valued solution $f$ for any nonzero integer $m$. A minimal subsystem of the extension has fibers that are cosets of its closed vertical translation stabilizer. If that stabilizer is proper, a nontrivial circle character trivial on it gives the forbidden equation. Conversely a solution makes $e^{2\pi imy}/f(x)$ a nonconstant continuous invariant function.

For irrational $\alpha$, the irrational skew shift is the skew product $T_{\alpha }(x,y)=(x+\alpha ,y+x)$ on the torus $(\mathbb{R}/\mathbb{Z}{)}^2$. It preserves normalized Lebesgue measure and has iterates $T_{\alpha }^n(x,y)=(x+n\alpha ,y+nx+\alpha n(n-1)/2)$ modulo one. Its Koopman operator sends the Fourier basis character $e_{r,s}=e^{2\pi i(rx+sy)}$ to $e^{2\pi ir\alpha }{e}_{r+s,s}$. The resulting infinite chains of Fourier coefficients show that it is an ergodic transformation. Moreover, uniform equidistribution of an irrational skew shift proves unique ergodicity.