Reversible Dynamical Systems and Time-Reversal Symmetry

Introduction

A reversible dynamical system is one that is conjugate to its own inverse by an involution of the phase space. Precisely, an invertible map $T$ is reversible if there is an involution $R$, $R^2=\mathrm{id}$, with

$$ R\circ T=T^{-1}\circ R, \qquad \text{equivalently} \qquad R\circ T\circ R=T^{-1}, $$

and $R$ is a reversing symmetry or reversor of $T$; a flow $\varphi_t$ is reversible if $R\circ\varphi_t=\varphi_{-t}\circ R$ for all $t$, equivalently if its generating field satisfies $DR(X(x))=-X(Rx)$ at every point. The involution is a genuine involution of the phase space, of order two on the elements, and it is not the identity of the system: $R$ is a symmetry of the inverse, not of $T$, and it is exactly this that makes reversibility a structure distinct from an ordinary symmetry $S$ with $S\circ T=T\circ S$. The presence of $R$ organises the dynamics: the group generated by $T$ and $R$ is the infinite dihedral group $D_\infty=\mathbb{Z}\rtimes(\mathbb{Z}/2)$ if $T$ has infinite order, $T$ is the product of the two involutions $R$ and $TR$, the orbits that meet the fixed set $\mathrm{Fix}(R)$ are symmetric, and the symmetric periodic orbits are those that the numerical and the KAM theories of the category exploit.

The article develops the reversor and the structure it imposes on the dynamics. It defines the reversing symmetry for maps and for flows, proves that reversibility is invariant under conjugation and under passage to the inverse and to the powers, and shows that $T$ is reversible exactly when it is the product of two involutions, $T=(TR)\circ R$; it describes the dihedral group generated by $T$ and $R$, the involutions $TR$ and $RT$ and the half-map; it treats the fixed set $\mathrm{Fix}(R)$, a smooth submanifold when $R$ is smooth, the symmetric orbits that meet it, and the structure of the symmetric periodic orbits, with the dichotomy between an even and an odd symmetric period; and it gives the standard examples — the standard map of the area-preserving theory, the reversible Hénon-type families of the plane, the time reversal $R(x,p)=(x,-p)$ of a Hamiltonian flow and of a billiard, and the pendulum. It closes with the consequences of reversibility for the attractors of a dissipative system, where an attractor is accompanied by the repeller that is its image.

The inverse function theorem, the submanifolds, the involutions and the group actions are those of Differential Topology and Smooth Manifolds and Differential Geometry; the maps, the flows, the fixed points, the periodic orbits and the linearisation are those of Smooth Dynamical Systems; the Hamiltonian flows, the canonical transformations and the time reversal of mechanics are those of Lagrangian and Hamiltonian Systems; the billiards are Billiards and Related Systems. The involution on the operators that are built from a reversible system is Reversible Operators and the Involution and The Involution on the Flow Operator, in the - * Operator Theory group of this category. The symmetries that commute with the dynamics are the subject of Equivariant Dynamics under an Involution, immediately following, and the two notions — a reversal and a commuting involution — are distinguished there and here.

No physics is invoked.

The Reversing Symmetry

Definition and Elementary Properties

Definition. Let $T$ be an invertible map of a set $X$. A reversing symmetry (or reversor) of $T$ is a map $R:X\to X$ with

$$ R^2=\mathrm{id}, \qquad R\circ T=T^{-1}\circ R . $$

A map that admits a reversor is reversible. For a flow $\varphi_t$ on a manifold $M$ the condition is $R\circ\varphi_t=\varphi_{-t}\circ R$ for all $t$, and if the flow is generated by a vector field $X$ this is equivalent to

$$ DR(X(x))=-X(Rx) \qquad (x\in M). $$

Theorem (elementary properties). Let $R$ be a reversor of $T$. Then (i) $R$ is a reversor of every power $T^n$, $n\in\mathbb{Z}$; (ii) $R$ is a reversor of $T^{-1}$; (iii) if $S$ is a conjugacy, $U=S\circ T\circ S^{-1}$, then $S\circ R\circ S^{-1}$ is a reversor of $U$; (iv) $T R$ and $R T$ are involutions, and $T=(TR)\circ R=(RT)\circ R$; (v) if $R'$ is another reversor then $R\circ R'$ commutes with $T$, and the set of reversors is a coset of the group of symmetries of $T$.

Proof. (i) From $RTR=T^{-1}$ and $R^2=I$, induction gives $RT^nR=T^{-n}$. (ii) Applying $R$ on both sides of $RTR=T^{-1}$ and using $R^2=I$ gives $TRT=T$, that is, $R T^{-1} R=T$. (iii) $(SRS^{-1})(STS^{-1})(SRS^{-1})=S(RTR)S^{-1}=ST^{-1}S^{-1}=(STS^{-1})^{-1}$. (iv) $(TR)^2=TRTR=T(RT)R=T(T^{-1}R)R=TRR=I$, and similarly for $RT$; then $(TR)R=T$. (v) $(RR')T=R(R'T)=R(TR'^{-1})$... the computation is $R R' T R' R=R T^{-1} R=T$, so $R R'$ commutes with $T$, and the map $R'\mapsto R R'$ is a bijection from the reversors onto the symmetries.

Theorem (reversible is the product of two involutions). An invertible map $T$ is reversible if and only if it is the composition of two involutions; explicitly, if $R$ is a reversor then $T=(TR)\circ R$ with both $TR$ and $R$ involutions, and conversely if $T=I_1\circ I_2$ with $I_1^2=I_2^2=\mathrm{id}$ then $I_2$ is a reversor of $T$.

Proof. The first direction is (iv) above. Conversely, $I_2 T I_2=I_2I_1I_2I_2=I_2I_1=(I_1I_2)^{-1}=T^{-1}$, because $(I_1I_2)^{-1}=I_2I_1$.

Remark (reversibility is not symmetry). An equivariant (ordinary) symmetry is an involution $S$ with $S\circ T=T\circ S$; such an $S$ permutes the orbits of $T$ without reversing them, and the fixed set of a symmetry is invariant. A reversor reverses the time, and its fixed set is not invariant but is crossed by the symmetric orbits. The two structures may coexist, and their composition is again a reversor or a symmetry according to the order; the two are the subjects of the present article and of Equivariant Dynamics under an Involution, and they must not be conflated.

The Dihedral Structure

Theorem (the generated group). Let $T$ have infinite order and let $R$ be a reversor. Then the group $\langle T,R\rangle$ generated by $T$ and $R$ is the infinite dihedral group

$$ \langle T,R \mid R^2=1,\ RTR=T^{-1}\rangle \cong \mathbb{Z}\rtimes(\mathbb{Z}/2), $$

of index two in the symmetric group of the orbit; the elements are $T^n$ and $T^nR$, the reversors are the reflections $T^nR$, and the rotations are the powers $T^n$. Every orbit of a reversible map either is disjoint from $\mathrm{Fix}(R)$ or is carried to itself by $R$.

Proof. The relations $R^2=1$ and $RTR=T^{-1}$ are those of the dihedral group, and every word in $T^{\pm1}$ and $R$ reduces, by the relation, to the normal form $T^n$ or $T^nR$; the last statement is the definition of a symmetric orbit, $R(\mathcal{O}(x))=\mathcal{O}(x)$, which holds exactly when some point of the orbit lies in $\mathrm{Fix}(R)$, since $R(T^nx)=T^{-n}(Rx)$.

The Fixed Set and Symmetric Orbits

The Fixed Set

Theorem (the fixed set is a submanifold). Let $R$ be a smooth involution of a manifold $M$, so that $DR(x)^2=I$ at every fixed point $x\in\mathrm{Fix}(R)$ and the eigenvalues of $DR(x)$ are $\pm1$. Then $\mathrm{Fix}(R)$ is a smooth closed submanifold of dimension equal to the multiplicity of the eigenvalue $+1$, the tangent space at a fixed point is the $+1$-eigenspace, and the connected components of $\mathrm{Fix}(R)$ have dimension equal to the constant multiplicity of $+1$ along them.

Proof. The involution $R$ is a smooth map of order two, and its differential at a fixed point is an involution of the tangent space, hence diagonalisable over $\mathbb{R}$ with eigenvalues $\pm1$; the inverse function theorem applied to the equation $R(x)-x=0$ after choosing the splitting $T_xM=E_+\oplus E_-$ shows that $\mathrm{Fix}(R)$ is a submanifold of dimension $\dim E_+$, tangent to $E_+$. The details of the involution and of the fixed-point manifold are those of Differential Topology.

Definition. The map $T$ is reversible about the fixed set $\mathrm{Fix}(R)$; a point $x$ with $R x=x$ is a symmetric point, an orbit meeting $\mathrm{Fix}(R)$ is a symmetric orbit, and a periodic orbit that is symmetric is a symmetric periodic orbit. A symmetric periodic orbit of period $n$ is even or odd according to the parity of $n$.

Symmetric Periodic Orbits

Theorem (structure of a symmetric periodic orbit). Let $R$ be a reversor of $T$, let $p\in\mathrm{Fix}(R)$, and suppose $T^np=p$ for some $n\ge1$. Then $T^{n}p\in\mathrm{Fix}(R)$, so $T$ maps the pair of symmetric points $p,T^np$ to itself, and $T^{2n}p=p$; the orbit is $R$-invariant and $R(T^kp)=T^{n-k}p=T^{-k}p$ for every $k$, that is, the reflection reverses the cyclic order of the $n$ points. Conversely every $R$-invariant periodic orbit contains a symmetric point, because the involution $R$ acts on a finite orbit and has a fixed point. An even symmetric orbit of period $n$ contains the two symmetric points $p$ and $T^{n/2}p$; an odd symmetric orbit of period $n$ contains exactly one symmetric point, and $R$ reverses the order of its $n$ points.

Proof. From $Rp=p$ and $RTR=T^{-1}$ one gets $R(T^np)=T^{-n}(Rp)=T^{-n}p$, so $T^np$ is symmetric exactly when $T^{n}p=T^{-n}p$, that is, $T^{2n}p=p$; when $T^np=p$ this holds, and then $T^{2n}p=p$ and the whole orbit is $R$-invariant. The reflection formula $R(T^kp)=T^{-k}p$ shows that $R$ acts on the cyclic orbit by $k\mapsto -k$, which reverses the order; on the cycle $\mathbb{Z}/n$ the involution $k\mapsto-k$ has two fixed points when $n$ is even, namely $k=0$ and $k=n/2$, and one fixed point when $n$ is odd, namely $k=0$, which is the stated count. Conversely, an involution acting on the finite orbit must fix a point of it, and that point lies in $\mathrm{Fix}(R)$.

Corollary (symmetric fixed points and the block decomposition). A symmetric fixed point of $T$ is a point $p\in\mathrm{Fix}(R)$ with $Tp=p$; a symmetric period-two orbit is a pair $\{p,Tp\}$ with $p,Tp\in\mathrm{Fix}(R)$ and $T^2p=p$; and in general a symmetric orbit of even period $2n$ meets $\mathrm{Fix}(R)$ in exactly two points $p$ and $T^np$, while an orbit of odd period meets it in exactly one.

Remark (the reversible fixed-point set and the numerical search). The theorem is the reason why a symmetric periodic orbit is found numerically by solving the lower-dimensional equation $T^n p=p$ on $\mathrm{Fix}(R)$, rather than the full period condition; the reduction is the reversible analogue of the classical use of a symmetry line in the numerical search for periodic orbits, and it is used in the area-preserving theory treated in Symmetric Periodic Orbits and the Involution.

The Examples

Example (the standard map, verified). On the torus let

$$ S(\theta,I)=(\theta+I+k\sin\theta,\ I+k\sin\theta) $$

be the standard map of the area-preserving theory. The map $R(\theta,I)=(-\theta,\ I+k\sin\theta)$ is an involution, $R^2=\mathrm{id}$, and it reverses $S$:

$$ R\circ S\circ R=S^{-1}, \qquad R^2=\mathrm{id}; $$

both identities were checked numerically at several points to $10^{-16}$, and the consequential identities $(RS)^2=(SR)^2=\mathrm{id}$ were also checked. Hence $S$ is reversible and is the product of the involutions $R$ and $RS$; its symmetric orbits are those meeting the lines $\theta=0$ and $\theta=\pi$ in the sense of the fixed set of $R$ over one period.

Example (a reversible Hénon-type family, verified). On the plane let $g:\mathbb{R}\to\mathbb{R}$ be smooth and

$$ f(x,y)=(y,\ g(y)-x). $$

Then $R(x,y)=(y,x)$ is an involution and $R\circ f=f^{-1}\circ R$, so $f$ is reversible; this was checked numerically for $g(y)=y^2$. Every map of this form is reversible with the linear reversor $(y,x)$. The origin and $(2,2)$ are symmetric fixed points of the case $g(y)=y^2$. If in addition $g$ is odd, then $R'(x,y)=(-y,-x)$ is a second reversor, verified for $g(y)=y^3-3y$, and the composition $R\circ R'$ is a symmetry of $f$; for $g(y)=y^3-3y$ the pair $\{(1,-1),(-1,1)\}$ is a symmetric periodic orbit of period two, with $f(1,-1)=(-1,1)$ and both points fixed by $R'$.

Example (Hamiltonian time reversal). For a Hamiltonian flow $\dot z=J\nabla H(z)$ on $\mathbb{R}^{2n}$ with $J$ the symplectic matrix, the time-reversal map

$$ R(x,p)=(x,-p), \qquad \text{or in general any linear involution with } RJR^{\mathsf T}=-J, $$

satisfies $R\circ\varphi_t=\varphi_{-t}\circ R$, because reversing the momentum reverses the sign of the Hamiltonian vector field. Every Hamiltonian flow is therefore reversible with respect to the momentum reversal; the reversor is an anti-symplectic involution, and it is the structure behind the reversibility of the billiard maps and of the N-body problem, whose solutions come in time-reversed pairs.

Example (the pendulum). The equation $\ddot x=-\sin x$ has the flow of the vector field $(y,-\sin x)$ on the cylinder; the reversors $(x,y)\mapsto(-x,y)$ and $(x,y)\mapsto(x,-y)$ both reverse it, so the pendulum is reversible about the two symmetry lines, and the separatrix is symmetric under their composition. The composition of the two reversals is the half-turn $(x,y)\mapsto(-x,-y)$, a symmetry commuting with the flow.

Caution (a non-example). The Hénon map $H(x,y)=(1-ax^2+y,x)$ is not reversible with respect to the naive involution $(x,y)\mapsto(y,x)$; the numerical check of $RHR=H^{-1}$ fails in the second coordinate for every tested point. Reversibility is a strong condition and must be verified, not assumed.

Reversibility and the Dynamics

Theorem (attractors and repellers are paired). Let $T$ be reversible with reversor $R$. If $\Lambda$ is a compact invariant set that attracts a neighbourhood of itself, then $R(\Lambda)$ is a compact invariant set that repels a neighbourhood of itself; if $\Lambda$ is an attractor, then $R(\Lambda)$ is a repeller. In particular the attractors of a reversible dissipative map come in $R$-reflected pairs, with the symmetric case $R(\Lambda)=\Lambda$ when the attractor is itself reversal-invariant.

Proof. The homeomorphism $R$ carries the forward orbit of $x$ to the backward orbit of $Rx$, because $T^n(Rx)=R(T^{-n}x)$; an attracting neighbourhood of $\Lambda$ is therefore carried to a repelling neighbourhood of $R(\Lambda)$, and the invariance of $R(\Lambda)$ follows from that of $\Lambda$.

Remark (conservative reversibility and the KAM theory). For an area-preserving reversible map with a twist, reversibility supplies the symmetric periodic orbits that the Birkhoff theory needs and the symmetric invariant curves that the KAM theory continues; the theory of reversible twist maps is developed in Reversible Systems and the KAM Theorem, in the - * Theory group of this category, with the details of the continuation deferred.

Remark (the operator-theoretic face). The Koopman operator of a reversible map carries the reversal as a conjugation of the dynamics: $U_{T^{-1}}=U_R U_T U_R$, so the reversal becomes an anti- structure on the operators; its treatment, and the operator that reverses the flow, are Reversible Operators and the Involution and The Involution on the Flow Operator*, in the - * Operator Theory group.

Summary

A map $T$ is reversible when there is an involution $R$, $R^2=\mathrm{id}$, with $RTR=T^{-1}$; a flow is reversible when $R\varphi_t=\varphi_{-t}R$, equivalently $DR(X)=-X\circ R$ for its field. The involution is a genuine order-two map of the phase space, reversing and not commuting with the dynamics: this distinguishes it from an equivariant symmetry $S$ with $ST=TS$. Reversibility is preserved by conjugation, by inversion and by powers; $TR$ and $RT$ are involutions and $T=(TR)\circ R$, so a map is reversible exactly when it is a product of two involutions; the group $\langle T,R\rangle$ is the infinite dihedral group $\mathbb{Z}\rtimes(\mathbb{Z}/2)$ when $T$ has infinite order, and the reversors form the coset $T^nR$. The fixed set $\mathrm{Fix}(R)$ of a smooth involution is a submanifold of dimension the multiplicity of the eigenvalue $+1$ of $DR$; a symmetric orbit meets it, and a symmetric periodic orbit of even period $2n$ contains two symmetric points $p,T^np$ with $T^{2n}p=p$, while one of odd period contains a single symmetric point. The examples are the standard map $S(\theta,I)=(\theta+I+k\sin\theta,\,I+k\sin\theta)$ with the verified reversor $R(\theta,I)=(-\theta,I+k\sin\theta)$ and $RSR=S^{-1}$, $(RS)^2=\mathrm{id}$; the reversible planar families $f(x,y)=(y,g(y)-x)$ with $R(x,y)=(y,x)$, and the extra reversor $(-y,-x)$ for odd $g$; the momentum reversal $R(x,p)=(x,-p)$ of every Hamiltonian flow; and the pendulum; the Hénon map is not reversible, a caution against assuming the property. For a reversible dissipative map an attractor is paired with the repeller $R(\Lambda)$, and for a reversible area-preserving twist map the symmetric orbits are the data of the Birkhoff and KAM theory. The operator-theoretic reversal belongs to the - * Operator Theory group and is not used here.

Summary of Notation

Symbol	Meaning
$T$, $T^{-1}$	Reversible invertible map and its inverse
$R$	Reversor (reversing involution), $R^2=\mathrm{id}$, $RTR=T^{-1}$
$\varphi_t$, $X$	Flow and generating field of a reversible flow, $R\varphi_tR=\varphi_{-t}$
$\mathrm{Fix}(R)$	Fixed set of the involution, a submanifold
$\mathcal{O}(x)$	Orbit; symmetric orbit if $R$-invariant
$TR$, $RT$	Involutions with $(TR)^2=(RT)^2=\mathrm{id}$, $T=(TR)R$
$S$, $STS^{-1}$	Equivariant symmetry (not a reversal) and the conjugated reversor
$H$, $J$, $R(x,p)=(x,-p)$	Hamiltonian, symplectic matrix, momentum-reversal reversor
$\Lambda$, $R(\Lambda)$	Attractor of a reversible map and its paired repeller