For a flow φ_t on a manifold M, a map R : M → M is a reversing symmetry (reversor) if it is an involution and satisfies R ∘ φ_t = φ_{−t} ∘ R for all t. This implies that if x(t) is a trajectory, then R(x(−t)) is also a trajectory, i.e. the system is invariant under time reversal implemented by R.[2]

A dynamical system is called reversible when it possesses a reversing symmetry R satisfying Θ ∘ T = T^{−1} ∘ Θ for discrete maps, or, for flows, the condition A(Θx) = −dΘ_x A(x), which is equivalent to Θ ∘ φ_t = φ_{−t} ∘ Θ. In Hamiltonian systems, the canonical time-reversal is typically Θ(q,p) = (q,−p).[3]

In a reversible dynamical system (M,φ_t,R), R is an involution with R ∘ φ_t = φ_{−t} ∘ R. Symmetric orbits are those which intersect Fix(R), the fixed-point set of the involution. If an orbit is symmetric, then it contains points fixed by R, and the study of such orbits reduces to the analysis of symmetric fixed points or symmetric periodic solutions.[4]

By definition, the map T is reversible if and only if T ∘ R is an involution, moreover the map T is reversible with respect to the involution T ∘ R. An orbit of the reversible diffeomorphism T is symmetric if and only if it contains a fixed point of one of the involutions, R or R′ = T ∘ R. If q is the period of the orbit, then this point is a symmetric fixed point of the map T^q (this map is reversible with respect to both R and R′, as the map T is).[5]

A smooth vector field X on M is then called R-reversible if TR(X) = −X∘R. In other words, R maps positive time trajectories of X into negative time trajectories. If φt denotes the flow of X then reversibility is equivalent to R∘φt = φ−t∘R for all t. Thus, for any point x, the orbit of R(x) is just the time-reversed orbit of x, and R gives a conjugacy between the flow restricted to the orbit of x and the flow restricted to the orbit of R(x), with reversed time direction.

In the definition used here (following Meiss), a reversible system is one for which there exists a diffeomorphism S that conjugates the flow with its inverse, i.e. (φ_{−t} ∘ S)(z) = (S ∘ φ_t)(z) for a dynamical system ż = f(z). Concretely this implies −f(S(z)) = D S(z) f(z). The map S is typically an involution (S ∘ S = Id) and is referred to as a reversor or reversing symmetry.[7]

We study a one-parameter family of time-reversible Hamiltonian vector fields in R^4. The system is reversible with respect to an involution R, in the sense that R∘Φ^t = Φ^{-t}∘R, where Φ^t is the flow. ... An orbit γ is R-symmetric if R(γ)=γ. In this case, R maps each point of γ to another point of γ and reverses the time parametrization. ... We analyze the two-dimensional invariant manifolds W^u(O) and W^s(O) associated with an equilibrium O and their intersections. The reversing symmetry R maps W^u(O) onto W^s(O) and vice versa, and acts on the invariant set formed by homoclinic and heteroclinic trajectories as a conjugacy between the forward and backward dynamics restricted to this set.

In reversible systems with reversing involution R, one has R∘φt = φ−t∘R. Consequently, whenever x(t) is an orbit, the curve R(x(−t)) is also an orbit. The map R defines a homeomorphism between the orbit O(x) and the orbit O(R(x)), conjugating the restricted dynamics but reversing the direction of time along the orbit.

We consider analytic vector fields X on R2 that are reversible with respect to the involution R(u,v) = (u,−v). Reversibility means R∗X = −X∘R, which implies R∘φt = φ−t∘R for the associated flow φt. In particular, if γ(t) is an orbit of X then R(γ(−t)) is also an orbit, and R gives an equivariant conjugacy between the flow restricted to γ and the flow restricted to R(γ), exchanging forward and backward time.

We consider planar vector fields that are orbital-reversible. Following standard terminology, a system is reversible if there exists an involution σ such that σ_* F = −F. We say that system (1.1) is orbital-reversible if there exist an involution σ and a function μ such that σ_*(μF) = − μF, meaning that the orbits are reversed under σ up to a time reparametrization.[1]

A reversible Hamiltonian system is a Hamiltonian vector field XH on a symplectic manifold (M,ω) together with an involution R such that R∗ω = −ω and R∗XH = −XH∘R. The flow φt of XH then satisfies R∘φt = φ−t∘R. Therefore for any point x, its orbit and the orbit of R(x) are mapped to each other by R, providing a conjugacy between the dynamics restricted to these orbits up to time reversal.

Similarly, a discrete-time dynamical system defined by an invertible map T is called R-reversible if it possesses a reversing symmetry R: R ◦ T = T−1 ◦ R. For continuous-time systems, a dynamical system is reversible if it is invariant under the combination (t → −t, z → I(z)) where I is an involution. This implies that if (x(t), x˙(t)) is a solution, then (x(−t), −x˙(−t)) is also a solution. Thus the reversor relates points on the same orbit by mapping forward-time trajectories to backward-time ones, which algebraically yields a conjugacy between the dynamics at x and at R(x) restricted to the reversible orbit.

We consider a reversible Hamiltonian system in R^4 with an involutory reversor R, satisfying R∘Φ^t = Φ^{-t}∘R. ... On the invariant manifolds W^u(O) and W^s(O), the restriction of the flow is intertwined by R in the sense that for any x in W^u(O) we have R(Φ^t(x)) = Φ^{-t}(R(x)). Thus R provides a bijection between forward trajectories in W^u(O) and backward trajectories in W^s(O). ... In particular, for any point x on a reversible orbit contained in W^u(O)∪W^s(O), the points x and R(x) lie on the same orbit with opposite time orientation, and the map R establishes a conjugacy between the dynamics restricted to the orbit considered as a two-sided trajectory.

Time-reversible Hamiltonian vector fields admit a reversing symmetry R satisfying R2 = Id and R∗XH = −XH∘R. As a consequence, if x(t) is a trajectory of XH, then R(x(−t)) is again a trajectory. The restriction of R to the union of such a pair of orbits establishes a topological conjugacy between the forward flow on one orbit and the backward flow on the other.

A dynamical equation system with dynamical variables z is called reversible if it is invariant under the combination (t → −t, z → I(z)) where I is an involution. Main example: Any autonomous Hamiltonian H(q,p) = H(q,−p) that is even in momenta describes a reversible system. Here the involution is I(q,p) = (q,−p). In geometric language, reversibility means there is an involutive map I on the phase space such that the flow Ψ satisfies I ◦ Ψt = Ψ−t ◦ I, so points x and I(x) lie on the same orbit and are intertwined by a time-reversing conjugacy of the flow restricted to that orbit.

For a reversible flow (M,φ_t,R), an orbit is called R-symmetric if it is invariant under the action of R, i.e. R(γ) = γ as a set. If an orbit passes through the origin at time t = 0 and touches the fixed-point set Z of R at time t = t*, it is a periodic orbit of period 4 t* and joins two points of Z which are symmetric with respect to R.[10]

The Hamiltonian systems (2) and (5) are reversible, the reversing involution in both the cases is given by the formula G:(u,φ,x,y,p,q)↦(u,−φ,x,−y,p,−q). ... The reversibility condition can be written as G∘Φ^t = Φ^{-t}∘G, where Φ^t is the Hamiltonian flow. ... Thus, if x lies on an orbit γ of Φ^t, then G(x) lies on the same orbit with reversed time parametrization. The map G defines a one-to-one correspondence between points related by time reversal on γ, and conjugates the forward and backward dynamics restricted to γ.

For a smooth reversible vector field X with reversor R, we have R∘φt = φ−t∘R for the local flow φt generated by X. This relation implies that R maps each orbit of X onto an orbit with reversed orientation. On the level of the orbit, R defines a conjugacy between the restricted dynamics, intertwining φt on one orbit with φ−t on the image orbit.

Hamiltonian systems with time-independent H(q,p) satisfy Liouville’s theorem: the flow ϕt preserves phase-space volume. If the Hamiltonian does not depend on time, trajectories remain on level sets of H, and the flow defines canonical transformations on these invariant sets. In reversible Hamiltonian systems with an involution R that reverses momenta, R maps each trajectory to the same geometrical orbit but traversed in reverse, so x and R(x) are related by a canonical (symplectic or anti-symplectic) transformation along the orbit.

Time reversibility in dynamical systems means that there exists an involutive transformation π such that U_{−t} = π U_t π, where U_t is the evolution operator. In this setting π implements a one-to-one mapping between any forward-time trajectory and a corresponding time-reversed trajectory, but this condition does not by itself require a conjugacy between x and π(x) restricted to a given orbit.[8]

Linear reversible Hamiltonian vector fields are considered, with reversor R satisfying R J = −J R and R^2 = I. ... The flow Φ^t of such a vector field is reversible: R∘Φ^t = Φ^{-t}∘R. ... If γ is a symmetric periodic orbit, meaning that R(γ)=γ and R reverses the direction of time, then for any point x∈γ the point R(x) lies on γ with opposite time orientation. The restriction of Φ^t to γ is thus related by R to the inverse flow, giving a conjugacy between forward and backward dynamics on the orbit.

Hamiltonian flows preserve the symplectic form and therefore define canonical transformations on phase space. The flow maps are solutions to x˙ = f(x) and can be viewed as symplectic diffeomorphisms sending an initial condition x0 to x(t). In systems with time-reversal symmetry given by an involution R, the map R intertwines the Hamiltonian flow with its inverse, effectively giving a conjugacy between the forward dynamics at x and the backward dynamics at R(x) on the same orbit.

Given an asymptotically linear symmetric reversible Hamiltonian system with reversing symmetry R, any brake orbit x(t) has a companion orbit R(x(−t)). The involution R sends the orbit x(t) onto the orbit R(x(−t)), which is dynamically equivalent with opposite time direction. Thus, restricted to the union of these orbits, R acts as a conjugacy between the corresponding dynamical flows.

A Hamiltonian system with N degrees of freedom is described by the Hamiltonian function H(q,p,t). The pair (q_i, p_i) is referred to as canonically conjugate, and the fundamental Poisson bracket equations are preserved under canonical transformations. Reversible Hamiltonian flows with an involution that reverses momenta generate Poincaré return maps that are symplectic (or volume-preserving) and relate points on the same energy surface, including x and its time-reversed image R(x), by symplectic conjugacies on invariant sets such as periodic orbits.

For a reversible flow φt with reversing involution R, the identity R∘φt = φ−t∘R holds. Each orbit either is mapped onto itself with reversed orientation (a symmetric orbit) or is paired with a distinct orbit under R. On a symmetric orbit, the restriction of R provides an involutive conjugacy between forward and backward time dynamics along that orbit.

We are interested in the Hamiltonian flows associated to these functions. Recall that if (X,ω) is a symplectic manifold and f : X → R is a smooth function, the Hamiltonian flow of f preserves ω. Such flows descend to quotient spaces and can cover Hamiltonian flows on moduli spaces, relating different points by symplectic equivalences. In the presence of additional symmetries such as involutions, flow lines can be paired so that points x and R(x) on symmetric orbits are related through induced conjugacies on the quotient dynamics.

In a reversible dynamical system with reversor R, any trajectory x(t) gives rise to the trajectory R(x(−t)), which traverses the same invariant sets in reverse. However, the generic structure is that R maps each orbit to another orbit (possibly the same setwise in the symmetric case), providing a bijection between orbits or between different segments of the same orbit, rather than a canonical topological conjugacy between the dynamics based at x and at R(x) along that orbit.[6]

In port-Hamiltonian system theory, the structure matrix J(x), the dissipation structure R(x) and Hamiltonian H determine the flow variables and effort variables. The geometric framework emphasizes how symmetries and involutive structures can define dual flows that are adjoint or conjugate to each other. For reversible Hamiltonian-like systems with an involution acting on state variables, this structure can induce conjugate dynamics between x and its image under the involution along the same energy flow.

We divide the contact phase space into three parts, which are corresponding to three differential invariant sets Ω_±, Ω_0. On the invariant sets Ω_±, under some geometric conditions, the contact Hamiltonian system is equivalent to a Hamiltonian system via a Hölder transformation. ... On such invariant sets, the flow admits time-reversal properties analogous to reversible Hamiltonian systems: if R is a reversor, then R∘Φ^t = Φ^{-t}∘R on the invariant set, so that R acts as a conjugacy between the restriction of Φ^t and its inverse on that set.