Energy translation symmetries and dynamics of separable autonomous two-dimensional ODEs

We study symmetries in the phase plane for separable, autonomous two-state systems of ordinary differential equations (ODEs). We prove two main theoretical results concerning the existence and non-triviality of two orthogonal symmetries for such systems. In particular, we show that these symmetries correspond to translations in the internal energy of the system, and describe their action on solution trajectories in the phase plane. In addition, we apply recent results establishing how phase plane symmetries can be extended to incorporate temporal dynamics to these energy translation symmetries. Subsequently, we apply our theoretical results to the analysis of three models from the field of mathematical biology: a canonical biological oscillator model, the Lotka--Volterra (LV) model describing predator-prey dynamics, and the SIR model describing the spread of a disease in a population. We describe the energy translation symmetries in detail, including their action on biological observables of the models, derive analytic expressions for the extensions to the time domain, and discuss their action on solution trajectories.


Introduction
In the most straightforward case, minimal biological models are analysed by means of a linear stability analysis in the phase plane [1]. Typically, this approach entails analysing a time-invariant system of two first order ODEs in the phase plane, which is the plane spanned by the two states of the model, to provide qualitative information about the long-term dynamics of the system. However, such analysis does not provide quantitative insight into the relationship between different solution trajectories of the same model and it cannot identify common structural properties of the solution trajectories. Even in the simplest case of separable models in the phase plane which are directly solvable, albeit usually implicitly, these questions about the properties of the model, including the relationship between different solution trajectories, cannot be answered by a linear stability analysis. Symmetry methods, however, provide a versatile and generalisable set of mathematical tools for answering these types of questions. They have been used with huge success in theoretical physics but are not yet widely used in mathematical biology.
For coupled systems of two or more first order ODEs there are few systematic methods for finding Lie symmetries. For this type of systems, the linearised symmetry conditions that must be solved in order to find the symmetries are generally undetermined, making the unknown functions defining the generators of symmetries, referred to as the infinitesimals [2], difficult to find. Typically, the equations must therefore be solved using ansätze for the infinitesimals but in general the structure of the infinitesimals is unknown rendering the task of designing the ansätze challenging.
For a single first order ODE, Cheb-Terrab and Kolokolnikov [3] have designed a set of ansätze for the infinitesimals that are capable of finding symmetries for a large class of models. Moreover, certain systems of first order ODEs can be formulated as a single higher order ODE [4], for which the linearised symmetry condition decomposes and can be systematically solved for the infinitesimals. Examples where the latter strategy is applicable are the Lorenz model [5] and various models describing disease transmission in epidemiology [6]. Nonetheless, many systems of first order ODEs cannot be cast as a single higher order ODE and accordingly symmetries of these systems cannot be determined using this methodology.
Recently, it was shown by Ohlsson et al. [7] that it is possible to extend the symmetries of the single phase plane ODE corresponding to a system of two autonomous ODEs to symmetries of the time-dependent system by solving the so-called lifting condition. If the symmetry condition for the phase plane ODE is easier to solve than those of the original system, this connection provides an alternative approach for determining time domain symmetries of autonomous two-dimensional systems.
However, finding symmetries of a dynamical system is only half the battle if the goal is to improve our understanding of it. In order to provide meaningful information about the structure of the system and a powerful way to represent its dynamics, the ability to interpret the action of the symmetries is also paramount.
In this paper, we address the challenges of finding symmetries and using them to derive insight into the dynamics of the particular class of models where the phase plane ODE is separable. Specifically, we present two main theoretical results establishing the existence of two non-trivial symmetry generators of any separable phase plane ODE (Theorem 1) and their interpretations in terms of translations of the internal energy of phase plane trajectories (Theorem 2). Using the methodology developed in [7] we also provide the explicit form of the lifting condition for these symmetries. The symmetries are connected to the Hamiltonian structure of the dynamical system through their action on the space of solutions; raising or lowering the energy of a trajectory.
We exemplify the analysis based on our theoretical developments using three biological models: a canonical biological oscillator, the Lotka-Volterra (LV) model describing predatorprey dynamics, and the SIR model describing the spread of a disease in a population. We construct the non-trivial energy translation symmetries and provide analytic solutions for the corresponding lifting conditions, extending them to incorporate the temporal dynamics of the models. Furthermore, we discuss the action of the symmetries on biologically meaningful observables and the structure of the space of solutions.
2 Orthogonal phase plane symmetries of separable models that act as translations in the internal energy Consider the autonomous two-state model and its corresponding (u, v) phase plane representation given by For separable ODEs the reaction terms are of the form where f u , f v , g u , g v , h are continuous functions, and for f u (u) ̸ = 0 , g v (v) ̸ = 0 the phase plane ODE in Eq. (2.2) can be directly integrated to give The arbitrary integration constant H parametrises the space of solutions of Eq. (2.2) and is often interpreted as the internal energy of a trajectory since it is a first integral of Eq. (2.1) and therefore conserved under time evolution of the original system.

Symmetries in the time domain
Consider the space M 3 parametrised by (t, u, v), which we will refer to as the time domain, and let Γ 3,ϵ : M 3 → M 3 be a family of Lie point transformations parametrised by ϵ, and generated by the vector field [2,8]. A transformation Γ 3,ϵ constitutes a symmetry of the system in Eq. (2.1) if X satisfies the linearised symmetry conditions [2,8,9,10]: where the first prolongation X (1) of the generator is given by 6) and the prolonged infinitesimals η are calculated using the total derivative D t = ∂ t +u∂ u +v∂ v according to The symmetry Γ 3,ϵ is non-trivial if it acts non-trivially on the space of solutions to Eq. (2.1).

Symmetries in
Then, Γ 2,ϵ is a symmetry of the phase plane ODE in Eq. (2.2) if it satisfies the linearised symmetry condition where the the first prolongation of the infinitesimal phase plane generator is given by and is the prolonged infinitesimal defined by the total derivative phase plane. As in the time domain, a symmetry transformation Γ 2,ϵ is non-trivial if it acts non-trivially on the space of solutions to Eq. (2.2).
The action of a phase plane symmetry generated by Y is made manifest by introducing the canonical coordinates [8] (s, r) in the phase plane, defined by Y s = 1 and Y r = 0, in terms of which the transformation Γ 2,ϵ becomes Γ 2,ϵ : (s, r) → (s + ϵ, r) . (2.11) The two canonical coordinates (s, r) can be interpreted as properties of the model in Eq. (2.2) that are changed and conserved, respectively, under the action of the symmetry. In fact, r is an example of a differential invariant of the generator Y [2,8].
Recently, it was shown by Ohlsson et al. [7] that it is possible to lift an infinitesimal generator Y of a symmetry in the phase plane to an infinitesimal generator X of a corresponding symmetry Γ 3,ϵ in the time domain. More precisely, the infinitesimal generator is

Two non-trivial phase plane symmetries of separable models
Our main theoretical result, established through the following two theorems, is the construction of two orthogonal, non-trivial symmetries for any separable model in the (u, v) phase plane.
Theorem 1 (Orthogonal phase plane symmetries of separable ODEs). Let the func-

3). Then the vector fields
generate two orthogonal symmetries of the phase plane ODE in Eq. (2.3).
Proof. The linearised symmetry condition in Eq. (2.8) is equivalent to 15), and the generators Y u and Y v are clearly orthogonal.
It is well-known that separable first order ODEs have a unidirectional symmetry in the independent variable [3,10]. However, we are free to choose how to parameterise the phase plane by considering either u or v as the independent variable, and the symmetries of the phase plane ODE are independent of this choice [7]. Based on this observation, Theorem 1 extends the results of [3,10] to include a symmetry generator Y v in the dependent coordinate direction.
Theorem 2 (Non-triviality of phase plane symmetries of separable ODEs). The symmetry transformations Γ u 2,ϵ and Γ v 2,ϵ generated by Y u in Eq. (2.13) and Y v in Eq. (2.14), respectively, are non-trivial and act on the internal energy H in Eq. (2.4) according to Proof. The canonical coordinates of Γ u 2,ϵ and Γ v 2,ϵ , respectively, are given by Consequently, the internal energy H in Eq. (2.4) of any phase plane trajectory can be expressed as For separable models and the non-trivial symmetry generators in Eqs. (2.13) and (2.14) the lifting condition in Eq. (2.12) generally simplifies. In particular, for the case h(u, v) = 1 which is common in applications, the lifting condition reduces to .

Application to biological models
We will now use the phase plane symmetries discussed in the previous section to analyse three separable models in biology: a canonical oscillator model [1], the Lotka-Volterra (LV) model [11,12,13], and the epidemiological SIR model [14].

A canonical oscillator model
Let u(t) and v(t) be the states at time t of the canonical oscillator model whose dynamics is where λ is a positive constant describing the angular frequency of the oscillator. In terms of the polar coordinates (σ, θ) defined by the model becomes separable and the corresponding (r, θ) phase plane ODE can be solved to produce the internal energy H Osc = ln σ − ln |1 − σ| − θ/λ.
The two orthogonal phase plane symmetries in Eq. (2.16) are generated by : (θ, σ) → θ, (3.7) All solutions of the oscillator model have the same frequency and are related by a constant shift in the angular coordinate θ, or equivalently a translation in time, implying that the only qualitative distinction between solutions is whether they lie inside (σ < 1) or outside (σ > 1) the limit cycle (see [7] for an in-depth discussion). Consequently, the action of the symmetries Γ Osc,θ 2,ϵ and Γ Osc,σ 2,ϵ is a rotation of the phase plane trajectory, or equivalently, a reparametrisation of the phase plane.
This fact is also reflected in the time domain symmetries obtained by solving the lifting condition in Eq. (2.12)

The Lotka-Volterra model
The dimensionless LV model [11,12,13] is given by where u(t) and v(t) correspond, respectively, to the number of prey and predators in the population at time t, and α is a rate parameter. The corresponding (u, v) phase plane ODE is directly solvable, yielding the internal energy 1 The two phase plane symmetry generators in Eqs. (2.13) and (2.14) are given by where W is the Lambert W function (see [15,16] where W 0 and W −1 are the real branches of the Lambert W function. Consequently, the action in Eq. (2.16) provides a direct way to interpret the action of Γ LV u and Γ LV v in terms of biologically meaningful quantities.
Lifting the phase plane symmetry generators of the LV model to the time domain yields

The SIR model
The original SIR model [14] describes the spread of a disease in a population and can be reduced to a dimensionless model for the sub-populations of susceptible, S(t), and infected, where ρ is a recovery rate parameter. The corresponding (S, I) phase plane ODE The two phase plane symmetry generators in Eqs. (2.13) and (2.14) are given by The solution trajectory in Eq. (3.20) has a maximum at S = ρ given by which implies that the symmetries generated by Y S and Y I can be interpreted as, respectively, decreasing and increasing the maximum number of infected individuals. In addition, the maximum and minimum of susceptible individuals is obtained from the intersection with the stable locus I = 0 as Similarly to the case for the LV model, these expressions relate the translation in internal energy H SIR to quantities of direct biological relevance.
Solving the lifting condition for Y SIR S and Y SIR I , we obtain the following vector fields as generators of symmetries in the time domain group of systems of first order ODEs is infinite (see [17] for a discussion on this). The challenge, then, is to find informative symmetries in the absence of a general strategy for solving the underdetermined linearised symmetry conditions. In the case of autonomous models, it is always possible to reformulate the original system consisting of, say, n equations as a system of n − 1 equations by considering one of the states as the independent variable (see the discussion of complete symmetry groups of second order systems in [18] The main limitation of the methodology in the present work is that the scope is restricted to autonomous models with separable reaction terms. Autonomy gives rise to a closed-form phase plane representation. Separability then allows us to immediately integrate the phase plane equation, and obtain two symmetries that can be readily understood in terms of altering the internal energy of the system. However, there are numerous examples, with applications in both physics and biology, of systems that are not immediately separable. In such cases, we must first find phase plane symmetries by solving the linearised symmetry condition for the phase plane ODE in Eq. (2.8). As discussed previously, solving this linearised symmetry condition remains a challenging problem in general as the problem is ill-posed.
In light of this difficulty, an interesting extension of this work would be to attempt to solve the linearised symmetry condition for the case of non-separable reaction terms.
The most straightforward approach is to design ansätze for the unknown infinitesimals ζ u and ζ v solving the linearised symmetry condition in Eq. (2.8). A set of ansätze for these infinitesimals for a large class of single first order ODEs has been previously designed [3], which serves as a natural staring point for this endeavour. Given a solution of the linearised symmetry condition for the phase plane ODE, the analysis presented in this work can be readily extended to the case of non-separable reaction terms. Moreover, the Hamiltonian structure of, e.g., the LV model has been considered before [19]. In future work, we intend to extend such an analysis to general separable systems based on the results presented in the present paper. In addition, an interesting avenue of investigation is to consider the algebraic structure of the space of solutions endowed by the symmetry generators X u and X v in Eqs. (2.13) and (2.14).
In order to show how the theoretical results developed in Section 2 can be applied to analyse the structure and dynamics of separable models, we consider three examples in detail.
In particular, for the LV and SIR models we show how the orthogonal energy translation From a biological modelling perspective, we would finally like to emphasise the benefit of the ability, afforded by the structure of the space of solutions of separable models, to conduct a complete analysis of the separable models in terms of the internal energy and interpret the corresponding translational symmetries and their action on biologically meaningful quantities. In contrast, any symmetry of the model may be used to, e.g., compute exact solutions but the action on solutions will generally be biophysically obscure.