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On the correspondence between symmetries of two-dimensional autonomous dynamical systems and their phase plane realisations

Fredrik Ohlsson, Johannes G. Borgqvist, Ruth E. Baker

Abstract

We consider the relationship between symmetries of two-dimensional autonomous dynamical system in two common formulations; as a set of differential equations for the derivative of each state with respect to time, and a single differential equation in the phase plane representing the dynamics restricted to the state space of the system. Both representations can be analysed with respect to the symmetries of their governing differential equations, and we establish the correspondence between the set of infinitesimal generators of the respective formulations. Our main result is to show that every generator of a symmetry of the autonomous system induces a well-defined vector field generating a symmetry in the phase plane and, conversely, that every symmetry generator in the phase plane can be lifted to a generator of a symmetry of the original autonomous system, which is unique up to constant translations in time. The process of lifting requires the solution of a linear partial differential equation, which we refer to as the lifting condition. We discuss in detail the solution of this equation in general, and exemplify the lift of symmetries in two commonly occurring examples; a mass conserved linear model and a non-linear oscillator model.

On the correspondence between symmetries of two-dimensional autonomous dynamical systems and their phase plane realisations

Abstract

We consider the relationship between symmetries of two-dimensional autonomous dynamical system in two common formulations; as a set of differential equations for the derivative of each state with respect to time, and a single differential equation in the phase plane representing the dynamics restricted to the state space of the system. Both representations can be analysed with respect to the symmetries of their governing differential equations, and we establish the correspondence between the set of infinitesimal generators of the respective formulations. Our main result is to show that every generator of a symmetry of the autonomous system induces a well-defined vector field generating a symmetry in the phase plane and, conversely, that every symmetry generator in the phase plane can be lifted to a generator of a symmetry of the original autonomous system, which is unique up to constant translations in time. The process of lifting requires the solution of a linear partial differential equation, which we refer to as the lifting condition. We discuss in detail the solution of this equation in general, and exemplify the lift of symmetries in two commonly occurring examples; a mass conserved linear model and a non-linear oscillator model.
Paper Structure (14 sections, 3 theorems, 58 equations, 7 figures)

This paper contains 14 sections, 3 theorems, 58 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Geometric framework for symmetries
  3. Symmetries in the time domain
  4. Symmetries in the phase plane
  5. Reduction to the phase plane
  6. Lifting to the time domain
  7. Lifting the generator
  8. Solving the lifting condition
  9. Examples of lifting phase plane symmetries
  10. Mass-conserved linear model
  11. A non-linear oscillator model
  12. Discussion
  13. Acknowledgements
  14. Author contributions

Key Result

Theorem 1

The push-forward $f_*\left( X^{(1)} \right)$ of the prolonged vector field $X^{(1)}$ in Eq. eqn:prolonged_vector_J5 by $f: J_5 \to J_3$ in Eq. eqn:reduction_map is a vector field on $J_3$, which coincides with the prolongation $\left( f_*X \right)^{(1)}$ of the push-forward of the vector field $X$ i

Figures (7)

  • Figure 1: The dynamics of the mass-conserved linear model. Multiple solutions of the mass-conserved linear model are illustrated in (A) the $(u,v)$ phase plane and (B) the time domain.
  • Figure 2: Action of the phase plane symmetries for the mass conserved model. Dashed arrows represent (A) the scaling symmetry $\Gamma_2^S$ generated by $Y_S$ and (B) the generalised rotation symmetry $\Gamma_2^G$ generated by $Y_G$.
  • Figure 3: Action of lifted phase plane symmetries for the mass-conserved linear model. Top row: Original solution curves $\left(u(t),v(t)\right)$ and transformed solution curves $\left( \hat{u}(t),\hat{v}(t) \right) = \Gamma_3^S\left(u(t),v(t)\right)$ generated by $\hat{Y}_S$ in Eq. \ref{['eqn:linear_model_scaling_X']} for $F(x)=0$ (A) and $F(x)=x$ (B). Bottom row: Original solution curves $\left(u(t),v(t)\right)$ and transformed solution curves $\left( \hat{u}(t),\hat{v}(t) \right) = \Gamma_3^G\left(u(t),v(t)\right)$ generated by $\hat{Y}_G$ in Eq. \ref{['eqn:linear_model_genrot_X']} for $F(x)=0$ (C) and $F(x)=x$ (D). Dashed arrows represent the transformations $\Gamma_3^S$ and $\Gamma_3^G$, respectively.
  • Figure 4: The dynamics of the non-linear oscillator model. Multiple solutions of the non-linear oscillator model are illustrated for $r>1$ in (A) the $(u,v)$ phase plane and (B) the time domain, and for $r<1$ in (C) the $(u,v)$ phase plane and (D) the time domain.
  • Figure 5: Action of the phase plane symmetry for the non-linear oscillator model. Dashed arrows represent the rotation symmetry $\Gamma_2^R$ acting on oscillating trajectories for (A) $r>1$ and (B) $r<1$.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 1
  • proof
  • Remark
  • Theorem 2
  • proof
  • Remark
  • Theorem 3
  • proof
  • Remark
  • Remark