5 research outputs found
On the correspondence between symmetries of two-dimensional autonomous dynamical systems and their phase plane realisations
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.Comment: 22 pages, 7 figure
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.Comment: 18 pages, 3 figure
Turing pattern formation on the sphere is robust to the removal of a hole
The formation of buds on the cell membrane of budding yeast cells is thought to be driven by reactions and diffusion involving the protein Cdc42. These processes can be described by a coupled system of partial differential equations known as the Schnakenberg system. The Schnakenberg system is known to exhibit diffusion-driven pattern formation, thus providing a mechanism for bud formation. However, it is not known how the accumulation of bud scars on the cell membrane affect the ability of the Schnakenberg system to form patterns. We have approached this problem by modelling a bud scar on the cell membrane with a hole on the sphere. We have studied how the spectrum of the Laplace–Beltrami operator, which determines the resulting pattern, is affected by the size of the hole, and by numerically solving the Schnakenberg system on a sphere with a hole using the finite element method. Both theoretical predictions and numerical solutions show that pattern formation is robust to the introduction of a bud scar of considerable size, which lends credence to the hypothesis that bud formation is driven by diffusion-driven instability
On the correspondence between symmetries of two-dimensional autonomous dynamical systems and their phase plane realisations
We consider the relationship between symmetries of two-dimensional autonomous dynamical systems 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 their symmetries, and we establish the correspondence between the set of infinitesimal generators of the respective formulations. We 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 system, which is unique up to constant translations in time. We exemplify the lift of symmetries in two cases; a mass conserved linear model and a non-linear oscillator
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