Abstract
We develop several formal analogies between the logistic equation and the spatially homogeneous and isotropic relativistic cosmology described by the Einstein–Friedmann equations. These analogies produce an effective Lagrangian and Hamiltonian and new symmetries for the logistic equation.
Highlights
The logistic equation is f(t) = r f (t)[1 − f (t)], (1)where r is a positive constant and an overdot denotes differentiation with respect to t
The evolution of the cosmic scale factor a(t) and energy density ρ(t) is ruled by the Einstein equations, which are greatly simplified by the high degree of symmetry of FLRW cosmology and reduce to a set of three ordinary differential equations, the Einstein–Friedmann equations
The third symmetry is enjoyed by an equation of the form (4) with a single perfect fluid or with a single fluid plus cosmological constant [33]
Summary
Where r is a positive constant and an overdot denotes differentiation with respect to t. There are two analogs, corresponding to the two different time coordinates widely used in cosmology These analogies generate an effective Lagrangian and Hamiltonian for the logistic equation or, in other words, solve its inverse Lagrangian problem. We develop the formal analogies between logistic equation and Friedmann equation, written using comoving time or conformal time, respectively. In both cases, we write down the corresponding Lagrangian and Hamiltonian for the logistic equation, and examine whether the symmetries of the Einstein–Friedmann equations generate new symmetries of the logistic equation.
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