Chapter 10
Useful Transformations from
Non-autonomous to Autonomous Systems
Alona Ben-Tal
Abstract Every non-autonomous vector field can be viewed as autonomous by treating the time as another dependent variable and augmenting the dynamical system with
an additional differential equation. This standard transformation of non-autonomous
vector fields to autonomous is, arguably, of little use in that common techniques for
the study of autonomous systems, such as looking for stationary solutions, cannot be
applied. This chapter presents alternative ways of transforming a non-autonomous
system to autonomous in the special case of periodic forcing, but it also makes the
argument that this could be done in other special cases. The argument is inspired
by mathematical modelling. Two examples of alternative transformations are given,
one in a system of ordinary differential equations and one in a Boolean network.
10.1 Introduction
Analysis of autonomous systems of ordinary differential equations, ˙
x = f(x), where
x ∈ R
n , ˙
x is the derivative of x with respect to time and f(x) is a vector of functions, usually starts by finding stationary solutions and their stabilities [13, 25]. The
stationary solutions, x
∗ , are solutions for which x(t) = x
∗ for all time t. In other
words, they satisfy the equation f(x
∗
) = 0. The stability of each stationary solution
can often be found by calculating the eigenvalues of the (n × n) Jacobian
∂ f i
∂ x j
x=x ∗
where i, j ∈ {1, . . . , n}. As an example, consider the simple, damped pendulum with
mass m and length L (see Fig. 10.1a). The rates of change of the angle, θ , and the
angular velocity, z are given by:
A. Ben-Tal (B)
School of Natural and Computational Sciences, Massey University,
Private Bag 102904, 0745 North Shore, Auckland, New Zealand
e-mail: a.ben-tal@massey.ac.nz
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_10
163
Useful Transformations from
Non-autonomous to Autonomous Systems
Alona Ben-Tal
Abstract Every non-autonomous vector field can be viewed as autonomous by treating the time as another dependent variable and augmenting the dynamical system with
an additional differential equation. This standard transformation of non-autonomous
vector fields to autonomous is, arguably, of little use in that common techniques for
the study of autonomous systems, such as looking for stationary solutions, cannot be
applied. This chapter presents alternative ways of transforming a non-autonomous
system to autonomous in the special case of periodic forcing, but it also makes the
argument that this could be done in other special cases. The argument is inspired
by mathematical modelling. Two examples of alternative transformations are given,
one in a system of ordinary differential equations and one in a Boolean network.
10.1 Introduction
Analysis of autonomous systems of ordinary differential equations, ˙
x = f(x), where
x ∈ R
n , ˙
x is the derivative of x with respect to time and f(x) is a vector of functions, usually starts by finding stationary solutions and their stabilities [13, 25]. The
stationary solutions, x
∗ , are solutions for which x(t) = x
∗ for all time t. In other
words, they satisfy the equation f(x
∗
) = 0. The stability of each stationary solution
can often be found by calculating the eigenvalues of the (n × n) Jacobian
∂ f i
∂ x j
x=x ∗
where i, j ∈ {1, . . . , n}. As an example, consider the simple, damped pendulum with
mass m and length L (see Fig. 10.1a). The rates of change of the angle, θ , and the
angular velocity, z are given by:
A. Ben-Tal (B)
School of Natural and Computational Sciences, Massey University,
Private Bag 102904, 0745 North Shore, Auckland, New Zealand
e-mail: a.ben-tal@massey.ac.nz
© Springer Nature Switzerland AG 2021
A. Stefanovska and P. V. E. McClintock (eds.), Physics of Biological
Oscillators, Understanding Complex Systems,
https://doi.org/10.1007/978-3-030-59805-1_10
163