10 Useful Transformations from Non-autonomous to Autonomous Systems
165
system of ordinary differential equations could not (assuming continuity and smoothness of f(x)) [15]. Indeed, the forced pendulum exhibits complicated motions (see
for example, [9, 10, 17, 26]). Arguably, this is where the usefulness of this transformation ends. The most basic analysis that can be done in native autonomous systems
(i.e. finding stationary solutions) cannot be done in the transformed system—there is
no solution to the equation f(y) = 0. One could argue that the absence of stationary
solutions in non-autonomous systems is not surprising - solutions of non-autonomous
systems are expected to be time-dependent. It could also be argued that techniques
used to study time-dependent solutions of autonomous systems, could be used to
study solutions of non-autonomous systems (for example, Poincaré map [16, 25],
averaging and perturbation methods [15–17]). Nevertheless, we show in Sect. 10.3
that in special cases, there exist other transformations from non-autonomous systems to autonomous that are more useful than the standard transformation in that
they allow analysis of the transformed system using theoretical results derived for
native autonomous systems. Our approach is inspired by mathematical modelling
and we begin by looking at a specific example: modelling the cardio-respiratory
system.
10.2 Appearance of Oscillations in Mathematical
Modelling of the Cardio-Respiratory System
The main function of the cardio-respiratory system is to ensure an adequate delivery
of oxygen to every cell in the body and the removal of carbon dioxide [11]. Over
a short period of time (several minutes), this is achieved by adjusting the breathing
pattern and the heart rate via a neural network located in the brainstem [12, 14, 21].
Assuming the external environment is large enough and insulated, conditions outside the body practically stay constant over a short period of time (e.g. temperature,
concentrations of oxygen and carbon dioxide). Hence, we can assume that there are
no external sources of time-dependent signals. Under these conditions, the cardiorespiratory system (and indeed the whole body) can be viewed as an autonomous
system. However, due to its complexity, parts of the system are often modeled separately. This leads to two sources of oscillations: intrinsic and forced.
10.2.1 Intrinsic Oscillations
Intrinsic oscillations can be found in mathematical models of the respiratory neural
system [7, 8, 18, 20, 23]. As an example, consider a model of a single neuron in
an area of the brainstem called the pre Bötzinger Complex (preBötC) which drives
breathing [7]:
165
system of ordinary differential equations could not (assuming continuity and smoothness of f(x)) [15]. Indeed, the forced pendulum exhibits complicated motions (see
for example, [9, 10, 17, 26]). Arguably, this is where the usefulness of this transformation ends. The most basic analysis that can be done in native autonomous systems
(i.e. finding stationary solutions) cannot be done in the transformed system—there is
no solution to the equation f(y) = 0. One could argue that the absence of stationary
solutions in non-autonomous systems is not surprising - solutions of non-autonomous
systems are expected to be time-dependent. It could also be argued that techniques
used to study time-dependent solutions of autonomous systems, could be used to
study solutions of non-autonomous systems (for example, Poincaré map [16, 25],
averaging and perturbation methods [15–17]). Nevertheless, we show in Sect. 10.3
that in special cases, there exist other transformations from non-autonomous systems to autonomous that are more useful than the standard transformation in that
they allow analysis of the transformed system using theoretical results derived for
native autonomous systems. Our approach is inspired by mathematical modelling
and we begin by looking at a specific example: modelling the cardio-respiratory
system.
10.2 Appearance of Oscillations in Mathematical
Modelling of the Cardio-Respiratory System
The main function of the cardio-respiratory system is to ensure an adequate delivery
of oxygen to every cell in the body and the removal of carbon dioxide [11]. Over
a short period of time (several minutes), this is achieved by adjusting the breathing
pattern and the heart rate via a neural network located in the brainstem [12, 14, 21].
Assuming the external environment is large enough and insulated, conditions outside the body practically stay constant over a short period of time (e.g. temperature,
concentrations of oxygen and carbon dioxide). Hence, we can assume that there are
no external sources of time-dependent signals. Under these conditions, the cardiorespiratory system (and indeed the whole body) can be viewed as an autonomous
system. However, due to its complexity, parts of the system are often modeled separately. This leads to two sources of oscillations: intrinsic and forced.
10.2.1 Intrinsic Oscillations
Intrinsic oscillations can be found in mathematical models of the respiratory neural
system [7, 8, 18, 20, 23]. As an example, consider a model of a single neuron in
an area of the brainstem called the pre Bötzinger Complex (preBötC) which drives
breathing [7]: