Non-autonomous mathematical formulations differ from autonomous ones in that their governing equations explicitly depend on time or another external variable. In natural systems, certain behaviors or processes are better modeled with non-autonomous formulations because they are influenced by external, time-dependent factors. Some examples of natural behaviors that qualify as non-autonomous include:

1. Seasonal Climate Variation:
Climate patterns, such as temperature changes or monsoon cycles, are influenced by external factors like the Earth’s orbit, axial tilt, and solar radiation, all of which vary over time. These changes make non-autonomous systems suitable for modeling long-term climate behavior.

2. Tidal Forces:
Tidal movements are driven by the gravitational pull of the Moon and the Sun, which vary as the positions of these celestial bodies change relative to Earth. Tidal equations thus have time-dependent forcing terms, making them non-autonomous.

3. Biological Rhythms:
Circadian rhythms in living organisms, which regulate daily cycles such as sleep and feeding, are influenced by the 24-hour light-dark cycle. These external light variations necessitate non-autonomous models.

4. Astronomical Geophysical Cycles:
Systems like the Chandler wobble (the irregular movement of Earth’s rotation axis) or the Quasi-Biennial Oscillation (QBO) in the equatorial stratosphere are influenced by periodic external factors, such as lunar cycles, making them non-autonomous. This also includes systems where lunar or Draconic cycles interact with annual cycles in non-linear ways, as explored in studies of Earth’s rotational dynamics.

5. Oceanographic Dynamical Phenomena:
Non-autonomous formulations are needed to model phenomena such as El Niño, which is influenced by complex interactions between atmospheric and oceanic conditions, themselves driven by seasonal and longer-term climatic variations.

6. Planetary Motion in a Varying Gravitational Field:
In astrophysical systems where a planet moves in the gravitational field of other bodies, such as a multi-body problem where external forces vary in time, non-autonomous dynamics become essential to account for these influences.

In contrast, autonomous systems are self-contained and their behavior depends only on their internal state variables, independent of any external time-varying influence. So that non-autonomous systems often better capture the complexity and variability introduced by time-dependent external factors.

However, many still want to find connections to autonomous formulations as they often coincide with resonant conditions or some natural damping rate.

Autonomous mathematical formulations are characterized by the fact that their governing equations do not explicitly depend on time or other external variables (they can be implicit via time derivatives though). These systems evolve based solely on their internal state variables. Many natural behaviors can be modeled using autonomous systems when external influences are either negligible or can be ignored. Here are some examples of natural behaviors that qualify as autonomous:

1. Radioactive Decay:
The decay of radioactive isotopes is governed by an internal process where the rate of decay depends only on the amount of the substance present at a given moment. The decay equation does not depend on time explicitly, making it an autonomous system.

2. Epidemiological Models (without external intervention):
Simplified models of disease spread, such as the SIR (Susceptible-Infected-Recovered) model, can be autonomous if no external factors (like seasonal effects or interventions) are considered. The evolution of the system depends only on the current number of susceptible, infected, and recovered individuals.

3. Predator-Prey Dynamics (Lotka-Volterra Model):
In the absence of external influences like seasonal changes or human intervention, predator-prey relationships, such as those described by the Lotka-Volterra equations, can be modeled as autonomous systems. The population changes depend solely on the interaction between predators and prey.

4. Chemical Reactions (closed systems):
In a closed system with no external input or removal of substances, the kinetics of chemical reactions can be modeled as autonomous. The rate of reaction depends only on the concentrations of reactants and products at any given time.

5. Newtonian Mechanics of Isolated Systems:
For an isolated mechanical system (e.g., a simple pendulum or two-body orbital system), the equations of motion can be autonomous. The system evolves based solely on the internal energy and forces within the system, without any external time-dependent influences. This relates to general oscillatory systems or harmonic oscillators — the simple harmonic oscillator (such as a mass on a spring) can be modeled autonomously if no external time-varying forces are acting on the system. The system’s behavior depends only on its position and velocity at any point in time. In the classical gravitational two-body problem in celestial mechanics, where two bodies interact only through their mutual gravitational attraction, the motion can be described autonomously. The positions and velocities of the two bodies determine their future motion, independent of any external time-dependent factors.

6. Thermodynamics of Isolated Systems:
In an isolated thermodynamic system, where there is no exchange of energy or matter with the surroundings, the internal state (e.g., pressure, temperature, volume) evolves autonomously based on the system’s internal conditions.

These examples illustrate systems where internal dynamics govern the evolution of the system, and time or external influences do not explicitly appear in the equations. However, in many real-world cases, external factors often come into play, making non-autonomous formulations more appropriate for capturing the full complexity of natural behaviors. A pendulum that is periodically synchronized as for example a child pushed on a swing set,  may be either formulated as a forced response in an autonomous set of equations or a non-autonomous description if the swing pusher carefully guides the cycle.

This is where the distinctions between autonomous vs non-autonomous and forced vs natural responses should be elaborated.

Understanding the Structure of the General Solution

In the case of a forced linear second-order dynamical system, the general solution to the system is typically the sum of two components:

Homogeneous (natural) solution: This is the solution to the system when there is no external forcing (i.e., the forcing term is zero).

Particular solution: This is the solution driven by the external forcing.

The homogeneous solution depends only on the internal properties of the system (such as natural frequency, damping, etc.) and is the solution when F(t) = 0.

The particular solution is directly related to the forcing function F(t), which can be time-dependent in the case of a non-autonomous system.

So let’s  consider the autonomous vs non-autonomous context.

Autonomous System: In an autonomous system, even though the system is subject to forcing, the forcing term does not explicitly depend on time but rather on internal state variables (such as x or dx/dt). Here, the particular solution would also be state-dependent and would not explicitly involve time as an independent variable.

Non-Autonomous System: In a non-autonomous system, the forcing term explicitly depends on time, such as F(t) = A sin(w t). This external time-dependent forcing drives the particular solution. While the homogeneous solution remains autonomous (since it’s based on the system’s internal properties), the particular solution reflects the non-autonomous nature of the system.

The key insight is that of the non-autonomous particular solution. Even though a system’s response can have components from the homogeneous solution (which are autonomous in nature), the particular solution in a non-autonomous system will be time-dependent and follow the time-dependence of the external forcing.

So consider the transition from autonomous to non-autonomous: when you introduce a periodic forcing function F(t), the particular solution becomes non-autonomous, even though the overall system response still includes the autonomous homogeneous solution. This results in the system being classified as non-autonomous, as the particular solution carries the time-dependent behavior, despite the autonomous structure of the homogeneous solution.

Summary: A forced response in a linear second-order system can include both autonomous and non-autonomous components. Even though the homogeneous solution remains autonomous, the particular solution introduces non-autonomous characteristics when the forcing term depends explicitly on time. In non-autonomous systems, the forcing introduces time dependence in the particular solution, making the overall system non-autonomous, even though part of the response (the homogeneous solution) is autonomous.