ChatGPT

Mathematical GeoEnergy 2018 vs ChatGPT 2025

/ Paul Pukite (@whut) / Leave a comment

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Paul Pukite (@whut) says

1 JUL 2025 AT 9:48 PM

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“If so, do you have an explanation why the diurnal tides do not move the thermocline, whereas tides with longer periods do?”

The character of ENSO is that it shifts by varying amounts on an annual basis. Like any thermocline interface, it reaches the greatest metastability at a specific time of the year. I’m not making anything up here — the frequency spectrum of ENSO (pick any index NINO4, NINO34, NINO3) shows a well-defined mirror symmetry about the value 0.5/yr. Given that Incontrovertible observation, something is mixing with the annual impulse — and the only plausible candidate is a tidal force.
So the average force of the tides at this point is the important factor to consider. Given a very sharp annual impulse, the near daily tides alias against the monthly tides — that’s all part of mathematics of orbital cycles. So just pick the monthly tides as it’s convenient to deal with and is a more plausible match to a longer inertial push.

Sunspots are not a candidate here.

Some say wind is a candidate. Can’t be because wind actually lags the thermocline motion.

So the deal is, I can input the above as a prompt to ChatGPT and see what it responds with

https://chatgpt.com/share/68649088-5c48-8010-a767-4fe75ddfeffc

Chat GPT also produces a short Python script which generates the periodogram of expected spectral peaks.

I placed the results into a GitHub Gist here, with charts:
https://gist.github.com/pukpr/498dba4e518b35d78a8553e5f6ef8114

I made one change to the script (multiplying each tidal factor by its frequency to indicate its inertial potential, see the ## comment)

At the end of the Gist, I placed a representative power spectrum for the actual NINO4 and NINO34 data sets showing where the spectral peaks match. They all match. More positions match if you consider a biennial modulation as well.

Now, you might be saying — yes, but this all ChatGPT and I am likely coercing the output. Nothing of the sort. Like I said, I did the original work years ago and it was formally published as Mathematical Geoenergy (Wiley, 2018). This was long before LLMs such as ChatGPT came into prominence. ChatGPT is simply recreating the logical explanation that I had previously published. It is simply applying known signal processing techniques that are generic across all scientific and engineering domains and presenting what one would expect to observe.

In this case, it carries none of the baggage of climate science in terms of “you can’t do that, because that’s not the way things are done here”. ChatGPT doesn’t care about that prior baggage — it does the analysis the way that the research literature is pointing and how the calculation is statistically done across domains when confronted with the premise of an annual impulse combined with a tidal modulation. And it nailed it in 2025, just as I nailed it in 2018.

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Non-autonomous mathematical models

/ Paul Pukite (@whut) / Leave a comment

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.

Does GPT understand geophysics?

/ Paul Pukite (@whut) / 12 Comments

Not sure what version of ChatGPT is being used on my browser at the present time, but I created a couple of GPT prompts (essentially queries) and compared and contrasted the results, and against expectations.

The first is to determine if ChatGPT could infer a wobble in a classical electromagnetic dynamics experiment. The setup is very simple and straightforward, illustrated by the schematic shown below:

As one can’t (as of yet) submit the schematic as a prompt to ChatGPT and illicit a response, we need to create a natural language prompt first:

Consider a levitating rotating magnetized sphere with north and south magnetic poles located at each endpoint of the axis of rotation. The rotation speed is 1 radian per second. An iron rod of the same length is placed in parallel with the axis of rotation, and wound with wire to form an electromagnet. At what frequency would the sphere wobble if the electromagnet was driven by a slight alternating current perturbation of 1 Herz, after the natural response subsided?

This is the response as of yesterday:

The motion of the levitating rotating magnetized sphere is governed by the interplay between the magnetic field of the sphere and the magnetic field of the iron rod, which is now an electromagnet due to the wire wound around it.

When the iron rod is magnetized by the alternating current, it generates a magnetic field that interacts with the magnetic field of the rotating sphere, causing a torque that can cause the sphere to wobble. The wobbling frequency can be determined by analyzing the equation of motion for the system.

Assuming the wobbling is small, we can linearize the equation of motion by taking the first-order Taylor series expansion. The resulting equation can be expressed as:

I * d^2θ/dt^2 + γ * dθ/dt + k * θ = M * B

where:

I is the moment of inertia of the sphere
θ is the angular displacement of the sphere from its equilibrium position
γ is the damping coefficient
k is the spring constant
M is the magnetic moment of the electromagnet
B is the magnetic field of the rotating sphere

The solution to this equation of motion is a damped harmonic oscillator, which can be expressed as:

θ(t) = A * exp(-γt/(2I)) * cos(ωt + φ)

where:

A is the amplitude of the wobbling motion
ω is the frequency of the wobbling motion
φ is the phase angle
The frequency of the wobbling motion can be found by substituting the solution into the equation of motion and solving for ω. After some algebraic manipulation, we get:

ω = sqrt((k/I) – (γ^2)/(4I^2)) – MB/(Isqrt((k/I) – (γ^2)/(4I^2)))

In this case, the sphere is rotating at a constant rate of 1 radian per second, so we can assume that the magnetic field of the sphere is also constant. Therefore, the only variable that changes due to the alternating current perturbation is the magnetic moment of the electromagnet, which oscillates at a frequency of 1 Hz.

Assuming that the natural response of the system has subsided, the wobbling motion will be dominated by the frequency of the alternating current perturbation. Therefore, the frequency of the wobbling motion will also be 1 Hz.

The response is to assert that the sphere will wobble at the same frequency as the applied forcing frequency, which is the intuitive response.

Next, change the prompt to create a similar behavior, but replacing the magnetic forces with gravitational forces.

Consider a rotating planet with massive extra weight at each pole. The rotation speed is 1 radian per Earth year. A small satellite moon orbits the planet with a significant declination. At what frequency would the planet wobble if the moon’s declination cycle was 250 days, after the natural response subsided?

This is a straightforward experiment to imagine, but difficult to perform in a lab. Submitting to ChatGPT today, this is the response:

The motion of the rotating planet with massive extra weight at each pole is governed by the conservation of angular momentum. The presence of the extra weight at the poles changes the distribution of mass of the planet, which in turn affects its moment of inertia.

Assuming that the extra weight is symmetric about the axis of rotation, we can approximate the planet as an oblate spheroid, which means that the moment of inertia of the planet is not constant but depends on the angle between the axis of rotation and the axis of symmetry of the spheroid. This angle is known as the obliquity of the planet and can change due to external perturbations such as the gravitational pull of a satellite moon.

The wobbling motion of the planet due to the gravitational pull of the satellite moon can be described by the following equation of motion:

I * d^2θ/dt^2 + γ * dθ/dt + k * θ = M * sin(ψ)

where:

I is the moment of inertia of the planet
θ is the angular displacement of the planet from its equilibrium position
γ is the damping coefficient
k is the spring constant
M is the mass of the satellite moon
ψ is the angle between the axis of rotation of the planet and the direction of the gravitational force exerted by the satellite moon

The solution to this equation of motion is a damped harmonic oscillator, which can be expressed as:

θ(t) = A * exp(-γt/(2I)) * cos(ωt + φ)

where:

A is the amplitude of the wobbling motion
ω is the frequency of the wobbling motion
φ is the phase angle

The frequency of the wobbling motion can be found by substituting the solution into the equation of motion and solving for ω. After some algebraic manipulation, we get:

ω = sqrt((k/I) – (γ^2)/(4I^2)) – Msin(ψ)/(Isqrt((k/I) – (γ^2)/(4I^2)))

In this case, the planet is rotating at a constant rate of 1 radian per Earth year, which corresponds to a period of approximately 2π × 365.25 days. The mass of the satellite moon and the gravitational force it exerts on the planet are also constant, so the only variable that changes due to the moon’s declination cycle is the angle ψ.

Assuming that the natural response of the system has subsided, the wobbling motion will be dominated by the frequency of the moon’s declination cycle, which is 1/250 Hz. Therefore, the frequency of the wobbling motion will also be 1/250 Hz.

Since the math of magnetic forcing and gravitational forcing follows the same inverse square law, the answers are similar as expected. ChatGPT gets the dimension wrong, using Hz instead of 1/(day), but other than that, this is also an intuitive result, as the forced response carries through after any natural response damps out.

Does GPT understand geophysics better than geophysicists?

I did this LLM AI experiment to find out whether ChatGPT would substantiate the model of the Chandler wobble that I had published in Mathematical Geoenergy (Wiley/AGU, 2018). The premise is that the nodal declination cycle of the Moon would force the Chandler wobble to a period of 433 days via a nonlinear interaction with the annual cycle. A detailed analysis of satellite peaks in the frequency spectrum strongly supports this model. Introducing the electromagnetic analog places it on an equal footing with a model that can be verified by a controlled lab experiment, see (A) vs (B) below.

Of course, if I had created a prompt to ChatGPT to simply inquire “What causes the Chandler wobble to cycle at 433 days?”, it would have essentially provided the Wikipedia answer, which is the hazy consensus culled from research since the wobble was discovered over 100 years ago.

Note that this “physics-free” answer provided by ChatGPT has nothing to do with the moon.

ADDED 3/26/2023

This blog post has a section on ChatGPT using MatPlotLib to create a visualization of orbiting planets.

https://andrewmayneblog.wordpress.com/2023/03/23/chatgpt-code-interpreter-magic/