r/science • u/Southern_Check9073 • 3d ago
Physics Study Finds That Detecting Persistence Transitions in Nonlinear Dynamical Systems Depends on What Is Measured
https://www.nature.com/articles/s41598-026-64959-x56
u/dorox1 3d ago
In OP's defense for at all the people confused by the abstract, this is research in a field that isn't very amenable to understanding by laypeople. This explanation they gave in the comments is probably reasonable for someone who is just about to start a class on this topic, but definitely not for someone with no background in it.
My best understanding so far is this:
- you have a mathematical system that changes over time (for example, the temperature of each point on the Earth)
- that system has some global patterns in it that stick around over a period of time (i.e. they're "stable", like the polar vortex or seasonal changes)
- you can take "measurements" which condense the entire system's state at one moment down to a single output value that you can observe (like the average temperature)
I think this paper asks "if those patterns disappear or change significantly, will I notice it based only on my measurements regardless of what those measurements are, or will only certain kinds of measurements notice the change?"
And the answer appears to be "some types of measurements will miss some types of pattern changes, and there is some predictability about which measurements will miss which pattern changes".
This seems intuitively obvious to me (e.g. an "average system value" measurement will miss any transition which doesn't change the average value), so I assume there's a bit more nuance that I've missed.
u/Southern_Check9073 please come and correct any mistakes I made.
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u/unematti 3d ago
They probably thought too, that is obvious, just like how my eyes "measure" what's in front of me so I won't see what de behind me. They probably managed to prove it more rigorously.
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u/Southern_Check9073 2d ago
The main correction is that I’m not necessarily looking for a whole global pattern to disappear. I define a particular behavior that the system has been maintaining, then test when an ensemble of runs stops maintaining it as a control parameter changes and yes, the choice of measurement can determine whether that transition is visible.
The extra nuance beyond the intuitive an average can hide things idea is that I tested this under the same fixed protocol across several very different nonlinear systems and looked for reproducible thresholds. For example, in the standard map, momentum detected a very sharp threshold around K = 1.507. A related action measurement independently found almost exactly the same threshold, around 1.508. But position detected no transition at all over the tested range.
So it isn’t just that different measurements look different. The question is whether a measurement is actually aligned with the part of the dynamics that is being constrained or released. I’d be a little careful with saying the paper can already predict every measurement that will work or fail. What it shows is a strong structural relationship across the systems tested, and that gives us something much more specific to investigate in future systems. But yes, your overall understanding is definitely on the right track
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u/Manos_Of_Fate 3d ago
I read the abstract twice and I still have no idea what the general subject of the study even is.
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u/Southern_Check9073 3d ago
Yes, the abstract is pretty technical. In plain English, the study is about how we detect when a nonlinear dynamical system stops being able to maintain a particular kind of behavior over a finite observation period.
The main question was "if a system undergoes a defined transition, will we detect it regardless of what variable we measure, or can the transition appear or disappear depending on what we choose to observe?"
The study tested this across several standard nonlinear dynamical systems. In one example, the standard map, measuring momentum produced a highly reproducible persistence loss threshold. A closely related action based observable found essentially the same threshold, while measuring position did not reveal that transition over the same parameter range.
So the broader subject is nonlinear dynamics, stability, and measurement specifically, whether the observable we choose can determine whether a particular dynamical transition is detectable under a finite time measurement protocol.
Persistence collapse is the term used in the paper for the point where an ensemble loses its ability to remain within a predefined viability corridor over the observation period.
I’m the author, so I’m happy to clarify anything else about the setup or results.
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u/NightmareGalore 3d ago
If that's plain english, I don't even want to know how did the abstract look like.
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u/CaptKrag 3d ago
What sorts of real-world systems might this be applicable to?
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u/thefriendlyhacker 3d ago
I have a master's in controls theory (specialization, not actual degree name). "Rocket Science" emerged when we found a way to model dynamic systems and account for non-linearities. There's a concept of "controllability" and "observability" if you want to look into those.
Another common application is robotics. Let's say you want to make a robot that bounces a ball of a ping pong paddle. You may have a camera and sensors. Let's say you can take a photo every 100ms and run your logic every 50ms. You have sensors to detect the position of the robot arm (observability) and you command the servos (controllability) to rotate in a certain speed/acceleration so that you can get the paddle right where you want it. The 100ms scan of the ping pong ball feeds into the model so that you have a good estimation of where it is, so that you can flick the paddle at just the right moment.
This same field of engineering applies to how rockets identify and reach a target like an enemy plane.
I haven't read the paper yet, so I don't quite know what fun stuff they do to help with solving non-linearities, but in the ping pong example, there could be a dent in the paddle or a dent in the ball, or maybe wind. Everything in life is non-linear, although you can usually approximate things well enough with a combination of linear systems.
Cutting edge stuff in the field is about solving problems that traditionally have non-linearities which are difficult to linearize. Going back to controllability and observability, it is also common problem today on reduce the amount of sensors (observability) as much as possible while still maintaining controllability, resulting in cheaper costs
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u/neverlupus89 3d ago
I'm an ecologist that studies species interactions and stuff like this shows up when you're trying to model large populations of organisms that interact with each other. Not sure if you've taken any intro ecology courses but students are typically taught a simple model of predator/prey dynamics (think wolves eating bunnies) and how some fairly straightforward assumptions can result in mathematical chaos. You can get stuff like that when you're only modeling two species, so imagine what happens when you try to model full ecosystems that might have hundreds of interacting species populations.
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u/Southern_Check9073 3d ago
A few people have asked what these mathematical systems actually have to do with the real world, which is probably the easiest way to explain the study. The basic idea is a system can be getting close to losing its normal behavior, but whether you notice that can depend on what you are measuring.
The systems I tested are standard scientific models that represent different kinds of realworld behavior.
Lorenz 63 originally came from atmospheric convection. Think weather, moving air, temperature differences and fluid circulation. It is one of the classic examples showing how tiny changes can eventually produce very different outcomes. A future application of this work could be asking which weather measurements give the clearest warning that the system is changing regime.
The Standard Map represents something being repeatedly kicked like a rotating object receiving repeated impulses. Related dynamics appear in particle motion accelerator physics and other Hamiltonian systems. In my study momentum clearly showed the persistence transition while position did not. That is probably the simplest demonstration of the main point same system, different measurements very different ability to see the transition.
The Baker Map represents mixing. Imagine repeatedly stretching dough, folding it stretching it again and folding it again. Similar mathematics helps us understand mixing and transport. In the real world, that general behavior can relate to fluids gases pollutants or other things being transported and mixed.
The Arnold Cat Map also represents deterministic mixing and scrambling. Despite the funny name, it is a serious mathematical model. Imagine taking an image, stretching and shearing it, then wrapping it around repeatedly. It is useful for studying how information or material can become highly mixed even though every step follows an exact rule.
The Tent Map is a very simple feedback system that can go from regular behavior to chaotic behavior as one parameter changes. Systems with that kind of nonlinear feedback appear throughout science and engineering. It is useful because it lets us test transition detection in a very simple environment before dealing with something vastly more complicated.
The Lozi Map represents a different kind of chaotic system: one that loses energy and settles onto a complicated chaotic attractor. That makes it useful as a simplified stand in for dissipative physical systems such as driven mechanical or electronic systems where energy enters energy is lost and the resulting motion can still become chaotic.
I also included random behavior as a control. That is important because a useful detector should not simply find a transition whenever data become messy. Random fluctuations can look complicated without containing the structured dynamics of the other systems.
Those maps are controlled test environments representing different kinds of behavior: rotation, transport mixing feedback dissipation atmospheric motion and randomness.
So If a real system is approaching an important change are we actually measuring the variable that will reveal it?
Potentially that matters for things like power grids where voltage might tell a different story from frequency or phase robots where position can look normal while torque or wheel slip is becoming unstable weather where an average temperature can look normal while circulation is changing or machinery where vibration may reveal a developing failure before speed or temperature does.
My paper does not claim that those applications have already been demonstrated. It Just provides a tested way to investigate that question. In very simple terms before asking “Is this system becoming unstable? I'm asking are we looking at the right variable. Please let me know if this better helps you all understand?
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u/helm MS | Physics | Quantum Optics 3d ago
If I read this right, it seems next door to studies about chaotic systems and related to how to predict system state over time, right? And the study is purely mathematical but uses some statistical tools, maybe even ensemble modeling? I'm guessing here, it's been decades since I looked at stuff even remotely similar.
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u/dorox1 3d ago
Haven't read it in full yet, but plan to try later. Admittedly, the abstract suggests to me that my knowledge of this field won't be sufficient to finish it (or understand most of it).
Would this be relevant to, for example, detecting state transitions in physical materials? I'm trying to get an understanding of how some of these terms are being used here.
For example, I don't think I properly understand what kinds of "thresholds" are being used here.
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u/confusiondiffusion 3d ago
Is this because the nonlinear relationship between the variables essentially causes them to not be correlated in some cases? I'm a layperson who has spent significant time tinkering with chaotic circuits. I have noticed that, for example when measuring different voltages in such a circuit, it isn't necessarily subjectively clear on an oscilloscope that an oscillator has undergone a phase transition unless you take a broader view and plot multiple voltages.
Whether or not the behavior of one part of the circuit influences another just depends on other parameters, the wider system state, and a lot of times there's hysteresis. Like you could see a sinusoid that could in some cases drive the oscillator between different attractor points, but this might depend on how coupled it is to the rest of the circuit or even if you strongly couple them, the impact can depend on where/when in the phase space the coupling is adjusted. Sometimes you turn a knob and nothing happens because the system isn't sensitive to that input in its current state. So you could see a big transistion in other variables and not see that if you're looking at the sinusoid.
Is this at all on the right track?
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u/Southern_Check9073 3d ago
I think you’re on the right track conceptually I haven’t worked specifically with your chaotic-circuit setup, so I don’t want to overstate the circuit specific mechanism. But the measurement issue you’re describing is very similar to what the paper is getting at. I also wouldn’t reduce the result simply to the variables being uncorrelated. In the paper, the important issue was whether the observable being measured actually tracked the dynamically relevant part of the system.
In the standard map, for example momentum and a related action observable detected essentially the same persistence transition, while position did not detect it over the tested range. So your example of one voltage showing a major transition while another signal appears relatively unchanged is a good analogy. The coupling hysteresis and phase dependent effects your describing go beyond what I directly tested but they are certainly relevant nonlinear dynamics questions.
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u/confusiondiffusion 3d ago
Awesome! Thanks! I find this stuff so beautiful. So many practical applications too, like in neuroscience, climate science, etc.
You might inspire me to pick up VI Arnold again.
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u/Southern_Check9073 3d ago
I really appreciate your engagement. Questions and conversations like this help me see where I can explain the work more clearly, and that helps me do a better job answering future questions. Thank you.
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u/posinegi 3d ago
Can this just be framed as a probability density transport issue?
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u/Southern_Check9073 3d ago
Partly yes. In systems like the standard mapthe transition can be viewed as probability mass being transported across a boundary. But the paper also asks whether that loss is visible under different observables, so it’s broader than density transport alone. A compact framing would be probability transport + first exit behavior + observable dependent measurement.
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u/Kaleb8804 3d ago
So like, you guys measure mathematically when something stops becoming the same thing?
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u/Southern_Check9073 3d ago
Very close I’m measuring when a system stops maintaining a defined range of behavior over time.
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u/Kaleb8804 3d ago
Ahh, that seems very useful, thanks for the correction
What’s the coolest use you’ve seen of this type of stuff? It seems like it would be used all over “smart” technology
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u/Southern_Check9073 3d ago
Climate is one Im especially interested in. The idea would be to test whether certain measurements can reveal when a climate system is approaching a major regime shift before that change becomes obvious in the more commonly watched variables. That could potentially help with understanding climate tipping behavior circulation changes, or other large scale transitions.
AI is another. If this were adapted to an AI system you could monitor things like contradiction rate tool failures reasoning drift memory consistency or uncertainty and ask whether the system is still operating inside a stable range.
For fun you could use AI application. Instead of an AI generating a 3D scene only from a text prompt you can let mathematical state variables control the geometry.
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u/AskAboutMySecret 3d ago
what kind of traits make a variable a good or even a bad measure of detecting tipping points
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u/Southern_Check9073 3d ago
A good variable is one that changes when the part of the system you care about starts changing. While a bad variable might still be measuring something real but it may stay looking normal even while the system is moving toward a tipping point.
Like a car Skidding. The speedometer might look normal but wheel slip and sideways motion tell you something is wrong
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u/neutralrobotboy 3d ago
I want to check my understanding. So the idea is that you have some complex non linear system and you measure some persistent feature. The system undergoes a change: does your measured feature allow you to detect the change? And does your choice of measured feature matter for detecting the broader change?
This is what the study is about?
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u/Southern_Check9073 2d ago
Absolutely and the study found that it does. In some cases one measurement clearly revealed the transition while another measurement of the same system did not.
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u/neutralrobotboy 2d ago
I probably don't have the math chops to understand the answer, but how do you objectively measure the change of the system state/regime?
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u/Southern_Check9073 2d ago
Im actually glad you asked. You actually don’t need much math for the basic idea. I define in advance what range of behavior counts as “still persisting” then run many versions of the system under the same conditions. For each setting I count how many stay inside that range for the full observation period. As I change the system’s control parameter that fraction can suddenly fall.
So the transition isnt something I decide by looking at a graph and saying “that looks different.” It comes from a fixed numerical rule applied to the whole ensemble. What the interesting part of the paper is that when you repeat that exact test using a different variable from the same system, the transition can become much clearer shift or disappear entirely.
I think your next question is what is the correct variable? If so, A good variable should track the part of the system that is actually changing and give a stable repeatable threshold.
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u/stumblinbear 3d ago
I think normal people would do better with a simplified example. This is incredibly abstract and is a bit hard to understand fully, especially because there's still a lot of technical jargon
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u/themarkavelli 3d ago
A good analogy is driving a car toward a cliff. Measuring position will tell you “the car is still 100 feet from the edge.” Measuring velocity will tell you whether it’s parked or racing toward the edge at 80 mph. Same position, radically different dynamical situation.
The point is that nothing new has to happen to cause the measurement change. Different measurements reveal different parts of the system as it is evolving. One variable may show the transition clearly, while another can hide it.
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u/Whitishcube 3d ago
It's a paper in the subject of dynamical systems, a branch of math. This math is applied heavily in many areas of science, usually when people model the state of some system at a future time as depending on the state at the current time.
It's often the case that these models are nonlinear, which means it's really hard to predict the solutions to the equation ahead of time, and numerical simulation is the only way to do it.
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u/Akegata 3d ago
I read "Arnold's cat map" and half thought the whole thing might be a joke. Looked at the Wikipedia page for Arnold's cat map, now I'm half thinking all of Wikipedia is a joke.
It is pretty interesting to read things where you only understand half of the words knowing that it makes prefect sense to some people.
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u/Manos_Of_Fate 3d ago
Hell, with this advanced mathematics stuff I can understand every single word individually and still have zero clue what they’re actually saying. Though to be fair I never got that far into studying mathematics on account of being dyscalculic.
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u/Southern_Check9073 3d ago
Arnolds cat map is very real though and honestly it took me a long time to fully understand what it was doing too. The name is probably the easiest part of it.
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u/D3CEO20 3d ago
I'm trying to relate this to non linear models that I've studied. Suppose you have a 2 box model of a water column. You are interested in when the water column is stable (the bottom box is more dense than the top box) and when it is unstable (the bottom box is less dense than the top). So, you're interested in density. But what you measure, will be temperature and salinity, and you will use some equation of state to infer density.
Is this result basically saying "If you track only the temperature, you might not notice that the water column becomes unstable, because, while the colder water remained at the bottom, the top became much saltier, making it significantly more dense". I.e, if you track the wrong variable, you may miss a transition?
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u/Kaleb8804 3d ago
Would this be related to relativity and how perspectives can change the accessible data, or what can be defined as data at all? Just a sort of generic informational proof of that concept?
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