We have been hearing that deterministic chaos was found in this or that natural phenomenon for over 20 years now, since the moment we opened our eyes almost suddenly to a reality that we had decided not to observe: that of real non-linear phenomena (ie almost all real phenomena tout court).
Personally one of the findings that impressed me most is chaos in the human heartbeat: in a healthy individual dynamics of the heartbeat as you can measure by means of an electrocardiogram is chaotic, while a regular dynamic is often symptom of a disease. Even the electroencephalogram of a healthy person has shown the typical and unambiguous marks of chaos.
What does it mean that a natural phenomenon shows chaotic behaviour? How is that a natural system is chaotic?
If we know the equations of a system then we can rely on some tools such as mathematics analysis, computer simulation that allows us to study how the behaviour of the system changes with its parameters.
The onset of chaos is always preceded by well detectable mathematical phenomena, points of transition from regular to complex behaviour: I mean, for example, period doubling that happens at given intervals and that ultimately flows into true chaos. These paths can be viewed in the so called bifurcations diagrams.
An universal characteristic of chaos is that the convergence ratio in the sequence of bifurcations (the so-called route to chaos), which is observed as the control parameter varies, is almost constant and is constant as the parameter approaches the infinte. This universal ratio is called delta of Feigenbaum, and allows us to know exactly when the next bifurcation will happen in a system that evolves towards chaos. So if we see a constant ratio that is close to the Feigenbaum's delta in the intervals between one bifurcation and the next one, we have an important clue that system has the characteristics of chaos in itself and is evolving towards a chaotic dynamic as consequence of the change of one of its structural parameters (eg tension of a diode).
The calculation of the so-called Lyapunov's exponents yields the conclusion that two close trajectories of the system differ on average: the mathematical consequence of this fact is the system's chaoticism for some given parameters (for other parameters the same system may not be chaotic). This is of course possible only if you know the equations of the system, in which case you can draw the exponents of Lyapunov mathematically.
If we do not know the system's equations, then we must carry out a measurement of its dynamic based on a series of samples long enough. On this series of measures we can quantitatively estimate some parameters: these are the coefficients of Lyapunov, or the Kolmogorov's entropy, or fractal dimension, or the size of correlation. These parameters can tell us whether we have chaos.
Finally we can say chaos has well detectable traces, and its traces have been discovered in many natural systems, whether living or not.
Personally one of the findings that impressed me most is chaos in the human heartbeat: in a healthy individual dynamics of the heartbeat as you can measure by means of an electrocardiogram is chaotic, while a regular dynamic is often symptom of a disease. Even the electroencephalogram of a healthy person has shown the typical and unambiguous marks of chaos.
What does it mean that a natural phenomenon shows chaotic behaviour? How is that a natural system is chaotic?
If we know the equations of a system then we can rely on some tools such as mathematics analysis, computer simulation that allows us to study how the behaviour of the system changes with its parameters.
The onset of chaos is always preceded by well detectable mathematical phenomena, points of transition from regular to complex behaviour: I mean, for example, period doubling that happens at given intervals and that ultimately flows into true chaos. These paths can be viewed in the so called bifurcations diagrams.
An universal characteristic of chaos is that the convergence ratio in the sequence of bifurcations (the so-called route to chaos), which is observed as the control parameter varies, is almost constant and is constant as the parameter approaches the infinte. This universal ratio is called delta of Feigenbaum, and allows us to know exactly when the next bifurcation will happen in a system that evolves towards chaos. So if we see a constant ratio that is close to the Feigenbaum's delta in the intervals between one bifurcation and the next one, we have an important clue that system has the characteristics of chaos in itself and is evolving towards a chaotic dynamic as consequence of the change of one of its structural parameters (eg tension of a diode).
The calculation of the so-called Lyapunov's exponents yields the conclusion that two close trajectories of the system differ on average: the mathematical consequence of this fact is the system's chaoticism for some given parameters (for other parameters the same system may not be chaotic). This is of course possible only if you know the equations of the system, in which case you can draw the exponents of Lyapunov mathematically.
If we do not know the system's equations, then we must carry out a measurement of its dynamic based on a series of samples long enough. On this series of measures we can quantitatively estimate some parameters: these are the coefficients of Lyapunov, or the Kolmogorov's entropy, or fractal dimension, or the size of correlation. These parameters can tell us whether we have chaos.
Finally we can say chaos has well detectable traces, and its traces have been discovered in many natural systems, whether living or not.
This should unanmbiguously answer the question: what is chaos. Yet it tells as little of what chaos means.
2 comments:
A nice post about chaos... I recently read about what "robust" chaos is... The absence of windows of periodicity in a chaotic regime... Indeed, I had built the logistic map in the past, and had found that the system tends to latch on to a nearby periodic window. The chaos was actually fragile... If chaos has robustness, it might be of more value for an engineer like me...
I am an engineer too, and I agree that the control of chaotic systems is a very interesting area of study abd applications, although I am not after it.
Have a look to this paper: discrete systems can show robust chaos indeed!
http://www.ee.iitkgp.ernet.in/~soumitro/prl-robust.pdf
It's just a starting reading. If you find out more please let me know! ;-)
Thanks.
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