3.2 Introduction to Nonlinear Concepts and Criticality Theory 

• The next part we'll take a look at understanding basic concepts underlying nonlinear dynamics. It's important to know what linear ad means linear sent systems tend to generate very predictable responses. So linear systems have a proportional response to its input. Nonlinear systems, however, tend to produce nonlinear responses, things that are not in direct proportion to the input. And these are various examples of nonlinear output.
• What differentiates high nonlinear systems from linear systems is that you can get all different kinds of output to a specific signal. A linear system, for instance, gives a very similar signal to the input here, which is sinusoidal. a nonlinear system responds differently to the Sun SOTL system, it can be periodic, quasi periodic or even chaotic. And these are for higher order nonlinear systems. And only dynamics has quite a rich history. And this even extends back to the 1800s to Porcari. But a lot of wonderful work rich, amazing work was done by the Soviet Russians. In the 1920s. Chaos Theory was started approximately 1960s by Webb and Lawrence Lawrence was at MIT, and popularized the term the butterfly effect. You had Winfrey Arthur Renfree, who studied nonlinear oscillators in biology in menupro, who popularized fractal geometry and fractals. But it was really in the 1980s, when you had widespread interest in chaos, fractals, oscillators and their applications, and this disseminated into the medical field in the 1990s.
• In the early 1990s, some scientists like RF Goldberger and Bruce West, speculated that certain characteristics and chaos theory and nonlinear dynamics might underlie some of the unique features seen in biologies. This includes the fractal characteristics that we see in the GI tract, and the lungs, and also the heart. And this extended also the temporal complexity as well in this is an example of a time so heart rate time series obtained in 300 minutes, where you can zoom in the 30 minutes range, and also into the three minute range and recognize that there is this fractal nature to it essentially, this repetition of patterns across spatial and time scales.
• Question is, how do you show that biological systems display nonlinear, plus or minus chaotic dynamics? And that was certainly a difficult question to answer. initial attempts were made to apply certain principles of chaos theory, such as period doubling or bifurcations, wiping off exponents, correlation dimensions, or embedding dimensions to biological systems. However, the problem is that deterministic chaos theory applies much more easily to simple systems that display complex behaviors and example this is a double pendulum that's shown here.
• On the other hand, biological systems are quite complex. It's not a simple system that demonstrates complex behaviors. It's a complex systems that demonstrates complex behaviors. And so what is the theory that is best applies to biological systems, such as heart rhythms that's quite complex and regulated by multiple processes.
• To me, probably the most useful analogy for theory from complexity theory is criticality theory. And an example of criticality theory is the icing model. Hopefully, you'll get an understanding of what the criticality theory means based on this example, the Ising model is a representative model of ferromagnetic magnetization. Each grid here represents an atom which can be aligned in one of two confirmations, either black or white, or in this case, up or down plus one or minus one. The total magnetization of the system is essentially the average of all the spins. And it's the dynamics of the interactions between the atoms is what makes this model interesting. So if you have two atoms aligned together, they share the same sign and the total energy is decreased and disfavored. If the two atoms are opposed or opposite, then the energy is increased and unfavored and H represents the total energy of the system, but each spin has its own energy which determines which conformation is favored. This relationship is further influenced by temperature, and the higher the temperature, the more likely the spin is likely to flip regardless of its intrinsic energy.
• The OSI model is very useful to understand the magnets response to temperature and this is a graph of magnetization versus temperature. And we can see here at the very low temperatures that the magnet has either a positive or negative confirmation, you know, either north or south. And so there is a net magnetization. However, at higher temperatures, because of the frequent flipping of the ferromagnetic atoms, you have a net magnetization of zero, ever it is at this criticality location where it's between absolute magnetization to no magnetization, where you see some very interesting patterns.
• This is where you begin to see fractal or self similarities. And is this an example of this, where you could take a segment from this critical threshold, and you could zoom in within each other. And these all four of these patterns look quite similar to each other, but they're existing at different scales. And so this explains the term self similarity or scale invariance. It's that criticality that we see this parallel relationship and this parallel relationship is described by this equation. An example of this is cluster size. And if you figure out the distribution of cluster size, you will notice that you know, you have much more frequent smaller clusters compared to larger clusters. And this logarithmic relationship shows this linear response because of this exponent here. And you see this parallel in multiple situations not only in the cluster size, but also the dynamic correlations between spans with respect to distance and also the temporal variation and overall payment Station.

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