Butterfly effect

Some Properties of Complex Systems Self-organization, the Butterfly Effect, Adaptability and Probabilistic Advantages... 

This second-level modeling of the feedback mechanisms leads to nonlinear models for processes, which, under some experimental conditions, may exhibit chaotic behavior. The previous equation is termed bilinear because of the presence of the b [y (/,)] r (I,) term and it is the general formalism for models in biology, ecology, industrial applications, and socioeconomic processes [601]. Bilinear mathematical models are useful to real-world dynamic behavior because of their variable structure. It has been shown that processes described by bilinear models are generally more controllable and offer better performance in control than linear systems. We emphasize that the unstable inherent character of chaotic systems fits exactly within the complete controllability principle discussed for bilinear mathematical models [601] additive control may be used to steer the system to new equilibrium points, and multiplicative control, either to stabilize a chaotic behavior or to enlarge the attainable space. Then, bilinear systems are of extreme importance in the design and use of optimal control for chaotic behaviors. We can now understand the butterfly effect, i.e., the extreme sensitivity of chaotic systems to tiny perturbations described in Chapter 3. 

The Lorenz equations may produce deterministic chaos because we know how it will instantaneously change. However, for high enough Rayleigh numbers, the system becomes chaotic. Small changes in the initial conditions can lead to very different behavior after long time interval, since the small differences grow nonlinearly with feedback over time (known as the Butterfly effect). These equations are fairly well behaved and the overall patterns repeat in a quasi-periodic fashion. 

As observed in Figs.3.14-4, all patterns generated by the C3-C1, C3-C2 and C2-C1 representations, remind, in one way or another, a butterfly. The latter stands for a basic phenomenon in the chaos model known as the butterfly effect, after the title of a paper by Edward N.Lorenz Can the flap of a butterfly s wing stir up a tornado in Texas An additional point may be summarized as follows, i.e., How come that relatively simple mathematical models create very complicated dynamic behaviors, on the one hand, and how Order, followed by esthetics patterns, may be created by the specific representation of the transient behavior, on the other ... 

Oscillations may involve periods with more than a single wave, and may be chaotic (with no recurring periods at all). Aperiodic behavior is called chaos, but is not random The seeming randomness results from the fact that minutes difference in starting conditions can lead to drastically different behavior (butterfly effect of meteorology). Even relatively simple hypothetical networks can produce chaotic behavior. However, while periodic oscillation are possible in networks with only two independent mathematical variables, chaos requires at least three. 

The reader can check this behavior in May s numerical equation of the logistic model. If we change slightly the initial value in the domain of stability (for example, p = 2.7), the population converges to the same value of 0.6296. This point acts as an attractor. In the chaotic region, a similar weak variation gives way to conq>letely different succ sive evolutions. This is an indication of the "butterfly" effect. 

Similarly, divergence will also occur if we have an infinitely precise computer to solve the chaotic problem, but the balloon experiences an unaccounted-for, infinitesimal fluctuation between finite time steps of the trajeaory calculation. The accumulation of errors resulting from the extreme sensitivity of a chaotic trajectory to its instantaneous environment was called the butterfly effect by Lorenz. The butterfly effect arises from the fact that the balloon s trajectory is dynamically unstable, which means that it is so sensitive to changes in its instantaneous environment that the perturbations caused by the fluttering of a butterfly s wing thousands of miles away are sufficient to cause the trajectory of the balloon to change from what it would otherwise have been. 

Chaos and randomness are no longer ideas of a hypothetical world, they are quite realistic. A basis for chaos is established in the butterfly effect, the Lorenz attractor, and there must be an immense world of chaos beyond the mdimentary fundamentals. This new form mentioned is highly complex, repetitive, and replete with intrigue. 

The flow of traffic along a multilane highway can be described by the same mathematics that describes the flow of water in a system of pipes. The behavior of an electron bound to a single proton is described mathematically as a particle in a box, because it must conform to specific conditions determined by the confines of the box. The seemingly uniform movement of individual birds in flocks of flying birds demonstrates the chaos theory, in which discrete large-scale patterns arise from unique small-scale actions. This is also known as the butterfly effect. 

Butterfly Effect Theory that small changes in the initial state of a system can lead to pronounced changes that affect the entire system. 

The butterfly effect is a large variance in possible outcomes in a situation that starts with changes on a much smaller scale. The phrase refers to a concept in probability that the flap of a butterfly s wings could, after a string of events, alter a tornado s path. 

Saito A, Miyamura Y, Nakajima M, Ishikawa Y, Sogo K, Kuwahara Y, Hirai Y (2006) Reproduction of the Morpho blue by nanocastmg lithography. J Vac Sci Technol B 24 3248 Simms RW, Cunningham MH (2008) High molecular weight poly(butyl methacrylate) via ATRP miniemulsitms. Macnunol Symp 2(X)8(261) 32-35 Starkey A (2005) The butterfly effect. New Sci 187 46... 

The well-known butterfly effect seems to typify much of today s supply chain turbulence. The idea is that a butterfly, flapping its wings somewhere over the Amazon basin, can cause a hurricane thousands of miles awayl Whilst this example of what is sometimes described as chaotic effects may be a little far-fetched, it provides a useful reminder of how the law of unintended consequences applies to today s highly interconnected supply chains. 

---