Lorenz weather toy was implementing the classical program. It used a purely deterministic system of equations and the small errors proved catastrophic. Given a particular starting point, the weather would unfold exactly the same way each time. Given a slightly different starting point, the weather at the next moment would unfold in a slightly different way.

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This is the new initial condition which results in totally different mature state values compared to the previous run. This variability continues in a scale-free manner to result in a weather pattern with no similarities. Now the question arises: what if we give Lorenz weather machine results accurate to six decimal places? An infinitesimal change in the input values can result in a totally different outcome. Therefore, the initial situation of a complex system cannot be accurately determined, and the evolution of a complex system can therefore not be accurately predicted.

As chaos theory progressed, the research community saw journals which compared strange dynamics of a ball bouncing on a table side by side with articles on quantum physics. The simplest systems now seemed to be unpredictable. Viewed in that context, entropy can be thought of as measuring disorder or randomness. The Universe is governed by entropy and constant changes and growing toward greater and greater disorder.

Yet, chaos is not simply disorder. Chaos explores the transitions between order and disorder, which often occur in surprising ways.

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An order arises from the ever-growing disorder of the Universe—chaos and order together. River networks, lightning bolts, and trees have a self-similar pattern. Snowflakes, metal crystals, and pinecones have similar structural pattern. In all these structures, the same pattern gets repeated.

They are said to have fractal-like features. Fractal, by definition, is a curve or geometrical figure, each part of which has the same statistical character as the whole. Fractal-like patterns are found throughout the nature. It can be seen in the formation of ice crystals and veins in the human body, in leaves, in water drops, in air bubbles, etc. It is often been said that nature is an excellent designer and fractals can be thought of as the design principle nature follows when putting things together. Fractals are the best existing mathematical descriptions of many natural forms, such as coastlines, mountains, or parts of living organisms.

Fractals are infinitely complex patterns that are self-similar across different scales and infinitely detailed. Most physical systems of nature and many human artifacts are not regular geometric shapes of the standard geometry derived from Euclid.

Fractal geometry offers almost unlimited ways of describing and measuring these natural phenomena. This self-similarity has also been found in DNA sequences. Even the rhythm of our heart is a fractal pattern as a function of time. Many image compression schemes use fractal algorithms to compress computer graphic files to less than a quarter of their original size. Together, they suggest the surprisingly similar patterns found in vastly different natural phenomena.

Fractals are related to chaos because they are complex systems that have definite properties. Chaotic systems and fractals are both iterated functions. Each iteration takes the state of the previous iteration as an input to produce the next state.

## Complex system

Although fractals can show beautiful patterns, we cannot actually predict what their value is going to be of a specific iteration other than by calculating all the preceding iterations. So in a sense, fractals are just beautiful chaos. The process of making a fractal is a type of self-ordering process. These self-organizing processes are very local in nature but lead to highly organized structures on very large scales. An order arises from the ever-growing disorder of the Universe.

We cannot see the order in chaos if we are looking moment to moment. We cannot see order if we are managing individual behaviors.

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However, if we stand back far enough, if we wait over time, scale, or distance, we will observe the order that is in chaos, a pattern emerges from the otherwise disordered system. In a deterministic linear system, variability is not significant. There is a definite answer, a mathematical proof and solution for its characteristics and problems. A perfectly periodic system has the concept of a regular attractor. The damped pendulum has two invariant points; one is the point of minimum height and the other of maximum height. Due to the dissipation of energy, point of minimum height behaves as a regular attractor.

The system tends to evolve toward the attractor.

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Into a structured linear system, if the amount of input force is increased, the system undergoes a transition to a chaotic system. The periodically repeating pattern previously present in the linear system breaks apart and gives rise to a pattern that never repeats. The system loses its predictability. A system in chaos is totally unpredictable from moment to moment. When converted into multidimensional space on high-speed computers, one could track many variables at once and could plot the movement of a system in chaos.

The system is in total disorder when it is observed from moment to moment. Prediction of the next state is impossible. However, over a period of time, we come to realize that the system conforms to a boundary. There is an inherent shape that it never violates and never will it move out of a defined boundary.

The plots that are based on nonlinear equations may appear to be random or chaotic but over many iterations will evolve to be simple and tend to produce patterns called strange attractors that confine to a boundary. These strange attractors need not be symmetrical always. The behavior strange attractor can be seen from the first chaotic system discovered by Edward Lorenz, the Lorenzian or chaotic waterwheel. A chaotic waterwheel is just like a normal waterwheel except for the fact that the buckets leak.

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The entire waterwheel system is mounted under a water sprout. In the working of the waterwheel, water pours in from the top at a steady state and gives the system energy while water leaks out from each bucket at a steady state and removes energy from the system. Water trickles into the top bucket and immediately trickles out through the hole in the bottom at low speeds. With an increase in the inflow of water, the waterwheel will begin to revolve as the buckets fill up faster than they can empty.

The heavier buckets containing more water on descending will let the water out, and when empty, it ascends on the other side to be refilled. The system is in a steady state; the wheel will continue to spin at a fairly constant rate. Rather than spinning in one direction at a constant speed, the wheel will speed up, slow down, change directions, and oscillate back and forth, between the combinations of behaviors in an unpredictable manner.

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However, the entire system is within some bounds. Because the system never exactly repeats itself, the trajectory of a particular bucket never intersects itself.

Instead it loops around and around forever in a strange pattern. It seems to be attracted toward a point in space. The system does not reach the ideal attraction point but is constantly pulled to it. Hence, simply put, a chaotic system is a deterministic and unpredictable system.

Chaotic patterns followed by the center of mass of the bucket in Lorenzian waterwheel with x, y, and z being the coordinates of the center of mass in a x-y plane, b x-z plane and c y-z plane [10]. Complex systems exhibit chaotic behavior that is characterized by extreme sensitivity to initial conditions, fractal geometry, and self-organized criticality [ 11 ]. Fractal geometry refers to self-similarity, which implies that an object looks the same at any scale.

Mathematically, self-similarity requires a power law distribution of representative objects.