IMHO In My Humble Opinion

The chaos theory is a set of ideas that attempts to explain the complex structures of unpredictable systems. The chaos theory is implied in many different areas including music, astronomy, chemistry, ecology, biology, physics, mathematics, etc. Some say that the chaos theory is random, but that is not true. In the book Chaos, by James Gleick, it says, “‘If the image is complicated, the rules will be complicated,’ Barnsley said.” (238). Although the theory of chaos is complex, it does have order.
The weather system is one of the many systems of chaos. A meteorologist named Edward Lorenz did an experiment on weather patterns. He concluded that it is impossible to predict the weather accurately because weather never repeats itself exactly. It takes paradigm shifts. Paradigm shifts tell us that things are always changing. If a rainy day was followed by a tornado, that does not mean that the next rainy day will also be followed by another tornado. It could instead be followed by a snowy day or even a sunny day. It is unpredictable.
Cloud formation is another system of chaos. Clouds only exist in planets that have an atmosphere. In order for clouds to form, there has to be water. Without an atmosphere, there is no water. Clouds are formed through the process of evaporation. Evaporation is when water changes from a liquid to a gas. Clouds are made out of water vapor. In a desert climate, there aren’t very many clouds because there isn’t much water. But in a rainforest climate, there are alot of clouds because there is alot of water. “One important feature of clouds is that the weather can be predicted to a certain level with the help of studying cloud formations.” (D’Silva, 2007)
There are many other chaotic systems. All these systems are represented by nonlinear equations. These nonlinear equations cannot be solved, as discovered by mathematician, Mitchell J. Feigenbaum in the 1970’s. There are 5 characteristics of a chaotic system. They are sensitivity to initial conditions, time irreversibility, strange attractors, fractal forms, and bifurcation.
Sensitivity to Initial Conditions- In complex systems, a slight change in the variables will change the whole outcome. Sometimes, even using the same exact variables will result in a different outcome. There is a rule that states that it is impossible for a model to accurately predict the outcome of complex systems. In 1961, a meteorologist named Edward Lorenz tried to predict weather patterns by putting weather conditions into a computer and looking at what would happen. As Lorenz printed out the information the computer provided, he noticed a familiar pattern. Later that year, Lorenz used the computer again to look closer at the weather sequence and saw different results. The pattern changed. In fact, the results were almost the opposite of the previous results. “He might as well have chosen two random weathers out of a hat.” (Price, 2008) He noticed that the results were in the shape of a butterfly. He called this the Butterfly Effect. “The ‘Butterfly Effect’, or more technically the ‘sensitive dependence on initial conditions’, is the essence of chaos.”(Cross, February 27, 2001) This means that the Butterfly Effect is the basic nature of chaos.
Time Irreversibility- Complex systems are always changing. Nothing will ever repeat itself exactly. For example, if you walk down the beach twice, you won’t walk on the same sand the second time as you did the first. Your feet pick up some sand and the ocean washes away some sand that you walked on the first time.
Strange Attractors- Strange attractors cause systems to change as time progresses. They were discovered by Edward Lorenz. One example is gravity. Gravity is what makes the earth rotate around the sun. This changes the weather patterns making some days hot, other days cold. Another example of an attractor is the Butterfly Effect discovered by meteorologist Edward Lorenz in the winter of 1961.
Fractal Forms- In 1963, a mathematician named Benoit Mandelbrot introduced the fractal concept. This concept says that if any magnified piece of an item looks the same as the whole item, it is a fractal. This means that the angles of the magnified piece would need to be the same as those angles of the original piece. An example of this is a molecule. No matter how magnified it is, it always shows the same image with the same angles. “Fractals are related to chaos because they are complex systems that have definite properties.” (Mendelson and Blumenthal, 2003)
Bifurcation- The solutions to nonlinear equations vary greatly. Bifurcation is when a main body splits into two separate parts. For example, a river might at some point separate into two different rivers.
The chaos theory has progressed alot through the years. The first person to discover it was meteorologist, Edward Lorenz in 1961. He knew the basic rules of it and discovered the Butterfly Effect. As years passed, scientists discovered more about the chaos theory. In 1963, mathematician, Benoit Mandelbrot introduced the fractal concept. In the 1970’s, mathematician, Mitchell J. Feigenbaum discovered that nonlinear equations are impossible to solve. We went from thinking that chaos is random to knowing chaos has order. We might eventually find out the complex patterns of weather, cloud formation, etc. The chaos theory and its systems are progressing very fast. Who knows how much more we will discover about the chaos theory 10 years from now? 50 years? 100 years? There is an unlimited amount of information on chaos that we can learn.

To answer some of your questions:

Energy is conserved, so you're right that it doesn't disappear. In your puddle example, I believe that what happens as time continues is the following:

• Whenever a ripple hits the edge of the puddle, it gets reflected back. It probably also slightly perturbs the edge of the puddle.
• Some of the kinetic energy of the motion of the ripples gets converted into heat energy (which is really just kinetic energy at a smaller scale.) And some of it is transferred to the air molecules in the same way. This is a very small amount of energy, though, so it's probably not enough to cause a perceptible temperature difference.

The problem with your second question (about the piano) is that waves don't propagate forever: as they're traveling through a medium, they lose energy by transmitting some of their energy to that medium. A sound wave is just a wave traveling through the air, causing air molecules to vibrate; as it does this, it's losing some of its energy to the air. Eventually, the magnitude of the sound wave becomes low enough that it's impossible to hear any more. Eventually the wave's magnitude will be small enough that the random oscillations of air molecules will be larger than the sound wave itself, and it will be swallowed up even if there is no ambient noise.

In theory I think it's possible for sound to somehow spontaneously get produced, but in practice I think that's so unlikely to ever occur that it's better to just say that it's never going to happen.

I think that what you might be asking is how the organized forms of energy in the universe end up losing their form and order, and the answer to that is the second law of thermodynamics, which states that in any closed system, entropy is always increasing. Energy is never really "lost", but the sort of low-level random oscillations of matter aren't a particularly useful form of energy and can't easily be converted to useful forms of energy (without losing additional energy as a side effect), which I think is what you might be trying to get at with your phrase "zero point energy".

If I understand what you're asking correctly.