### Chaos Theory (popularisation)

The main idea behind chaos theory is that even simple deterministic systems can sometimes produce completely unpredictable results. This happens when a deterministic system has sensitivity to initial conditions: basically, that if we take two starting states which are very similar to each other, over time the states will diverge and re-converge in unpredictable ways.

In other words, the main percept behind this theory is the underlying notion of small currencies significantly affecting the outcomes of seemingly unrelated events. This is one of the reason why Chaos theory is often referring to a “non-linear dynamics systems”.

### Deep Precisions on Markets Movements

If Chaos-Theory is today known as the most precise way to quantitatively forecast the future markets movements, it is still underused by institutions and represents today less than 5% of quantitative strategies on the market due to the complexity of the model. We apply Chaos and Multi-Fractals algorithm in order to see and understand the underlying order of complex systems that may appear to be without order at first glance.

### Non-liner Dynamics Systems

Relative to Financial Markets, we believe that price is the very last thing to change for a stock or any other securities listed on the markets. Price changes can be determined by stringent mathematical differential equations based on predefined assumptions as following:

**Financial Markets are irrational:**if investors are looking for maximizing their gain, turbulent movements such market collapse are stimulated and exacerbated by emotional and irrationals trading decisions.**Market Prices do not follow a linear and normal distribution:**The Modern Portfolio Theory or Mean Variance Theory as described by the famous Economist Harry Markowitz in 1952 base its theory on the assumption that markets movements are following a Gaussian curve which implied that the probability to have extreme markets fluctuations are almost non-existent.**The Capital Market is imperfect:**with a Gaussian approach, it is impossible to consider markets imperfections, which implies that historical prices do not impact future prices. The Chaos-Fractals approaches demonstrate the opposite. Indeed, Financial Markets have a memory and current prices have a direct influence on future prices. An important number of volatility on many and different time frame increase the risk of high volatility on the same future time frame.

### Core factors behind the Chaos-Fractals Theory

**The Butterfly Effect:** According to the Butterfly Effect, a butterfly in California can cause a hurricane in China materialize if the butterfly flaps its wings at just the right point in space and time. While the result isn’t immediate, the causal connection is real. The hurricane would not have happened if not for the butterfly. Expressed more generically, small changes in initial conditions can lead to drastic changes in final results. Human civilization is an ongoing demonstration of this principle in action.

**Unpredictability:** Because all the initial conditions of a complex system are not fully knowable, it is impossible to predict the fate of a complex system. Even if all the conditions can be measured, slight errors in measurements will be amplified dramatically.

As Isaac Newton said : *“I can calculate the motion of heavenly bodies, but not the madness of people”*.

**Order / Disorder:** Chaos is not simply disorder. Chaos comprises the transitions between order and disorder, which often occur in surprising ways. *“From where we stand the rain seems random. If we could stand somewhere else, we would see the order in it.”* Tony Hillerman Coyote Waits.

**Mixing:** Turbulence ensures that, in time, two adjacent points in a complex system will eventually end up in very different positions. For example, if you place one fluid into another fluid, making the potential for a mixture, then there are two main ways in which the fluids can become mixed. One is simple diffusion. However, in some fluid mixtures, diffusion is not, at least initially is not, a very fast method of mixing, on the largest length scales.

**When the flow is chaotic,** there may not be any stable streamlines at all describing the flow, and there is an extremely strong dependence of the endpoint on the initial position and velocity

### Mixing flow and Financial Markets

So when mixing is dominated by advection rather than by diffusion, the differences in the character of the mixing depend on the nature of the flow – either it is regular, or it is chaotic, or it is somewhere in between. Regarding the Financial Markets, mixing market prices on different time frame is dominated by advection and depend on a chaotic nature meaning that future market prices are extremely dependent on the endpoint of its initial position and velocity.

Geoffrey Ducournau Executive Director of Dimtech