If the Lyapunov exponent is positive then the system is chaotic and unstable. Nearby points will diverge irrespective of how close they are.
It needs to be calculated for each and every trigger nearby in order to find out which direction the system is unstable.
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The Lyapunov exponent
If the Lyapunov exponent is less than zero then the system attracts to a fixed point or stable periodic orbit. Such systems are dissipative.
The Lyapunov exponent
In order for a system to exhibit chaotic behavior it must be non linear.
Why is the Lyapunov exponent so important?
The Lyapunov exponent can show the future behavior of a chaotic system.
We do not want to use a stoploss which is good for stochastic systems. We want to avoid the discomfort of opening a position in the wrong direction rather than to determine how big our probable loss can be.
We do not want to deal with probablilities or gambling. Chaos is on our side, provided we know what to do with the new information which is being extracted from the (chaotic) system.
Note: the term " new information " should be understood in a mathematical way, it has nothing to do with the popular term "information" used by mass media. The new information in terms of chaos has its unique bit expression.
We do not want to use a stoploss which is good for stochastic systems. We want to avoid the discomfort of opening a position in the wrong direction rather than to determine how big our probable loss can be.
We do not want to deal with probablilities or gambling. Chaos is on our side, provided we know what to do with the new information which is being extracted from the (chaotic) system.
Note: the term " new information " should be understood in a mathematical way, it has nothing to do with the popular term "information" used by mass media. The new information in terms of chaos has its unique bit expression.
Last edited by Paul&Paul on Fri Nov 02, 2012 9:04 am, edited 1 time in total.
The Lyapunov exponent
For stable fixed points and cycles the Lyapunov exponent is negative. This mode of the system prompts contrarian trades all the time. Such trades are open to unlimited risk if you do not know when to stop it.
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The Lyapunov exponent
For chaotic attractors the Lyapunov exponent is positive. In other words when the exponent turns positive a trend begins. Therefore positive Lyapunov exponents are associated with trends of various strength and of various length (of time).
Chaotic systems enable to predict a trend's potential strength but nothing can be said about their length (of time).
Chaotic systems enable to predict a trend's potential strength but nothing can be said about their length (of time).
Last edited by Paul&Paul on Fri Nov 02, 2012 8:49 am, edited 1 time in total.
Unimodal maps
No matter what unimodal map is iterated, the same fractal expansion appears.
F=4.669... and the other multipliers are universal.
The algebraic form of the map is irrelevant.
Metropolis proved that all unimopdal maps have periodic attractors occurring in the same sequence.
Real systems often have tremendously many degrees of freedom. All that complexity can be captured by a 1dimensional map.
The Lyapunov exponent captures the average rate of divergence (or convergence) from a UPO.
F=4.669... and the other multipliers are universal.
The algebraic form of the map is irrelevant.
Metropolis proved that all unimopdal maps have periodic attractors occurring in the same sequence.
Real systems often have tremendously many degrees of freedom. All that complexity can be captured by a 1dimensional map.
The Lyapunov exponent captures the average rate of divergence (or convergence) from a UPO.
In conclusion
The Lyapunov exponent enables traders to open their positions
(1) in the direction of local instability, and
(2) in the place drawdown risk is minimum.
Traditionally business cycle research treats fluctuations as deviations from a steady state caused by exogenous "shocks" (like monetary policy changes, for example).
Chaotic systems dynamics research treats fluctuations as escapes from Unsteady Periodic Orbits caused by endogenous "triggers".
(1) in the direction of local instability, and
(2) in the place drawdown risk is minimum.
Traditionally business cycle research treats fluctuations as deviations from a steady state caused by exogenous "shocks" (like monetary policy changes, for example).
Chaotic systems dynamics research treats fluctuations as escapes from Unsteady Periodic Orbits caused by endogenous "triggers".
Predictability
If there is nonlinearity or chaos, then the exciting possibility of forecasting asset ptices exists.
Example:
For EURUSD M30 chart, the Lyapunov exponent corresponding to the trigger down (17pips) turns positive at 1.2982. It is a sell signal.
The drawdown was nil.
The return to it, typical for UPOs, reached exactly that 1.2982 and not a pip more and the pair slid lower again.
Similarly, for the trigger up marked earlier, the Lyapunov exponent turned positive at 1.2924. It was a buy signal. Again, the drawdown was nil. That trigger up expanded 9.1299 times its size.
The drawdown was nil.
The return to it, typical for UPOs, reached exactly that 1.2982 and not a pip more and the pair slid lower again.
Similarly, for the trigger up marked earlier, the Lyapunov exponent turned positive at 1.2924. It was a buy signal. Again, the drawdown was nil. That trigger up expanded 9.1299 times its size.
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