I am modifying much of this discussion from a paper which is currently under consideration for publication.

In the last installment we saw that certain complex systems are characterized by multiple modes of operation. Assuming that our system can be defined as a continuous system of differential equations, then it will evolve deterministically from each uniquely defined initial point to a unique sequence of successor states, implying that two states which lie on different trajectories will remain so—hence, no line crossings occur [Hirsh and Smale, 1974].

Any two different trajectories may evolve toward successor states which are arbitrarily close to one another. Thus they may converge toward a single state which does not change in time. This unchanging state is called an attractor, and the behavior is described as asymptotic stability, as the system tends to evolve asymptotically towards some immovable point. Alternatively, they may not converge onto a particular point, but the successor states from any state within a small region of phase space may stay within a small (but possibly larger) region of phase space. Such behavior is described as Lyapunov stability.

The conditions by which asymptotic stability occurs are extremely specific, and it is difficult on the basis of field observations of the climate system or of the stock market to be certain that our observed system demonstrates such behaviour.

A qualitative approach to interpreting dynamic systems includes describing the phase space in terms of the type, order, and number of attractors (or areas of stability) that are traced out as the system evolves through time. The ice volume phase space portrait of last time is an example of a system with a number of disjoint Lyapunov-stable areas (LSA), each separated in phase space by a separatrix. At any given time, the state of the system occupies only one such LSA, so that their number therefore constitutes the total number of alternative long-term behaviors, or equilibrium states, of the system.

Since an LSA is likely to be smaller than the total allowable range of states, the system tends to become boxed into an LSA unless it is subjected to external forcing. When the state approaches a separatrix, small perturbations can trigger a change to a nearby state, which can result in chaotic changes in the evolution of the system [Parker and Chua, 1989]. Thus very complex behavior can arise in multistable systems.

The typical approach is to label the LSAs and characterize the system as a series of steps from one LSA to another. If the LSAs in the figure are labelled (from lower left to upper right) A2, A3, A4, and A5, then the curve above might be characterized as evolving from A2 to A4 to A5, and is currently heading in the general direction of A2 once again.

The evolution from one LSA to another would not occur except in the presence of external forcing, which successively drives the system across a separatrix after which it evolves quickly towards some other LSA. Now let us consider application of this approach to the stock market.

Over long periods of time, a stock will typically trade within a range. Provided there is to be no change in the fortunes of the company, we would normally expect that small perturbations in the price will be countered. In this way, the stock price exhibits Lyapunov or even asymptotic stability. The market has a "perception" of the value of the stock, and any deviation from that value is arbitraged away. Arbitrageurs will therefore act as the negative feedback cycles that we infer for a complex system.

Nevertheless, the external forcing (information in the form of money) may be sufficient to perturb the stock price over a separatrix at which point it suddenly accelerates toward some new area of phase space. We would probably say that the stock has become a "momentum play", dominated by the momentum players who continue to push the stock rapidly in whatever direction it happens to be moving.

All momentum plays eventually come to an end, and if there have actually been no changes in the fortunes of the company, the reasonable expectation is for the price of the stock to return to its previous trading range. But there is no guarantee that it will do so by the most direct path.

We see two possible scenarios after the breakout. Infinite variety is possible, especially in terms of the excursion through phase space. For an example of a wild one, let's look at one stock. Should I name it? In the interests of full disclosure, the stock in question is Nautilus Minerals Inc. (NUS-V), which was possibly the object of a recent price manipulation. (Kudos also to IKN for this story). I have no position in Nautilus, but am on management of a company that might be perceived as a competitor (but isn't really).

We see a general downward trend until that rather singular spike corresponding to the punch line of an interesting promotion. Let's look at the two dimensional reconstructed phase space portrait.

Two dimensional phase space portrait of the NUS-V stock price since April. Lag is four trading days. There is a lot of dynamical information going on here, which I will go through in part 3 of this post, but note the two highlighted areas which may represent areas of Lyapunov stability, and that prior to the unusual spike of early July, the price trajectory appeared to be returning to the LSA that the price state held in April. DYODD.

References

Gipp, M. R., 2001. Interpretation of climate dynamics from phase space portraits: Is the climate system strange or just different?

Hirsh, M. W. and S. Smale, 1974.

Parker, T. S., and L. O. Chua, 1989.

In the last installment we saw that certain complex systems are characterized by multiple modes of operation. Assuming that our system can be defined as a continuous system of differential equations, then it will evolve deterministically from each uniquely defined initial point to a unique sequence of successor states, implying that two states which lie on different trajectories will remain so—hence, no line crossings occur [Hirsh and Smale, 1974].

Any two different trajectories may evolve toward successor states which are arbitrarily close to one another. Thus they may converge toward a single state which does not change in time. This unchanging state is called an attractor, and the behavior is described as asymptotic stability, as the system tends to evolve asymptotically towards some immovable point. Alternatively, they may not converge onto a particular point, but the successor states from any state within a small region of phase space may stay within a small (but possibly larger) region of phase space. Such behavior is described as Lyapunov stability.

The conditions by which asymptotic stability occurs are extremely specific, and it is difficult on the basis of field observations of the climate system or of the stock market to be certain that our observed system demonstrates such behaviour.

A qualitative approach to interpreting dynamic systems includes describing the phase space in terms of the type, order, and number of attractors (or areas of stability) that are traced out as the system evolves through time. The ice volume phase space portrait of last time is an example of a system with a number of disjoint Lyapunov-stable areas (LSA), each separated in phase space by a separatrix. At any given time, the state of the system occupies only one such LSA, so that their number therefore constitutes the total number of alternative long-term behaviors, or equilibrium states, of the system.

Since an LSA is likely to be smaller than the total allowable range of states, the system tends to become boxed into an LSA unless it is subjected to external forcing. When the state approaches a separatrix, small perturbations can trigger a change to a nearby state, which can result in chaotic changes in the evolution of the system [Parker and Chua, 1989]. Thus very complex behavior can arise in multistable systems.

*Probability density plot of reconstructed phase space portrait of the ice volume system compiled from the past 750 thousand years (filled solids) superimposed on the phase space plot of the last 120 thousand years (dashed curve). Regions of high probability (darker) represent LSAs and result from multiple visits to the same region of phase space, or by a drop the rate of evolution of the system.**From Gipp (2001).*The typical approach is to label the LSAs and characterize the system as a series of steps from one LSA to another. If the LSAs in the figure are labelled (from lower left to upper right) A2, A3, A4, and A5, then the curve above might be characterized as evolving from A2 to A4 to A5, and is currently heading in the general direction of A2 once again.

The evolution from one LSA to another would not occur except in the presence of external forcing, which successively drives the system across a separatrix after which it evolves quickly towards some other LSA. Now let us consider application of this approach to the stock market.

Over long periods of time, a stock will typically trade within a range. Provided there is to be no change in the fortunes of the company, we would normally expect that small perturbations in the price will be countered. In this way, the stock price exhibits Lyapunov or even asymptotic stability. The market has a "perception" of the value of the stock, and any deviation from that value is arbitraged away. Arbitrageurs will therefore act as the negative feedback cycles that we infer for a complex system.

*Reconstructed phase space portrait (price vs. lagged price) showing the trajectory traced out by one stock near a Lyapunov-stable area (LSA). Small arrows show the evolution of the system through time.*Nevertheless, the external forcing (information in the form of money) may be sufficient to perturb the stock price over a separatrix at which point it suddenly accelerates toward some new area of phase space. We would probably say that the stock has become a "momentum play", dominated by the momentum players who continue to push the stock rapidly in whatever direction it happens to be moving.

*Price chart for the stock in the above phase space. The momentum players have carried it out of a trading range. How high can it go?*All momentum plays eventually come to an end, and if there have actually been no changes in the fortunes of the company, the reasonable expectation is for the price of the stock to return to its previous trading range. But there is no guarantee that it will do so by the most direct path.

*Possible scenarios by which stock price may return back to a trading range after breaking out and momentum later fails. In scenario 1 the stock falls back to the LSA. In scenario 2 the stock goes on an excursion through phase space before returning to the trading range. In this example we are assuming that there has been no change in the perceived value of the stock.*We see two possible scenarios after the breakout. Infinite variety is possible, especially in terms of the excursion through phase space. For an example of a wild one, let's look at one stock. Should I name it? In the interests of full disclosure, the stock in question is Nautilus Minerals Inc. (NUS-V), which was possibly the object of a recent price manipulation. (Kudos also to IKN for this story). I have no position in Nautilus, but am on management of a company that might be perceived as a competitor (but isn't really).

*Nautilus Minerals Inc. share price for the past four months in pence (sorry about that!)*We see a general downward trend until that rather singular spike corresponding to the punch line of an interesting promotion. Let's look at the two dimensional reconstructed phase space portrait.

Two dimensional phase space portrait of the NUS-V stock price since April. Lag is four trading days. There is a lot of dynamical information going on here, which I will go through in part 3 of this post, but note the two highlighted areas which may represent areas of Lyapunov stability, and that prior to the unusual spike of early July, the price trajectory appeared to be returning to the LSA that the price state held in April. DYODD.

References

Gipp, M. R., 2001. Interpretation of climate dynamics from phase space portraits: Is the climate system strange or just different?

*Paleoceanography*,**16:**335-351.Hirsh, M. W. and S. Smale, 1974.

__Differential Equations, Dynamical Systems and Linear Algebra__, Academic, San Diego, Calif., 1974.Parker, T. S., and L. O. Chua, 1989.

__Practical Numerical Algorithms for Chaotic Systems,__Springer-Verlag, New York.
## No comments:

## Post a Comment