Partial control of chaotic systems

F. Sanjuán M. When the firm prevents the crash: Avoiding market collapse with partial control

Market collapse is one of the most dramatic events in economics. Such a catastrophic event can emerge from the nonlinear interactions between the economic agents at the micro level of the economy. Transient chaos might be a good description of how a collapsing market behaves. In this work, we apply a new control method, the partial control method, with the goal of avoiding this disastrous event. Contrary to common control methods that try to influence the system from the outside, here the market is controlled from the bottom up by one of the most basic components of the market—the firm. This is the first time that the partial control method is applied on a strictly economical system in which we also introduce external disturbances. We show how the firm is capable of controlling the system avoiding the collapse by only adjusting the selling price of the product or the quantity of production in accordance to the market circumstances. Additionally, we demonstrate how a firm with a large market share is capable of influencing the demand achieving price stability across the retail and wholesale markets. Furthermore, we prove that the control applied in both cases is much smaller than the external disturbances.

Partially controlling transient chaos in the Lorenz equations

Transient chaos is a characteristic behaviour in nonlinear dynamics where trajectories in a certain region of phase space behave chaotically for a while, before escaping to an external attractor. In some situations, the escapes are highly undesirable, so that it would be necessary to avoid such a situation. In this paper, we apply a control method known as partial control that allows one to prevent the escapes of the trajectories to the external attractors, keeping the trajectories in the chaotic region forever. We also show, for the first time, the application of this method in three dimensions, which is the major step forward in this work. To illustrate how the method works, we have chosen the Lorenz system for a choice of parameters where transient chaos appears, as a paradigmatic example in nonlinear dynamics. We analyse three quite different ways to implement the method. First, we apply this method by building an one-dimensional map using the successive maxima of one of the variables. Next, we implement it by building a two-dimensional map through a Poincaré section. Finally, we built a three-dimensional map, which has the advantage of using a fixed time interval between application of the control, which can be useful for practical applications.This article is part of the themed issue 'Horizons of cybernetical physics'.

Dynamics of partial control

Safe sets are a basic ingredient in the strategy of partial control of chaotic systems. Recently we have found an algorithm, the sculpting algorithm, which allows us to construct them, when they exist. Here we define another type of set, an asymptotic safe set, to which trajectories are attracted asymptotically when the partial control strategy is applied. We apply all these ideas to a specific example of a Duffing oscillator showing the geometry of these sets in phase space. The software for creating all the figures appearing in this paper is available as supplementary material.

Exploring partial control of chaotic systems

In this paper we make a thorough exploration of the technique of partial control of chaotic systems. This control technique allows one to keep the trajectories of a dynamical system close to a chaotic saddle even if the control applied is smaller than the effects of environmental noise in the system, provided that the chaotic saddle is due to the existence of a horseshoelike mapping in phase space. We state this here in a mathematically precise way using the Conley-Moser conditions, and we prove that they imply that our partial control strategy can be applied. We also give an upper bound of the control-noise ratio needed to achieve this goal, and we describe how this technique can be applied for large noise values. Finally, we study in detail the effect of imperfect targeting in our control technique. All these results are illustrated numerically with the paradigmatic Hénon map.