Dynamics of partial control

Control of escapes in two-degree-of-freedom open Hamiltonian systems

We investigate the possibility of avoiding the escape of chaotic scattering trajectories in two-degree-of-freedom Hamiltonian systems. We develop a continuous control technique based on the introduction of coupling forces between the chaotic trajectories and some periodic orbits of the system. The main results are shown through numerical simulations, which confirm that all trajectories starting near the stable manifold of the chaotic saddle can be controlled. We also show that it is possible to jump between different unstable periodic orbits until reaching a stable periodic orbit belonging to a Kolmogorov-Arnold-Moser island.

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.

Horizons of cybernetical physics

The subject and main areas of a new research field-cybernetical physics-are discussed. A brief history of cybernetical physics is outlined. The main areas of activity in cybernetical physics are briefly surveyed, such as control of oscillatory and chaotic behaviour, control of resonance and synchronization, control in thermodynamics, control of distributed systems and networks, quantum control.This article is part of the themed issue 'Horizons of cybernetical physics'.

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'.

Avoiding healthy cells extinction in a cancer model

We consider a dynamical model of cancer growth including three interacting cell populations of tumor cells, healthy host cells and immune effector cells. For certain parameter choice, the dynamical system displays chaotic motion and by decreasing the response of the immune system to the tumor cells, a boundary crisis leading to transient chaotic dynamics is observed. This means that the system behaves chaotically for a finite amount of time until the unavoidable extinction of the healthy and immune cell populations occurs. Our main goal here is to apply a control method to avoid extinction. For that purpose, we apply the partial control method, which aims to control transient chaotic dynamics in the presence of external disturbances. As a result, we have succeeded to avoid the uncontrolled growth of tumor cells and the extinction of healthy tissue. The possibility of using this method compared to the frequently used therapies is discussed.

Introduction to the focus issue: fifty years of chaos: applied and theoretical

The discovery of deterministic chaos in the late nineteenth century, its subsequent study, and the development of mathematical and computational methods for its analysis have substantially influenced the sciences. Chaos is, however, only one phenomenon in the larger area of dynamical systems theory. This Focus Issue collects 13 papers, from authors and research groups representing the mathematical, physical, and biological sciences, that were presented at a symposium held at Kyoto University from November 28 to December 2, 2011. The symposium, sponsored by the International Union of Theoretical and Applied Mechanics, was called 50 Years of Chaos: Applied and Theoretical. Following some historical remarks to provide a background for the last 50 years, and for chaos, this Introduction surveys the papers and identifies some common themes that appear in them and in the theory of dynamical systems.