Stabilizing periodic orbits above the elliptic plane in the solar sail 3-body problem

We consider periodic orbits high above the ecliptic plane in the Elliptic Restricted Three-Body Problem where the third massless body is a solar sail. Periodic orbits above the ecliptic are of practical interest as they are ideally positioned for the year-round constant imaging of, and communication with, the poles. Initially we identify an unstable periodic orbit by using a numerical continuation from a known periodic orbit above the ecliptic in the circular case with the eccentricity as the varying parameter. This orbit is then used to construct a reference trajectory for the sail to track. In addition we illustrate an alternative method for constructing a periodic reference trajectory based on a time-delayed feedback mechanism. The reference trajectories are then tracked using a linear feedback regulator (LQR) where the control actuation is delivered by varying the solar sails orientation. Using this method it is shown that a 'near term' solar sail is capable of performing stable periodic motions high above the ecliptic

The dynamics of the leverage cycle

We present a simple agent-based model of a financial system composed of leveraged investors such as banks that invest in stocks and manage their risk using a Value-at-Risk constraint, based on historical observations of asset prices. The Value-at-Risk constraint implies that when perceived risk is low, leverage is high and vice versa, a phenomenon that has been dubbed pro-cyclical leverage. We show that this leads to endogenous irregular oscillations, in which gradual increases in stock prices and leverage are followed by drastic market collapses, i.e. a leverage cycle. This phenomenon is studied using simplified models that give a deeper understanding of the dynamics and the nature of the feedback loops and instabilities underlying the leverage cycle. We introduce a flexible leverage regulation policy in which it is possible to continuously tune from pro-cyclical to countercyclical leverage. When the policy is sufficiently countercyclical and bank risk is sufficiently low the endogenous oscillation disappears and prices go to a fixed point. While there is always a leverage ceiling above which the dynamics are unstable, countercyclical leverage can be used to raise the ceiling. We also study the impact on leverage cycles of direct, temporal control of the bank's riskiness via the bank's required Value-at-Risk quantile. Under such a rule the regulator relaxes the Value-at-Risk quantile following a negative stock price shock and tightens it following a positive shock. While such a policy rule can reduce the amplitude of leverage cycles, its effectiveness is highly dependent on the choice of parameters. Finally, we investigate fixed limits on leverage and show how they can control the leverage cycle.Comment: 35 pages, 9 figure

Stabilizing equilibrium by linear feedback control for controlling chaos in Chen system

Stabilization of a chaotic system in one of its unstable equilibrium points by applying small perturbations is studied. A two-stage control strategy based on linear feedback control is applied. Improvement of system performance is addressed by exploiting the ergodicity of the original dynamics and using Lyapunov stability results for control design. Extension to the not complete observability case is also analyzed.Facultad de Ingenierí

Stabilizing equilibrium by linear feedback control for controlling chaos in Chen system

Stabilization of a chaotic system in one of its unstable equilibrium points by applying small perturbations is studied. A two-stage control strategy based on linear feedback control is applied. Improvement of system performance is addressed by exploiting the ergodicity of the original dynamics and using Lyapunov stability results for control design. Extension to the not complete observability case is also analyzed.Facultad de Ingenierí

Nonlinear Analysis and Control of Interleaved Boost Converter Using Real-Time Cycle to Cycle Variable Slope Compensation

Switched-mode power converters are inherently nonlinear and piecewise smooth systems that may exhibit a series of undesirable operations that can greatly reduce the converter's efficiency and lifetime. This paper presents a nonlinear analysis technique to investigate the influence of system parameters on the stability of interleaved boost converters. In this approach, Monodromy matrix that contains all the comprehensive information of converter parameters and control loop can be employed to fully reveal and understand the inherent nonlinear dynamics of interleaved boost converters, including the interaction effect of switching operation. Thereby not only the boundary conditions but also the relationship between stability margin and the parameters given can be intuitively studied by the eigenvalues of this matrix. Furthermore, by employing the knowledge gained from this analysis, a real-Time cycle to cycle variable slope compensation method is proposed to guarantee a satisfactory performance of the converter with an extended range of stable operation. Outcomes show that systems can regain stability by applying the proposed method within a few time periods of switching cycles. The numerical and analytical results validate the theoretical analysis, and experimental results verify the effectiveness of the proposed approach

Exploitation of renewable resources with differentiated technologies: An evolutionary analysis

In this paper, we propose a dynamical model of technology adoption for the exploitation of a renewable natural resource. Each technology has a different efficiency and environmental impact. The process of technology adoption over time is modeled through an evolutionary game employed by profit maximizing exploiters. The loss in profits due to lower efficiency levels of environmentally-friendly technologies can be counterbalanced by the higher consumers' propensity to pay for greener goods. The dynamics of the resource take place in continuous time, whereas the technology choice can be revised either in continuous-time or in discrete-time. In the latter case, the model assumes the form of a hybrid system, whose dynamics is mainly explored numerically. We shows that: (1) overexploitation of the resource arises whenever the reduction in harvesting due to a lower efficiency of clean technology is more than compensated by a higher propensity to pay for greener goods; (2) the difference between the fixed costs of these technologies can be exogenously fixed to provide an incentive for adopting clean technology without affecting the long-run level of the resource; and (3) in some cases, discrete switching of the technology causes overshooting in the dynamics whereas in others it enhances the stability of the system

Control of chaotic dynamics in an OLG economic model

WOS:000273142000019 (Nº de Acesso Web of Science)This paper deals with the control of chaotic economic motion. We show that very complicated dynamics arising, e.g., from an overlapping generations model (OLG) with production and an endogenous intertemporal decision between labour and leisure, which produces chaos, can in fact be controlled with relative simplicity. The aperiodic and very complicated motion that stems from this model can be subject to control by small perturbations in its parameters and turned into a stable steady state or into a regular cycle. Therefore, the system can be controlled without changing of its original properties. To perform the control of the totally unstable equilibrium (both eigenvalues with modulus greater than unity) in this economic model we apply the pole-placement technique, developed by Romeiras, Grebogi, Ott and Dayawansa (1992). The application of control methods to chaotic economic dynamics may raise serious reservations, at least on mathematical and logical grounds, to some recent views on economics which have argued that economic policy becomes useless in the presence of chaotic motion (and thus, that the performance of the economic system cannot be improved by public intervention, i.e., that the amplitude of cycles can not be controlled or reduced). In fact, the fine tuning of the system (that is, the control) can be performed without having to rely only on infinitesimal accuracy in the perturbation to the system, because the control can be performed with larger or smaller perturbations, but neither too large (because these would lead to a different fixed point of the system, therefore modifying its original nature), nor too small because the control becomes too inefficient

Stabilizing equilibrium by linear feedback control for controlling chaos in Chen system

Stabilization of a chaotic system in one of its unstable equilibrium points by applying small perturbations is studied. A two-stage control strategy based on linear feedback control is applied. Improvement of system performance is addressed by exploiting the ergodicity of the original dynamics and using Lyapunov stability results for control design. Extension to the not complete observability case is also analyzed.Facultad de Ingenierí