Observations of Chaotic Behaviour in Nonlinear Inventory Models

This article describes the use of simulation to investigate incipient chaotic behaviour in inventory models. Model structures investigated were either capacity limited or of variable delay time, implemented in discrete and continuous transform algebras. Results indicate the absence of chaos for a continuous time model but gave limited evidence for chaos in both unrestricted discrete models and those with a positive orders only limit. The responses where interaction with the capacity limit occurred did not confirm chaotic behaviour at odds with published results. Using the Liapunov exponent as a measure of chaotic behaviour, the results indicated, where the delay varies in proportion to order rate, a larger fixed delay reduced the Liapunov exponent as did increasing the dependence of delay on order rate. The effect of the model structures showed that the IOBPCS model, produced the largest Liapunov exponent. Reducing the discrete model update time reduced the Liapunov exponent.

TWO-DIMENSIONAL DYNAMICAL SYSTEMS WITH PERIODIC COEFFICIENTS

This paper is devoted to the properties conferred on phase space by a vector field of generic second order autonomous dynamical systems with periodic coefficients, called parametric autonomous dynamical systems (PADS). At first, an associated periodical parametric linear equation (APPLE) is defined at every point of the phase plane. The exact value of the Floquet–Liapunov exponent of the APPLE is computed using a fast algorithm, without integration. The role of Floquet–Liapunov exponents is known to establish the stability of periodic solutions. In this work, it is pointed out that, under certain conditions, they bring information on local characteristics of PADS solutions according to their location in the phase plane, such as sensitivity to initial conditions, oscillation frequency, period doubling, parametric resonance, funneling. Then, an invariant manifold of an associated constant coefficients equivalent system (ACCES) is defined. It is shown that this manifold is periodically crossed by solutions of the PADS. This manifold crossing property contributes to the structure of the PADS solutions in phase plane. The implementation of this method is shown on a Van der Pol equation with a periodic coefficient in order to illustrate all kinds of solution patterns near the manifold in the phase plane according to the Floquet–Liapunov exponent local value. The manifold crossing property can be observed in all cases. Then, a parametric Duffing equation is processed. A numerical study shows some chaos routes, their bifurcation diagram and the top Liapunov exponent variations. The ACCES of the Duffing equation does not have any slow manifold. However, the Floquet–Liapunov exponent computation allows to specify the locus in the phase plane where the curvature of the trajectories changes, giving rise to chaos.

Synchronization of chaotic symmetric gyros by one-way coupling conditions

A simple satellite is a symmetric gyro. Satellites are used for communications, weather forecasts, surveying and in several other scientific and military applications. In particular, for military use secure communications are very important. As chaotic signals are usually broadband and noise like, synchronized chaotic systems can be used as cipher generators for secure communication. This paper presents chaos synchronization of two identical chaotic motions of symmetric gyros. It has been demonstrated that applying four different kinds of one-way coupling conditions can synchronize two identical chaotic systems. The sign of the sub-Liapunov exponent has been used as an indicator for the occurrence of chaos synchronization. It has also been found that when chaos synchronization occurs the positive major sub-Liapunov exponent transverses the zero value and becomes negative for the last time. Chaos synchronization can also be shown by phase trajectory. In addition, synchronization time is also examined. Furthermore, it has been found that different distances of the initial conditions between the master system and slave system do not affect the occurrence of chaos synchronization.

Coexistence of Periodic Roll Motion and Chaotic One in a Forced Flooded Ship

This letter describes the coexistence of periodic and chaotic roll motion of a flooded ship in waves. We found experimentally, both with a flooded ferry model and with a simplified box-shaped model, that the two types of roll motion can coexist under the same wave condition. A trajectory reconstructed in a delay-coordinate state space from the time series data of the measured roll angle looks like a low-dimensional strange attractor. Moreover, a mathematical model for the simplified box-shaped ship shows the coexistence of a periodic solution and a chaotic one with a positive maximum Liapunov exponent.