Lyapunov Exponent Map Research Articles

Understanding flow around planetary moons via finite-time Lyapunov exponent maps

Understanding flow around planetary moons via finite-time Lyapunov exponent maps

Hoeffding’s independence test for an ion dynamics characterization in the octupole trap

In the present work, we propose a rapid analysis of the trapped ion dynamics regimes. The Hoeffding’s Independence Test (HIT) mapping technique has been considered as a promising approach for the detection of dynamic regimes. As a model problem, we have applied the proposed method to ion dynamics in an octupole ion trap. We compare the results of HIT mapping with the results of direct calculation of the non-trivial Lyapunov exponent. The HIT mapping result well agree with a Lyapunov exponent mapping. The computational time for plotting the HIT maps was significantly less than the computational time for plotting the Lyapunov exponent maps.

Period-doubling bifurcation in surface radio-frequency trap: Transition to chaos through Feigenbaum scenario.

We have numerically investigated the dynamics of charged microparticles in a "five-wire" surface radio-frequency trap. The period-doubling bifurcation conditions have been shown to depend on the particle, the trap, and the alternating voltage parameters. For a comprehensive study of the dynamics chaotization through a cascade of period doubling, we have used Fourier analysis of a particle trajectory as well as the calculations of a non-trivial Lyapunov exponent map. We have demonstrated that the period-doubling bifurcation is consistent with a Feigenbaum scenario. A new approach to particle property determination can, thus, be based on observing a period-doubling bifurcation.

Transfers Between Moons with Escape and Capture Patterns via Lyapunov Exponent Maps

This contribution focuses on the design of low-energy transfers between planetary moons and presents an efficient technique to compute trajectories characterized by desirable behaviors in the vicinities of the departure and destination bodies. The method utilizes finite-time Lyapunov exponent maps in combination with the moon-to-moon analytical transfer method previously proposed by the authors. The integration of these two components facilitates the design of direct transfers between moons within the context of the circular restricted three-body problem, and allows the inclusion of a variety of trajectory patterns, such as captures, landings, transits, and takeoffs, at the two ends of a transfer. The foundations and properties of the technique are illustrated through an application based on impulsive direct transfers between Ganymede and Europa. However, the methodology can be employed to assist in the design of more complex mission scenarios, such as moon tours.

Nonlinear Dynamics and Control of Time-Delay Supercavitating Vehicle

This paper investigates the nonlinear dynamical behaviors of a time-delayed supercavitating vehicle system varying with the system parameters, and proposes control strategies to stabilize the trajectory. A simplified dynamical model of the supercavitating vehicle is firstly constructed and then analyzed on its local and global stability in terms of different parameter values and initial conditions. Due to the forces resulting from the fins and the afterbody plane causing one of the system parameters to become a piecewise constant function of position, the system is described as a linear switched one with time delay. Both theoretical analysis and numerical simulations are adopted to describe the system behaviors. Several special phenomena such as bifurcations, deviation behaviors and period-doubling processes emerging from the system for a given set of parameters are discussed in detail in this paper. Besides, the equilibrium and chaotic solutions are accurately located by plotting the Lyapunov exponent map. Chaotic attractors present different strange forms with the variation of the parameters. Finally, based on the system’s global property, the control and anti-control strategies are proposed to stabilize the system or generate desirable chaos and it is shown that the system can be controlled by selecting appropriate control parameters.

Capturing an Asteroid via Triangular Libration Points

This paper develops a new asteroid capture mechanism via triangular libration points based on their stable structure and chaotic motion nearby. Different from collinear libration points, the triangular libration points are linearly stable surrounded by long- and short-period orbit families in the planar case. First, their dynamical structure is explored by the Lie series method and an analytical approximation of long-period orbits is proposed. Then, massive escaping trajectories from the parking orbits are generated by a chaotic-assisted method on the finite time Lyapunov exponent map. The trajectories from the asteroid’s natural orbit to the Earth–moon system are constructed by modified Lambert transfers. Next, the fuel consumption for all potential near-Earth asteroids is evaluated ergodically in both perihelion and aphelion cases based on their initial condition, which guides the selection of the target. Finally, it is applied to 20 asteroids, and the minimum fuel consumption () appears capturing asteroid 2015DU. Furthermore, a postcapture maneuver strategy based on a Poincaré section is proposed using long-period orbits to adjust the parking orbits and to satisfy the mission requirements.

Embodied neuromechanical chaos through homeostatic regulation.

In this paper, we present detailed analyses of the dynamics of a number of embodied neuromechanical systems of a class that has been shown to efficiently exploit chaos in the development and learning of motor behaviors for bodies of arbitrary morphology. This class of systems has been successfully used in robotics, as well as to model biological systems. At the heart of these systems are neural central pattern generating (CPG) units connected to actuators which return proprioceptive information via an adaptive homeostatic mechanism. Detailed dynamical analyses of example systems, using high resolution largest Lyapunov exponent maps, demonstrate the existence of chaotic regimes within a particular region of parameter space, as well as the striking similarity of the maps for systems of varying size. Thanks to the homeostatic sensory mechanisms, any single CPG "views" the whole of the rest of the system as if it was another CPG in a two coupled system, allowing a scale invariant conceptualization of such embodied neuromechanical systems. The analysis reveals chaos at all levels of the systems; the entire brain-body-environment system exhibits chaotic dynamics which can be exploited to power an exploration of possible motor behaviors. The crucial influence of the adaptive homeostatic mechanisms on the system dynamics is examined in detail, revealing chaotic behavior characterized by mixed mode oscillations (MMOs). An analysis of the mechanism of the MMO concludes that they stems from dynamic Hopf bifurcation, where a number of slow variables act as "moving" bifurcation parameters for the remaining part of the system.

Surface mixing and biological activity in the North-West African upwelling.

Near-shore water along the North-West African margin is one of the world's major upwelling regions. It is associated with physical structures of oceanic fronts which influence the biological productivity. The study of these coherent structures in connection with chlorophyll concentration data is of fundamental importance for understanding the spatial distributions of the plankton. In this work, we study the horizontal stirring and mixing in different upwelling areas using Lagrangian coherent structures (LCSs). These LCSs are calculated using the recent geodesic theory of LCSs. We use these LCSs to study the link between the chlorophyll fronts concentrations and surface mixing, based on 10 years of satellite data. These LCSs move with the flow as material lines, thus the horizontal mixing is calculated from the intersection of these LCSs with the finite time Lyapunov exponent maps. We compare our results with those of a recent study conducted over the same area, but based on finite size Lyapunov exponents (FSLEs), whose output is a plot of scalar distributions. We discuss the differences between FSLE and geodesic theory of LCS. The latter yields analytical solutions of LCSs, while FSLEs can only provide LCSs for sharp enough ridges of nearly constant height.

Analysis of vibration characteristics and adaptive continuous perturbation control of some torsional vibration system with backlash

The dynamic equation of some nonlinear torsional vibration system with two masses is established, which contains backlash. In the case of primary resonance, the frequency response equation of the system is deduced with the modified Lindstedt–Poincare method combined with the multiple scales method. The influence of the backlash change on the amplitude of the torsional vibration system is analyzed by the amplitude frequency response map, and by using the method of numerical analysis, the influence of the backlash change on the system entering the chaotic motion is analyzed by bifurcation diagram, the maximum Lyapunov exponent map, phase diagram and Poincare map. After adding the method of the adaptive continuous perturbation control, the amplitude of the system decreases, and there is a transformation from chaotic motion to periodic motion.

Robust nonlinear feedback control of a chaotic permanent-magnet synchronous motor with a load torque disturbance

The permanent-magnet synchronous motor (PMSM) system, which is a nonlinear dynamic system, will demonstrate a variety of chaotic phenomena when its parameters or external inputs fall into a certain area, which will lead to a deterioration of its performance. Thus, chaos should be suppressed or eliminated. In this paper, the property of equilibrium points is analyzed, and the condition for the occurrence of a Hopf bifurcation in a PMSM system is given based on a mathematical model of the PMSM system with a bifurcation diagram, a Lyapunov exponent map and phase plane diagrams given. After the drawbacks of the existing control methods have been analyzed, a robust nonlinear feedback controller is designed to control the chaos in the PMSM system with a load torque disturbance. The object is to eliminate the chaos and to drive the system speed to a desired value, Numerical simulation proves the validity of this control method.

Adaptive nonlinear feedback control of chaos in permanent-magnet synchronous motor system with parametric uncertainty

The permanent-magnet synchronous motor (PMSM) system, which is a nonlinear dynamic system, will exhibit a variety of chaotic or limit-cycle phenomenon under some choices in system parameters and external disturbances and its chaotic characteristics will become obvious. Based on the mathematical model of the PMSM system, the property of equilibrium points is analyzed and the relationship between Hopf bifurcation and the system parameters associated with control parameters is illustrated. In addition, bifurcation diagram, Lyapunov exponent map, and phase plane diagram are also presented in this paper. An adaptive nonlinear feedback controller, which could estimate the system parameters online, is then designed to eliminate the chaos and drive the speed of PMSM to a desired value in presence of system parametric uncertainty. Numerical simulation proves that the proposed control method has a better controlling effect than the general nonlinear feedback controller.

Synchronisation in the phase model of three coupled lasers

The problem of synchronisation of three lasers is considered within the phase approximation. The domains of complete synchronisation, partial synchronisation, two-frequency resonant regimes, and three-frequency quasi-periodicity have been found using bifurcation analysis, the method of Lyapunov exponent maps, and construction of phase portraits. The differences in the properties of a three-element chain and ring, as well as the influence of the coupling type, are discussed.

Chaotic operation and chaos control of travelling wave ultrasonic motor

The travelling wave ultrasonic motor, which is a nonlinear dynamic system, has complex chaotic phenomenon with some certain choices of system parameters and external inputs, and its chaotic characteristics have not been studied until now. In this paper, the preliminary study of the chaos phenomenon in ultrasonic motor driving system has been done. The experiment of speed closed-loop control is designed to obtain several groups of time sampling data sequence of the amplitude of driving voltage, and phase-space reconstruction is used to analyze the chaos characteristics of these time sequences. The largest Lyapunov index is calculated and the result is positive, which shows that the travelling wave ultrasonic motor has chaotic characteristics in a certain working condition Then, the nonlinear characteristics of travelling wave ultrasonic motor are analyzed which includes Lyapunov exponent map, the bifurcation diagram and the locus of voltage relative to speed based on the nonlinear chaos model of a travelling wave ultrasonic motor. After that, two kinds of adaptive delay feedback controllers are designed in this paper to control and suppress chaos in USM speed control system. Simulation results show that the method can control unstable periodic orbits, suppress chaos in USM control system. Proportion-delayed feedback controller was designed following and arithmetic of fuzzy logic was used to adaptively adjust the delay time online. Simulation results show that this method could fast and effectively change the chaos movement into periodic or fixed-point movement and make the system enter into stable state from chaos state. Finally the chaos behavior was controlled.

Chaos Characteristics and Control of Permanent Magnet Synchronous Motors

The permanent magnet synchronous motor (PMSM), a nonlinear dynamic system, can exhibit prominent chaotic characteristics under some choices of system parameters and external inputs. Based on a mathematical model of the permanent magnet synchronous motor, the existence of chaotic attractor is verified by the phase trajectory, Lyapunov exponent map and the bifurcation diagram. Chaotic phenomenon, such as a strong oscillation of speed and torque, unstable operating performance, affects the normal operation of motor. It makes the PMSM in a stable state to control chaos of the PMSM with a control strategy of infinitesimal geometry, which can eliminate chaos well.

Analysis of chaos and circuit implementation of a permanent magnet synchronous motor

The permanent magnet synchronous motor, which is a nonlinear dynamic system, can exhibit a variety of chaotic or limit cycle phenomenon under some choices of system parameters and external inputs, and its chaotic characteristics are more prominent. Based on the mathematical model of a permanent magnet synchronous motor, its nonlinear characteristics are analysed with respect to the bifurcation diagram, Lyapunov exponent map and the nature of its equilibrium point in this paper, and the results in well demonstrated by numerical simulations. Finally, an analog electronic circuit is designed to implement the mathematical model of a permanent magnet synchronous motor, and the experimental results of the system well agreed with the simulation results.

Frequency versus Lyapunov exponent map: A new approach to investigate dynamics of nonlinear magnetic systems

The complex Lyapunov exponent λ plays a vital role in characterizing the dynamics of a physical system. The real part of λ has frequently been related to as just the Lyapunov exponent and has been used for decades to characterize the stability of the system. The imaginary part or the frequency of oscillations can also give valid information about the dynamics of the system, particularly how it behaves near the equilibrium points. In this article we will show that the frequency versus Lyapunov exponent map can give additional information about the very nature of the system and provide background for detailed analysis concerning the applicability of the control technique and its robust nature. As an example of the applicability of the map, an appropriate model to investigate the origin and growth of the auto-oscillations are the circular YIG films. Starting with the low power ferromagnetic resonance spectrum and analyzing the behavior as a function of power the creation and evolution of “shoots” in the map have been demonstrated. The resulting map gives new insights about the relationship between the underlying dynamics of the system and the “growth” of the shoots into auto-oscillation fingers. This approach can explain many features of the auto-oscillation behavior and gives new insights into investigating techniques to control and synchronize chaos as well as to explain desynchronization bursts.