Interpolated Cell Mapping Research Articles

Parallel Cell Mapping Method for Global Analysis of High-Dimensional Nonlinear Dynamical Systems1

The cell mapping methods were originated by Hsu in 1980s for global analysis of nonlinear dynamical systems that can have multiple steady-state responses including equilibrium states, periodic motions, and chaotic attractors. The cell mapping methods have been applied to deterministic, stochastic, and fuzzy dynamical systems. Two important extensions of the cell mapping method have been developed to improve the accuracy of the solutions obtained in the cell state space: the interpolated cell mapping (ICM) and the set-oriented method with subdivision technique. For a long time, the cell mapping methods have been applied to dynamical systems with low dimension until now. With the advent of cheap dynamic memory and massively parallel computing technologies, such as the graphical processing units (GPUs), global analysis of moderate- to high-dimensional nonlinear dynamical systems becomes feasible. This paper presents a parallel cell mapping method for global analysis of nonlinear dynamical systems. The simple cell mapping (SCM) and generalized cell mapping (GCM) are implemented in a hybrid manner. The solution process starts with a coarse cell partition to obtain a covering set of the steady-state responses, followed by the subdivision technique to enhance the accuracy of the steady-state responses. When the cells are small enough, no further subdivision is necessary. We propose to treat the solutions obtained by the cell mapping method on a sufficiently fine grid as a database, which provides a basis for the ICM to generate the pointwise approximation of the solutions without additional numerical integrations of differential equations. A modified global analysis of nonlinear systems with transient states is developed by taking advantage of parallel computing without subdivision. To validate the parallelized cell mapping techniques and to demonstrate the effectiveness of the proposed method, a low-dimensional dynamical system governed by implicit mappings is first presented, followed by the global analysis of a three-dimensional plasma model and a six-dimensional Lorenz system. For the six-dimensional example, an error analysis of the ICM is conducted with the Hausdorff distance as a metric.

Nonlinear dynamics of a submerged floating moored structure by incremental harmonic balance method with FFT

A submerged floating moored structure has a great potential in ocean engineering applications. The nonlinear dynamics of a submerged floating moored structure subjected to vertical excitation with possible slackness in the mooing system are investigated by incremental harmonic balance (IHB) method. Heaviside step function is introduced to describe the nonlinearity in axial stiffness arising from loss of mooring tension. The dimensionless governing equation is derived, and three parameters, frequency ratio η, damping ratio ζ and dimensionless net buoyancy W, are found to be independent. Due to the fact that the restoring force term is function of the unknown displacement and could barely be expressed in an explicit form of time, a fast Fourier transformation (FFT) is implemented in IHB method to simplify the Galerkin average procedure. Both stable and unstable solutions and both period-1 and bifurcated solutions are obtained by IHB method. The stability of the periodic solutions is investigated by Floquet theory. Parameter study is carried out. Results indicate that the system nonlinearity becomes stronger as dimensionless the net buoyancy W and damping ratio ζ decrease. A path to chaotic motions though a series of period doubling bifurcations is found. Multiple solutions are observed, and the domains of attraction are investigated by interpolated cell mapping (ICM) technique.

3D Monte Carlo modelling of edge plasmas in fusion experiments

A three-dimensional model for the numerical simulation of transport in the edge plasma of modern fusion devices is developed. The model is based on the Braginskii set of equations which are solved with the help of a Monte Carlo method by three-dimensional plasma transport code E3D. The multiple coordinate system approach and interpolated cell mapping technique used in E3D allow us to avoid artificial numerical cross-field transport due to a high anisotropy of Braginskii equations in the case of general magnetic configurations including ergodic regions. The complete self-consistent set of fluid equations (equations for the density, parallel momentum, ion and electron temperatures) has been successfully implemented in E3D. A stabilization method for iterating coupled momentum and density equations is discussed.

Comments on “Parametric analysis and fractal-like basins of attraction by modified interpolated cell mapping”

Comments on “Parametric analysis and fractal-like basins of attraction by modified interpolated cell mapping”

Bifurcation and chaos of a harmonically excited oscillator with both stiffness and viscous damping piecewise linearities by incremental harmonic balance method

In this paper, an explicit formulation of the incremental harmonic balance (IHB) scheme for computation of periodic solutions of a harmonically excited oscillator which is asymmetric with both stiffness and viscous damping piecewise linearities is derived. Analysis of dynamical behavior as bifurcation and chaos of the non-linear vibration system considered is effectively carried out by the IHB procedure, showing that the system exhibits chaos via the route of period-doubling bifurcation, with coexistence of multiple periodic attractors observed and analyzed by the interpolated cell mapping method. In addition, numerical simulation by the IHB method is compared with that by the fourth order Runge–Kutta numerical integration routine, which shows that this method is in many respects distinctively advantageous over classical approaches, and especially excels in performing parametric studies.

Transport in complex magnetic geometries: 3D modelling of ergodic edge plasmas in fusion experiments

Both stellarators and tokamaks can have quite complex magnetic topologies in the plasma edge. Special complexity is introduced by ergodic effects producing stochastic domains. Conventional numerical methods from fluid dynamics are not applicable in this case. In the present paper, we discuss two alternative possibilities. Our multiple coordinate system approach (MCSA) [Phys. Plasmas 8 (2001) 916] originally developed for the TEXTOR DED allows modelling of plasma transport in general magnetic field structures. The main idea of the concept is: magnetic field lines can exhibit truly stochastic behavior only for large distances (compared to the Kolmogorov length), while for smaller distances, the field remains regular. Thus, one can divide the computational domain into a finite set of sub-domains, introduce local magnetic coordinate systems in each and use an ‘interpolated cell mapping’ technique to switch between the neighboring coordinate systems. A 3D plasma fluid code (E3D, based upon MCSA) is applied to realistic geometries. We also introduce here some new details of the algorithm (stellarator option). The results obtained both for intrinsic (stellarator) and external (tokamak with ergodic divertor) perturbations of the magnetic field are discussed. Another approach, also using local coordinate systems, but based on more conventional finite difference methods, is also under development. Here, we present the outline of the algorithm and discuss its potential as compared to the Lagrangian Monte-Carlo approach.

CHAOS, CHAOS CONTROL AND SYNCHRONIZATION OF A GYROSTAT SYSTEM

The dynamic behavior of a gyrostat system subjected to external disturbance is studied in this paper. By applying numerical results, phase diagrams, power spectrum, period-T maps, and Lyapunov exponents are presented to observe periodic and choatic motions. The effect of the parameters changed in the system can be found in the bifurcation and parametric diagrams. For global analysis, the basins of attraction of each attractor of the system are located by employing the modified interpolated cell mapping (MICM) method. Several methods, the delayed feedback control, the addition of constant torque, the addition of periodic force, the addition of periodic impulse torque, injection of dither signal control, adaptive control algorithm (ACA) control and bang–bang control are used to control chaos effectively. Finally, synchronization of chaos in the gyrostat system is studied.

Periodic Response and Chaos in Nonlinear Systems with Parametric Excitation and Time Delay

In this paper, the periodic motions of a nonlinear system with quadratic,cubic, and parametrically excited stiffness terms and with time-delayterms are obtained by the incremental harmonic balance (IHB) method. Theelements of the Jacobian matrix and residue vector arising in the IHBformulation are derived in closed form. A mechanism model representingthe one-mode oscillation of beams and plates is considered as anexample. A path-following algorithm with an arc-length parametriccontinuation procedure is used to obtain the response diagrams. Thesystem also exhibits chaotic motion through a cascade of period-doublingbifurcations, which is characterized by phase planes, Poincare sectionsand Lyapunov exponents. The interpolated cell mapping (ICM) procedure isused to obtain the initial condition map corresponding to multiplesteady-state solutions.

NON-LINEAR DYNAMICS AND CHAOS CONTROL OF A PHYSICAL PENDULUM WITH VIBRATING AND ROTATING SUPPORT

The dynamic behavior of a physical pendulum system of which the support is subjected to both rotation and vertical vibration are studied in this paper. Both analytical and computational results are employed to obtain the characteristics of the system. By using Lyapunov's direct method the conditions of stability of the relative equilibrium position can be determined. Melnikov's method is applied to identify the existence of chaotic motion. The incremental harmonic balance method is used to find the stable and unstable periodic solutions for the strong non-linear system. By applying various numerical results such as phase portrait, Pioncaré map, time history and power spectrum analysis, a variety of the periodic solutions and the phenomena of the chaotic motion can be presented. The effects of the changes of parameters in the system could be found in the bifurcation and parametric diagrams. Further, chaotic motion can be verified by using Lyapunov exponent and Lyapunov dimension. The global analysis of basin boundary and fractal structure are observed by the modified interpolated cell mapping method. Besides, non-feedback control, delayed feedback control, adaptive control, and variable structure control are used to control the chaos effectively.

NON-LINEAR DYNAMIC ANALYSIS AND CONTROL OF CHAOS FOR A TWO-DEGREES-OF-FREEDOM RIGID BODY WITH VIBRATING SUPPORT

The dynamic behaviors of a two-degrees-of-freedom rigid body with vibrating support are studied in this paper. Both analytical and computational results are employed to obtain the characteristics of the system. By using the Lyapunov direct method the conditions of stability of the relative equilibrium position can be determined. The incremental harmonic balance method (IHB) is used to find the stable and unstable periodic solutions for the strongly non-linear system. By applying various numerical results such as phase plane, Poincaré map, time history and power spectrum analysis, a variety of periodic solutions and the phenomena of the chaotic motion is presented. The effects of the changes of parameters in the system can be found in the bifurcation diagrams. Further, the chaotic behavior is verified by using Lyapunov exponents. The modified interpolated cell mapping method (MICM) is used to study the basins of attraction of periodic attractors and the fractal structure. Besides, additions of a constant torque, a periodic torque, addition of dither signals, delayed feedback control, adaptive control, and bang–bang control are used to control the chaos phenomena effectively.

Bifurcation and chaos of an articulated loading platform with piecewise non-linear stiffness using the incremental harmonic balance method

The bifurcation behavior of an articulated loading platform subjected to harmonic excitation is investigated by the incremental harmonic balance (IHB) method. The platform is modeled as a single-degree-of-freedom (SDOF) non-linear system with piecewise non-linear restoring force characteristics. The elements of the Jacobian matrix and the residue vector arising in the IHB formulations are derived in closed form. The path-following procedure using the arc length continuation method is used to trace the response curves and bifurcation diagrams. The periodic solutions and the subharmonic solutions obtained by the IHB method compare very well with the numerically integrated solutions. The bifurcation points also compare well with the numerically obtained results. The system exhibits chaotic motion through a sequence of period doubling bifurcations. Isolated period 3 solutions are also present. The Lyapunov exponents are computed and the initial condition map corresponding to coexistent attractors are obtained by the interpolated cell mapping (ICM) method.

Bifurcation and chaos in escape equation model by incremental harmonic balancing

Periodic motions of the nonlinear system representing the escape equation with cosine and sine parametric excitations and external harmonic excitations are obtained by the incremental harmonic balance (IHB) method. The system contains quadratic stiffness terms. The Jacobian matrix and the residue vector for the type of nonlinearity with parametric excitation are explicitly derived. An arc length path following procedure is used in combination with Floquet theory to trace the response diagram and to investigate the stability of the periodic solutions. The system undergoes chaotic motion for increase in the amplitude of the harmonic excitation which is investigated by numerical integration and represented in terms of phase planes, Poincaré sections and Lyapunov exponents. The interpolated cell mapping (ICM) method is used to obtain the initial condition map corresponding to two coexisting period 1 motions. The periodic motions and bifurcation points obtained by the IHB method compare very well with results of numerical integration.

REGULAR AND CHAOTIC DYNAMIC ANALYSIS AND CONTROL OF CHAOS OF AN ELLIPTICAL PENDULUM ON A VIBRATING BASEMENT

The dynamic behavior of an elliptical pendulum subjected to external disturbance is studied in this paper. The Lyapunov direct method is applied to obtain conditions of stability of the equilibrium points of the system. By applying numerical results, phase diagrams, power spectrum, period-T maps, and Lyapunov exponents are presented to observe periodic and chaotic motions. The effect of the parameters changed in the system can be found in the bifurcation and parametric diagrams. For global analysis, the basins of attraction of each attractor of the system are located by employing the modified interpolated cell mapping (MICM) method. Finally, several methods, the delayed feedback control, the addition of constant torque, the addition of periodic force, adaptive control algorithm and bang-bang control are used to control chaos effectively.

Global stability analysis of fuzzy controllers using cell mapping methods

Linear and nonlinear stability analysis of a nonlinear fuzzy logic controller is considered. The linear analysis is carried out by Lyapunov's first method (i.e. by determination of the eigenvalues of the linearized system). The nonlinear global stability analysis is executed by a combination of two methods: simple and interpolated cell mapping. Examples of controllers with a complex dynamic behavior are presented. These include controllers with multiple solutions such as (i) stable equilibrium point and stable limit circle, (ii) five unstable equilibrium points and three stable limit circles, and (iii) examples where the linear analysis is completely insufficient and only nonlinear investigation provides desirable results. The structure of domains of attraction of stable solutions is determined. Comparison of the results obtained by the cell mapping method with those provided by direct numerical integration, are presented.

Regular and chaotic dynamics of a simplified fly-ball governor

The dynamics of a simplified model of a fly-ball speed governor undergoing a harmonic variation about its rotational speed is studied in this paper. This system is a non-linear damped system subjected to parametric excitation. The harmonic balance method is applied to analyse the stability of period attractors and the behaviour of bifurcations. The time evolutions of the response of the non-linear dynamic system are described by time history, phase portraits and Poincaré maps. The regular and chaotic behaviour is observed by various numerical techniques such as power spectra, Lyapunov exponents and Lyapunov dimension. Finally, the domains of attraction of periodic and stranger attractors of the system are located by applying the interpolated cell mapping (ICM) method.

NON-LINEAR DYNAMICS AND CHAOS CONTROL OF A DAMPED SATELLITE WITH PARTIALLY-FILLED LIQUID

The dynamic behaviors of a damped satellite with partially-filled liquid which is subjected to external disturbance are studied in this paper. The Lyapunov direct method is used to obtain conditions of stability of the equilibrium point of the system. A co-dimension one bifurcation analysis for the autonomous system is carried out near the degenerate point. It is found that Hopf bifurcation occurs in the system by center manifold theory. By applying numerical results, phase diagrams, power spectrum, Poincaré maps, and Lyapunov exponents are presented to observe periodic and chaotic motions. The effect of the parameter changes in the system can be found in the bifurcation and parametric diagrams. For global analysis, the basins of attraction of each attractor of the system are located by employing the modified interpolated cell mapping (MICM) method. Finally, several methods, the delayed feedback control, the addition of constant motor torque, the addition of periodic force, and adaptive control algorithm are used to control chaos effectively.

Evaluating the global characteristic of a nonlinear dynamical system with recursive boundary enhancement

This paper describes a new algorithm for the production of global stability phase space diagrams in periodically forced differential systems. These equations being derived from the dynamics of structures with nonlinear responses. The overall system characteristic can be investigated with much greater speed than before. The algorithm uses a process of boundary grid refinement to produce a greatly enhanced procedure which is accurate and less computationally expensive than the standard grid of starts (GOS) method. The algorithm focuses on the boundaries of the catchment basins which need the most attention. This concentration on the boundaries cannot be made in an a priori manner as the boundaries are initially unknown. While the algorithm is proceeding concepts of parent cells, child cells and cell division are used to determine the location of the boundaries. The necessity for recursion in the algorithm is discussed. The procedure is valid for both non-fractal and fractal boundaries. A comparison with other methods such as SCM (simple cell mapping) and ICM (interpolated cell mapping) methods are made to evaluate computational efficiency and accuracy.

A MODIFIED INTERPOLATED CELL MAPPING METHOD

A modified interpolated cell mapping (MICM) method is developed for global analysis of deterministic dynamical systems with smooth basin boundaries. By improving the algorithm of the interpolated cell mapping (ICM) and including the idea of sampling mappings of cells from the generalized cell mapping (GCM), MICM gives a more complete and precise global analysis of systems that ICM. In addition, based on analysis using MICM, a method using exact mappings is proposed to remedy the inaccuracy and incompleteness of global analysis due to interpolated mappings and the assumption of the sink cell. For the three systems studied in this paper, this method locates identical attractors and basins of attraction as the integration of a grid of points (IGP) method, with a 100-fold improvement in computational efficiency.

ANALYSIS OF RANDOM DYNAMICAL SYSTEMS BY INTERPOLATED CELL MAPPING

A method of interpolated cell mapping (ICM) is developed for global analysis of random dynamical systems. In this method, one first analyzes a random system by constructing a uniform number of sample interpolated trajectories for each cell. More sample interpolated trajectories are then constructed for critical cell sensitive to random effects to characterize the global behaviour of the system. When applied to the global analysis of non-linear systems with uncertain external loadings or system parameters, the proposed method yields more precise results than the generalized cell mapping (GCM) method. These results agree well with those obtained by Monte Carlo simulation. Additionally, the proposed method allows global analyses of random systems to be improved easily when the original regions of study are too large to give precise analyses.

APPLICATION OF INTERPOLATED CELL MAPPING TO AN ANALYSIS OF THE LORENZ EQUATIONS

The Lyapunov exponents of the Lorenz equations are calculated using interpolated cell mapping. The effects of mapping time step and cell size on accuracy and computation time are analyzed, leading to the identification of optimal cell size and time step values. A higher order interpolation method for interpolated cell mapping is described. The accuracy and computation time of this method are compared to numerical integration for calculating the basins of attraction of the Lorenz equations and the dimension of the basin boundaries. Regions of state space with high levels of deformation are shown to exist where the basin boundaries are most dense.