Generalized Cell Mapping Method Research Articles

Fuzzy optimal control of nonlinear systems by fuzzy generalized cell mapping method with Bellman’s principle

A novel method is proposed in this paper to obtain global solutions of fuzzy optimal control with fixed state terminal conditions and control bounds. The global solution implies that the optimal control solutions are valid for all the initial conditions in a region of the state space. The method makes use of Bellman’s principle of optimality and fuzzy generalized cell mapping method (FGCM). A discrete form of fuzzy master equation with a control dependent transition membership matrix is generated by using the FGCM. This allows to evaluate both the transient and the steady-state responses of the controlled system. The method, simply called FGCM with BP, is applied to three nonlinear systems leading to excellent control performances.

The Birth of a Hidden Attractor Through Boundary Crisis

Hidden attractors have been reported in a significant number of dynamical systems, according to their definition related to the existence of saddle points. Since it has now become common knowledge that there are different types of saddle invariant sets besides a fixed saddle point, it is of great interest to investigate the subject. In this paper, a sudden appearance of a hidden chaotic attractor is observed with the variation of a control parameter in some two-dimensional mapping systems. In order to reveal the mechanism behind the sudden appearance of hidden attractors, the generalized cell mapping method with GPU parallel computing is employed in order to obtain very accurate invariant sets. It is found that hidden attractors are generated from an existing chaotic saddle through a boundary crisis, where a hidden chaotic attractor is touching a periodic or chaotic saddle on its basin boundary. Three examples of two-dimensional mapping systems are given to demonstrate the birth of hidden attractors through a boundary crisis.

Generalized Cell Mapping Method with Deep Learning for Global Analysis and Response Prediction of Dynamical Systems

A novel method that combines generalized cell mapping and deep learning is developed to analyze the global properties and predict the responses of dynamical systems. The proposed method only requires some prior knowledge of the system governing equations and obtains dynamical properties of the system from observed data. By combining the theoretical demonstration and empirical inference results, appropriate network structure and training hyperparameters are computed. Then a robust and efficient neural network approximation with the estimated mapping parameters is obtained. By using the approximate dynamical system model, we construct the one-step transition probability matrix and introduce the digraph analysis method to analyze the global properties. System responses at any time can be obtained with the trained model on the basis of the property of Markov chain. Several examples with periodic or chaotic attractors are presented to validate the proposed method. The influence of the number of hidden layers and the size of training data on calculated results is discussed, and an admissible architecture of the neural network is found. Numerical results indicate that the proposed method is quite effective for both global analysis and response prediction.

Stochastic dynamics of dielectric elastomer balloon with viscoelasticity under pressure disturbance

Abstract Dielectric elastomers are widely used in many fields due to their advantages of high deformability, light weight, biological compatibility, and high efficiency. In this study, the stochastic dynamic response and bifurcation of a dielectric elastomer balloon (DEB) with viscoelasticity are investigated. Firstly, the rheological model is adopted to describe the viscoelasticity of the DEB, and the dynamic model is deduced by using the free energy method. The effect of viscoelasticity on the state of equilibrium with static pressure and voltage is analysed. Then, the stochastic differential equation about the perturbation around the state of equilibrium is derived when the DEB is under random pressure and static voltage. The steady-state probability densities of the perturbation stretch ratio are determined by the generalized cell mapping method. The effects of parameter conditions on the mean value of the perturbation stretch ratio are calculated. Finally, sinusoidal voltage and random pressure are applied to the viscoelastic DEB, and the phenomenon of P-bifurcation is observed. Our results are compared with those obtained from Monte Carlo simulation to verify their accuracy. This work provides a potential theoretical reference for the design and application of DEs.

Stationary response of colored noise excited vibro-impact system* *Project supported by the National Natural Science Foundation of China (Grant No. 11772149), the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, China (Grant No. MCMS-I-19G01), and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions

The generalized cell mapping (GCM) method is used to obtain the stationary response of a single-degree-of-freedom. Vibro-impact system under a colored noise excitation. In order to show the advantage of the GCM method, the stochastic averaging method is also presented. Both of the two methods are tested through concrete examples and verified by the direct numerical simulation. It is shown that the GCM method can well predict the stationary response of this noise-perturbed system no matter whether the noise is wide-band or narrow-band, while the stochastic averaging method is valid only for the wide-band noise.

An efficient approach to obtaining the exit location distribution and the mean first passage time based on the GCM method

In this paper, according to the probability evolution analysis, we developed a new procedure based on the Generalized Cell Mapping method to get the exit location distribution and the mean first passage time of the weak noise excited system. With the Generalized Cell Mapping method, the original stochastic process is converted into a Markov chain. Moreover, the eigenvalue problem of the elliptic differential operator for the escape problem is simplified into an eigenvalue problem of the probability transition matrix. Thus, the exit location distribution and the mean first passage time of the system can be easily derived by solving the transition matrix’s eigenvalues and eigenvectors. By applying to the Maier–Stein system, Kramers problem, and the Vibro-impact system, shows the Generalized Cell Mapping method could save us a lot of time and provide us much better results compared with the directed Monte Carlo simulation.

Diagnosis of an Alternator System Using Quantized Approaches

In this paper, the Generalized Cell Mapping (GCM) method for a linear system is compared with a new stochastic method for novel cell-to-cell mapping. The authors presented the new stochastic method in Mohon and Pisu (2013). The two methods are compared in an application example of a vehicle alternator. The alternator may experience three faults including belt slippage, a faulty diode connection, or incorrect controller reference voltage. Fault detection and isolation (FDI) is performed using the two cell-to-cell mapping methods. The results show that the new stochastic method is slower but yields better isolation results than the GCM method.

Global dynamic analysis of the North Pacific Ocean by data-driven generalized cell mapping method

A data-driven generalized cell mapping method devoting to conquer the difficulty that how to carry out the global analysis for a model-free and complex system, without prior knowledge of the underlying system, is proposed in this paper to uncover the hidden flow structure existing in the North Pacific Ocean. By this way, the transient attracting region near the 30 degrees north latitude, acting as a short-term attractor in the North Pacific Ocean, is unveiled by investigating the topology of the season-aware transition probability matrix created from the historical data bank of drifters. The predicted probability distributions of drifters are presented to illustrate the evolutions and concentration of ocean currents, which coincides with the real responses of drifters. Furthermore, the purpose of this paper is also to promote the applications of the data-driven cell mapping method in practical engineering.

Statistical Characteristics of the First Passage Time Analysis for the Gene Regulatory Circuit in Bacillus subtilis by Cell Mapping Method

In this paper, we will explore the stochastic exit problem for the gene regulatory circuit in B. subtilis affected by colored noise. The stochastic exit problem studies the state transition in B. subtilis (from competent state to vegetative state in this case) through three different quantities: the probability density function of the first passage time, the mean of first passage time, and the reliability function. To satisfy the Markov nature, we convert the colored noise system into the equivalent white noise system. Then, the stochastic generalized cell mapping method can be used to explore the stochastic exit problem. The results indicate that the intensity of noise and system parameters have the effect on the transition from competent to vegetative state in B. subtilis. In addition, the effectiveness of the stochastic generalized cell mapping method is verified by Monte Carlo simulation.

Dynamic Analysis of Stochastic Friction Systems Using the Generalized Cell Mapping Method

Friction systems are a kind of typical non-linear dynamical systems in the actual engineering and often generate abundant dynamics phenomena. Because of non-smooth characteristics, it is difficult to handle these systems by conventional analysis methods directly. At the same time, random perturbation often affects friction systems and makes these systems more complicated. In this context, we investigate the steady-state stochastic responses and stochastic P-bifurcation of friction systems under random excitations in this paper. And in order to retain the non-smooth of friction system, the generalized cell mapping (GCM) method is first used to the original stochastic friction systems without any approximate transformation. To verify the accuracy and validate the applicability of the suggested approach, we present two classical nonlinear friction systems, i.e., Coulomb force model and Dahl force model as examples. Meanwhile, this method is in good agreement with the Monte Carlo simulation method and the computation time is greatly reduced. In addition, further discussion finds that the adjustable parameters can induce the stochastic P-bifurcation in the two examples, respectively. The stochastic P-bifurcation phenomena indicate that the stability of the friction system changes very sensitively with the parameters. Research of responses analysis and stochastic P-bifurcation has certain significances for the reliability and stability analysis of practical engineering.

On the Data-Driven Generalized Cell Mapping Method

Global analysis is often necessary for exploiting various applications or understanding the mechanisms of many dynamical phenomena in engineering practice where the underlying system model is too complex to analyze or even unavailable. Without a mathematical model, however, it is very difficult to apply cell mapping for global analysis. This paper for the first time proposes a data-driven generalized cell mapping to investigate the global properties of nonlinear systems from a sequence of measurement data, without prior knowledge of the underlying system. The proposed method includes the estimation of the state dimension of the system and time step for creating a mapping from the data. With the knowledge of the estimated state dimension and proper mapping time step, the one-step transition probability matrix can be computed from a statistical approach. The global properties of the underlying system can be uncovered with the one-step transition probability matrix. Three examples from applications are presented to illustrate a quality global analysis with the proposed data-driven generalized cell mapping method.

Fuzzy Noise-Induced Codimension-Two Bifurcations Captured by Fuzzy Generalized Cell Mapping with Adaptive Interpolation

In this paper, the Fuzzy Generalized Cell Mapping (FGCM) method is developed with the help of the Adaptive Interpolation (AI) in the space of fuzzy parameters. The adaptive interpolation on the set-valued fuzzy parameter is introduced in computing the one-step transition membership matrix to enhance the efficiency of the FGCM. For each of initial points in the state space, a coarse database is constructed at first, and then interpolation nodes are inserted into the database iteratively each time errors are examined with the explicit formula of interpolation error until the maximal errors are just under the error bound. With such an adaptively expanded database on hand, interpolating calculations assure the required accuracy with maximum efficiency gains. The new method is termed as Fuzzy Generalized Cell Mapping with Adaptive Interpolation (FGCM with AI), and is used to investigate codimension-two bifurcations in two-dimensional and three-dimensional nonlinear dynamical systems with fuzzy noise. It is found that global changes in fuzzy dynamics are dominated by the underlying deterministic counterparts, and the fuzzy attractor expands along the unstable manifold leading to a collision with a saddle when a bifurcation occurs. The examples show that the FGCM with AI has a thirtyfold to fiftyfold efficiency over the traditional FGCM to achieve the same analyzing accuracy.

Global Invariant Manifolds of Dynamical Systems with the Compatible Cell Mapping Method

An iterative compatible cell mapping (CCM) method with the digraph theory is presented in this paper to compute the global invariant manifolds of dynamical systems with high precision and high efficiency. The accurate attractors and saddles can be simultaneously obtained. The simple cell mapping (SCM) method is first used to obtain the periodic solutions. The results obtained by the generalized cell mapping (GCM) method are treated as a database. The SCM and GCM are compatible in the sense that the SCM is a subset of the GCM. The depth-first search algorithm is utilized to find the coarse coverings of global stable and unstable manifolds based on this database. The digraph GCM method is used if the saddle-like periodic solutions cannot be obtained with the SCM method. By taking this coarse covering as a new cell state space, an efficient iterative procedure of the CCM method is proposed by combining sort, search and digraph algorithms. To demonstrate the effectiveness of the proposed method, the classical Hénon map with periodic or chaotic saddles is studied in far more depth than reported in the literature. Not only the global invariant manifolds, but also the attractors and saddles are computed. The computational efficiency can be improved by up to 200 times compared to the traditional GCM method.

Stochastic response and bifurcations of a dry friction oscillator with periodic excitation based on a modified short-time Gaussian approximation scheme

In this paper, a modified short-time Gaussian approximation (STGA) scheme is incorporated into the generalized cell mapping (GCM) method to study the stochastic response and bifurcations of a nonlinear dry friction oscillator with both periodic and Gaussian white noise excitations. Firstly, the Fokker–Planck–Kolmogorov equation and the moment equations with Gaussian closure are derived. Because of the periodic force and the singularity of moment equations in the initial condition being a Dirac delta function, a set of novel STGA solutions of the conditional probability density functions (PDFs) over each small fraction of one period is then constructed in order to form the solution over the whole period. At last, the transient and steady-state response PDFs are computed by the GCM method. The accuracy of the results is demonstrated by the direct Monte Carlo simulations. The transient PDFs are presented when evolving from a Gaussian initial distribution to a non-Gaussian steady-state one. The effects of periodic excitation and dry friction damping on the steady-state stochastic response are particularly discussed. When the amplitude of periodic excitation is changed, both stochastic P-bifurcation and D-bifurcation are detected. It is also found that the increase in the dry friction damping coefficient makes the chaotic response disappear gradually, inducing stochastic D-bifurcation but no stochastic P-bifurcation.

Non-stationary response of MDOF dynamical systems under combined Gaussian and Poisson white noises by the generalized cell mapping method

A block matrix analysis procedure of the generalized cell mapping (GCM) method is proposed in this paper. The proposed method solves the storage problem in the response analysis of nonlinear stochastic dynamical system with the GCM method, and makes it possible to compute the non-stationary and stationary probability density functions (PDFs) of multi-degree-of-freedom (MDOF) nonlinear systems without using supercomputing. Two examples of two-degree-of-freedom systems under external or parametric excitations of combined Gaussian and Poisson white noises are presented to demonstrate the efficiency of the proposed method. Monte Carlo (MC) simulations are used to verify the accuracy of the solutions obtained with the proposed method.

The stochastic dynamical behaviors of the gene regulatory circuit in Bacillus subtilis

In recent years, the gene regulatory circuit in biological systems has attracted a considerable interest. In this paper, we consider the dynamical behaviors of the gene regulatory circuit in Bacillus subtilis, including deterministic system and stochastic system effected by colored noise. First of all, the global dynamical behaviors of the deterministic system are exhibited by numerical method. Moreover, we give an effective method to explore the stochastic response and bifurcation by means of the stochastic generalized cell mapping method. To satisfy the Markov nature, we transform the colored-noise system into the equivalent white-noise system. And the stochastic generalized cell mapping method can be used to obtain the dynamical behaviors of the stochastic system. We found that the system parameters and noise can induce the occurrence of the stochastic P-bifurcation in the stochastic system, which means that the conversion between competent state and vegetative state in Bacillus subtilis is possible. In addition, the effectiveness of the stochastic generalized cell mapping method is verified by Monte Carlo simulation.

Analysis of global properties for dynamical systems by a modified digraph cell mapping method

A modified procedure is proposed in this paper to solve the limitations of digraph cell mapping method in accurately analyzing the global properties of dynamical system, such as fractal basin boundaries. Firstly a rough cell structure is applied to the generalized cell mapping (GCM) method to obtain the general global properties of dynamical systems with digraph algorithm, and then a procedure based on the composite cell coordinate system method is proposed to increase the calculation accuracy. In order to further increase the calculation speed, a simple and feasible parallel strategy is applied during the creation process of one-step transition probability matrix for the GCM method. Meanwhile, the memory consumption can be greatly reduced by storing the one-step transition probability matrix as finite separate data files. The accurate global properties of two examples, a nonlinear oscillator with fractal basin structure and the Lorenz system, are demonstrated to show the effectiveness of our proposed improvement strategy.

The Stochastic Response of a Class of Impact Systems Calculated by a New Strategy Based on Generalized Cell Mapping Method

In this paper, a new strategy based on generalized cell mapping (GCM) method will be introduced to investigate the stochastic response of a class of impact systems. Significant difference of the proposed procedure lies in the choice of a novel impact-to-impact mapping, which is built to calculate the one-step transition probability matrix, and then, the probability density functions (PDFs) of the stochastic response can be obtained. The present strategy retains the characteristics of the impact systems, and is applicable to almost all types of impact systems indiscriminately. Further discussion proves that our strategy is reliable for different white noise excitations. Numerical simulations verify the efficiency and accuracy of the suggested strategy.

Studying the Global Bifurcation Involving Wada Boundary Metamorphosis by a Method of Generalized Cell Mapping with Sampling-Adaptive Interpolation

In this paper, a new method of Generalized Cell Mapping with Sampling-Adaptive Interpolation (GCMSAI) is presented in order to enhance the efficiency of the computation of one-step probability transition matrix of the Generalized Cell Mapping method (GCM). Integrations with one mapping step are replaced by sampling-adaptive interpolations of third order. An explicit formula of interpolation error is derived for a sampling-adaptive control to switch on integrations for the accuracy of computations with GCMSAI. By applying the proposed method to a two-dimensional forced damped pendulum system, global bifurcations are investigated with observations of boundary metamorphoses including full to partial and partial to partial as well as the birth of fully Wada boundary. Moreover GCMSAI requires a computational time of one thirtieth up to one fiftieth compared to that of the previous GCM.

The response analysis of fractional-order stochastic system via generalized cell mapping method.

This paper is concerned with the response of a fractional-order stochastic system. The short memory principle is introduced to ensure that the response of the system is a Markov process. The generalized cell mapping method is applied to display the global dynamics of the noise-free system, such as attractors, basins of attraction, basin boundary, saddle, and invariant manifolds. The stochastic generalized cell mapping method is employed to obtain the evolutionary process of probability density functions of the response. The fractional-order ϕ6 oscillator and the fractional-order smooth and discontinuous oscillator are taken as examples to give the implementations of our strategies. Studies have shown that the evolutionary direction of the probability density function of the fractional-order stochastic system is consistent with the unstable manifold. The effectiveness of the method is confirmed using Monte Carlo results.