Extrapolating tipping points and simulating non-stationary dynamics of complex systems using efficient machine learning

Stochastic bifurcation

Among the irregular responses of nonlinear dynamic systems, chaotic responses of nonlinear systems are probably the most attractive phenomena along with the new observations in the last decades. A nonlinear deterministic system may behavior chaotically under regular such as periodic excitations. Regular motion of a system subjected to periodic exertions is usually periodic. In contrast with regular motions, final states of chaotic vibrations are extremely nonperiodic. This research is to analyzing the irregular behavior of dynamic systems with implementation of a newly developed criterion named Periodicity-Ratio. The development of a methodology for diagnosing the irregular motions from the regular motions of a dynamic system is presented. The Periodicity-Ratio describes the degree of periodicity of motion and can be conveniently used to distinguish a nonperiodic motion from a regular vibration or oscillation and to diagnose whether or not a motion is chaotic and the other irregular responses of the nonlinear dynamic systems, without plotting any figures. The analyses on the irregular behavior of nonlinear dynamic systems with the implementation of the Periodicity-Ratio will be demonstrated.

Three instruments which have proven useful for studying nonlinear dynamical systems are demonstrated. The 3D rotator enables one to display in real time any two-dimensional projection of an attractor in R/sup 3/; it can also be used to take Poincare sections of attractors. The laser projector is a laser-based X-Y projection oscilloscope for demonstrating bifurcation phenomena. The nonlinear dynamics analyzer characterizes steady-state behaviors of nonlinear dynamical systems by estimating the power spectrum, dimension, and Lyapunov exponents of the corresponding attractors from their time series. >

Complex behavior in nonlinear dynamical systems often arises from resonances, which enable intricate energy transfer mechanisms among modes that otherwise would not interact. Theoretical, numerical and experimental methods are available to study such behavior when the resonance arises among modes of the linearized system. Much less understood are, however, resonances arising from nonlinear modal interactions, which cannot be detected from a classical linear analysis. Academic examples of such phenomena have been known, but no systematic method has been developed to detect and model nonlinear resonant interactions purely from numerical or experimental data. Here, we develop such a data-driven methodology that identifies nonlinear resonant response on low-dimensional spectral submanifolds (SSMs) of the dynamical system. Our approach is generally applicable to nonlinear resonances, but we specifically focus here on one particular behavior: subharmonic response in forced nonlinear systems without any resonance among the linearized frequencies of the unforced system. We first illustrate analytically how such a response is born out of a nonlinear resonance hidden in the conservative limit of the system. We then show how this effect can be identified and modeled purely from data. As our main example, we isolate and model previously unexplained response patterns in fluid sloshing experiments.

This chapter introduces the basic concepts of dynamical systems theory and several basic mathematical methods for controlling chaos. The main goal of this chapter is to provide an introduction to and a summary of the theory of dynamical systems, with particular emphasis on fractal theory, chaos theory, and chaos control. We first define what is meant by a dynamical system, then we define an attractor, and then the concept of the fractal dimension of a geometrical object. We also define the Lyapunov exponents as a measure of the chaotic behavior of a dynamical system. On the other hand, the fractal dimension can be used to classify geometrical objects because it measures the complexity of an object. The chapter also describes mathematical methods for controlling chaos in dynamic systems. These methods can be used to control a real dynamic system; however, due to efficiency and accuracy requirements we were forced to use fuzzy logic to model the uncertainty, which is present when numerical simulations are performed. We also describe in this chapter a new theory of chaos using fuzzy logic techniques. Chaotic behavior in nonlinear dynamical systems is very difficult to detect and control. Part of the problem is that mathematical results for chaos are difficult to use in many cases, and even if one could use them there is an underlying uncertainty in the accuracy of the numerical simulations of the dynamical systems. For this reason, we can model the uncertainty of detecting the range of values where chaos occurs, using fuzzy set theory. Using fuzzy sets, we can build a theory of fuzzy chaos, where we can use fuzzy sets to describe the behaviors of a system. We illustrate our approach with two cases: Chua’s circuit and Duffing’s oscillator.

In this paper a new balanced realization method for nonlinear systems is proposed which is based on singular value analysis of Hankel operators. The proposed method balances the relationship between the input-to-state behavior and the state-to-output behavior of nonlinear dynamical systems, whereas the existing results only balance the relationship among the coordinate axes of the state-space. This result is expected to be a basis for new model reduction and system identification of nonlinear systems.

Rotorcraft and their associated flight dynamics are highly non-linear systems, especially near extremities of the flight envelope. A global picture of the flight dynamics can be obtained by incorporating in the equations of motion non-linearities in aerodynamics, control system features and inertial terms. This paper deals with the coupling of these phenomena with the non-linear dynamics of an underslung load system. Dynamical systems theory provides the basis for the continuation and bifurcation method of analysing non-linear systems of ordinary differential equations, and forms the basis of this study. The principal strength of the bifurcation-based approach is to identify and give insight into phenomena that globally affect behaviour of an aircraft model, as the full system of non-linear equations is used in analysis. Implemented alongside appropriate numerical simulation runs, the technique promotes an understanding of the underlying structure governing the behaviour of non-linear dynamical systems. Bifurcation and continuation analysis is carried out here on a 21st order non-linear flight dynamics model representing a utility helicopter. Also, the equations of motion for an underslung load system are formulated and then analyzed in isolation. Thereafter they are appended to the helicopter model and a non-linear analysis of the complete system is performed. The effect of the underslung load dynamics on the nominal helicopter system is revealed by the bifurcation-based studies. The influence of varying parameters of the underslung model on the underlying equilibria of the rotorcraft is also investigated. The results reveal changes in helicopter and underslung load stability once the two systems are coupled. The underslung system reduces the extent of the stable equilibrium solution branches, and causes the stability augmentation system to saturate closer to the straight and level trim initial condition. The parameter dependance of the combined dynamics are presented and highlight areas of instability, which may be missed using conventional methods, and show how the transition between stable and unstable solutions is exactly determined as an underslung parameter is varied. This characterisation of the coupled system in terms of stability and response illustrates the potential for bifurcation-based methods to contribute to such non-linear analysis and synthesis problems.

To model the behavior of nonlinear dynamical systems using wireless sensor networks (WSNs), the development of computation and energy efficient distributed modeling techniques is of crucial importance. In this work, Volterra model is considered for the modeling of nonlinear systems because of its significant modeling capability and generalized nonlinear structure. For real-time estimation of the Volterra model parameters, a simple distributed algorithm is designed for a particular framework of WSNs having predefined bridge sensor nodes. The pertinent cost function is expressed as an unconstrained minimization problem using a decomposable augmented Lagrangian form. To facilitate the distributed convex optimization, the augmented Lagrangian form is minimized using alternating direction method of multipliers. The communication and computational complexities involved in the proposed methodology are provided to show its effectiveness in the real-time applications over centralized and non-cooperative solutions. The convergence analysis is provided to guarantee the mean stability of the proposed algorithm. Simulation results obtained under the noisy environment are plotted to demonstrate the effective performance of the proposed algorithm.

The cell mapping method has attractive features for studying global behavior of nonlinear dynamical systems, especially for locating stable periodic solutions and their domains of attractions. A major hurdle in practical implementation of the method for analyzing higher order system is due to its computational memory requirements. The storage of the cell map itself forms a significant portion of the overall memory requirement for cell mapping analysis. In this paper, we outline a procedure for eliminating the storage requirement for the cell map by modifying the existing simple cell mapping unraveling algorithm. The key factor in the modification comes from the analysis of an individual cell sequence originating from a given cell and how its ultimate mapping sequence is determined before a new sequence is generated. Since such a sequence must terminate in either a sink cell or a periodic motion, we propose a modified algorithm which combines this feature effectively with the existing algorithm. The utility of the proposed unraveling algorithm is demonstrated by computing bifurcation of periodic solutions of a strongly nonlinear mechanical system. Some features of the dynamic behavior are analyzed to show the existence of periodic solutions and chaotic behavior.

We investigate the effectiveness of connectionist architectures for predicting the future behavior of nonlinear dynamical systems. We focus on real-world time series of limited record length. Two examples are analyzed: the benchmark sunspot series and chaotic data from a computational ecosystem. The problem of overfitting, particularly serious for short records of noisy data, is addressed both by using the statistical method of validation and by adding a complexity term to the cost function ("back-propagation with weight-elimination"). The dimension of the dynamics underlying the time series, its Liapunov coefficient, and its nonlinearity can be determined via the network. We also show why sigmoid units are superior in performance to radial basis functions for high-dimensional input spaces. Furthermore, since the ultimate goal is accuracy in the prediction, we find that sigmoid networks trained with the weight-elimination algorithm outperform traditional nonlinear statistical approaches.

Chaos control is used to design a controller that is able to eliminate the chaotic behavior of nonlinear dynamic systems that experience such phenomena. This paper discuss the use of the FPGA as a controller of a parallel-connected DC-DC buck-boost converter, the goal of this paper is to build a controller that is capable of controlling the output current of a photovoltaic cells and minimize the effect of the module buck-boost converter chaotic behavior on the output voltage. To achieve this goal this paper presents two different methods, FPGA control the duty cycle and the frequency of the output controlling signal, this technique is done through software (FPGA code), and a delayed feedback control scheme in a module converter in the continuous-current conduction mode (CCM) using MATLAB/SIMULINK simulation. Thus, this paper shows the FPGA capabilities in the power industry and it’s specifies a guideline to overcome some of the obstacles when dealing with an FPGA as a buck-boost converter controller, and MATLAB/SIMULINK simulation results show the effectiveness and robustness of the scheme.

Applications of deterministic chaos theory to corrosion

A difficult matter for researchers is to develop numerical methods to analyze the behavior of nonlinear dynamical systems that arise in mathematical physics due to their nonlinearity and multi-dimensional nature. In this affection, a novel and effective coupling of spectral and one-step approaches is suggested and used to inspect the robust solutions to the mathematical problems. This proposed method is grounded on the one-step sixth-order Runge–Kutta (RK) method and the shifted Vieta–Fibonacci (sVF) polynomial-based spectral approach. First, use the sixth-order one-step technique for the temporal approximation, then utilize the sVF polynomials to estimate the spatial variables. The sVF polynomial-based new operational matrices are used to calculate the derivative terms of the mathematical models. Related theorems are presented in the paper to provide a mathematical validation of the technique. The suggested semi-spectral techniques transform the investigated nonlinear models into a set of linear algebraic equations, making them simpler to solve. We conduct theoretical and numerical research into the stability, convergence, and error-bound analysis to support the computational algorithm's mathematical formulation. Several mathematical models, including the Hartmann numbers ranging from 0 to 106, are taken into consideration to demonstrate the precision and efficacy of the proposed computational technique. A thorough comparison analysis illustrates the recommended computational approach's validity, correctness, and dependability. The suggested construction of sixth-order in the temporal direction and exponential order around 3.5545e0.1515M in the spatial direction is determined to be exceptional in handling nonlinear problems and examining the exact solutions.

A universal mechanism underlying generalized synchronization conditions in unidirectionally coupled stochastic oscillators is considered. The consideration is carried out in the framework of a modified system with additional dissipation. The approach developed is illustrated with model examples. The conclusion is reached that two types of the behavior of nonlinear dynamic systems known as generalized synchronization and noise-induced synchronization, which are viewed as different phenomena, actually represent a unique type of the synchronous behavior of stochastic oscillators and are caused by the same mechanism.

In reality, the behavior and nature of nonlinear dynamical systems are ubiquitous in many practical engineering problems. The mathematical models of such problems are often governed by a set of coupled second-order differential equations to form multi-degree-of-freedom (MDOF) nonlinear dynamical systems. It is extremely difficult to find the exact and analytical solutions in general. In this paper, the homotopy analysis method is presented to derive the analytical approximation solutions for MDOF dynamical systems. Four illustrative examples are used to show the validity and accuracy of the homotopy analysis and modified homotopy analysis methods in solving MDOF dynamical systems. Comparisons are conducted between the analytical approximation and exact solutions. The results demonstrate that the HAM is an effective and robust technique for linear and nonlinear MDOF dynamical systems. The proof of convergence theorems for the present method is elucidated as well.