Dynamical Movement Primitives: Learning Attractor Models for Motor Behaviors

Understanding and characterizing complex dynamical behaviors in nonlinear systems require the identification of their intrinsic coordinates or modes, a challenge distinct from linear systems where modal transformation is universal. Koopman operators and nonlinear normal modes (NNMs) serve as essential tools in the study of nonlinear dynamics. Koopman operators offer a method for depicting nonlinear dynamics within a linear framework. On the other hand, NNMs, which are an extension of linear normal modes (LNMs), are designed to capture the inherent invariance properties present in nonlinear dynamics. Because of the challenges involved in representing the Koopman operator and NNMs analytically, utilizing data-driven approaches could be a suitable solution for approximating the nonlinear modal spaces of nonlinear dynamical systems. In this dissertation, our goal is to utilize data-driven methods to approximate these two nonlinear modal analysis techniques for different nonlinear dynamical systems, such as fluid flows. Additionally, we will explore data-driven approaches for representing parametric nonlinear dynamical systems as data-driven surrogate models are suitable alternatives for modeling of parameterized nonlinear dynamical systems. The second chapter of this dissertation explores the representation capabilities of Koopman operators and nonlinear normal modes (NNMs) for nonlinear dynamical systems. Addressing the realistic absence of closed-form models, we introduce a physics-integrated deep learning-based approach to identify Koopman eigenfunctions and NNMs. We assess their representation accuracy by reconstructing and predicting nonlinear dynamical responses. Numerical experiments on Duffing systems reveal that NNMs outperform Koopman operators in accuracy and computational efficiency. The observation of significant coherent structures in fluid flows emphasizes the importance of discovering appropriate coordinates or modes. This discovery is essential for understanding and characterizing the complex dynamic features of fluid flows, which are inherently highly nonlinear systems. In chapter three, we present a physics-constrained deep learning method to discover invariant NNMs, encapsulating the spatiotemporal dynamics of fluid flows with strong nonlinearity. The NNM-physics-constrained convolutional autoencoder (NNM-CNN-AE) captures the nonlinear modal transformation, NNMs, and enables reduced-order reconstruction and long-term future-state prediction of flow fields. Testing on various flow regimes around a cylinder demonstrates that NNMs reveal spatiotemporal dynamics effectively and outperform linear proper orthogonal decomposition (POD) and Koopman-based methods in accuracy and efficiency. Moreover, in chapter four, we address the challenge of modeling parameterized nonlinear systems without complete knowledge of underlying physics. The dynamics-embedded conditional generative adversarial network (Dyn-cGAN) identifies temporal dynamics and their dependence on system parameters, facilitating long-term prediction of responses. Numerical studies on various parameterized nonlinear systems

Introduction. Problems and methods of finding Н∞ – control are the basis of modern control theory. They are actively used to develop robust controllers, especially in aircraft control systems under limited external actions. These methods allow for adapting control systems to changing environmental conditions, which is critically important for providing the reliability and safety of aircraft operation. Current research is aimed at improving approaches to the synthesis of controllers covering both linear and nonlinear dynamic systems. In this context, special attention is paid to the integration of new mathematical methods, such as linear matrix inequalities and frequency analysis, which allows for optimizing the system response to various external actions and providing protection against unexpected conditions. It is important to note that, despite the progress made in this area, significant problems remain unsolved regarding the analysis and synthesis of controllers for nonlinear systems. This necessitates further research and development in this promising area. In this paper, in order to fill the existing gap, sufficient conditions for the existence of control for one of the frequently encountered classes of nonlinear systems are formulated and proven, which will then be used as a theoretical basis for developing approximate algorithms for finding it.Materials and Methods. The basic research tool was the Н∞ – control synthesis methods based on the minimax approach, which consisted in finding the control law under the worst external action. In this context, it was proposed to prove sufficient conditions for the existence of control using the extension principle. However, due to the computational difficulties that might arise when applying those conditions, it was decided to simplify the initial formulation of the problem. The simplification process was performed by approximate replacing the nonlinear system with another nonlinear system, which was similar in structure to the linear one, using the factorization procedure. This approach made it possible to use the solution of the Riccati equation, whose coefficients depended on the state vector, for the synthesis of controllers. To solve model examples and applied problems, a software package was developed using the MATLAB mathematical package.Results. The article solved the problem of synthesis of Н∞ – control of the state of nonlinear continuous dynamic systems, linear in control and disturbance. Sufficient conditions for the existence of Н∞ – control were formulated and proved on the basis of the extension principle. An approximate method was proposed that provided solving the problem of finding control laws for dynamic systems that were nonlinear in state, similar to the methods used for linear systems. Analytical solutions were found for two model examples, which were illustrated by graphs of transient processes to demonstrate the results of numerical modeling of the considered nonlinear dynamic systems in the presence of external actions.Discussion and Conclusion. The proposed approximate algorithm for synthesizing state and output controllers guarantees the required quality of transient processes and asymptotic stability of closed nonlinear control systems. This significantly expands the class of dynamic systems for which it is possible to synthesize controllers capable of resisting various external actions. The methods presented in this paper can be effectively applied to solve a variety of control problems, including the design of autopilots and automatic navigation systems for aircraft, even under conditions of limited external actions.

The behavior of nonlinear dynamical systems can differ completely from that of linear dynamical systems. Regular oscillatory motion as well as seemingly irregular motion can emerge in these systems without resort to exogenous influences or peculiar parameter values. Recent mathematical advances made in the theory of nonlinear dynamical systems allow the restrictive linear approach to dynamical phenomena in economics to be discarded and observable fluctuations of economic variables to be modelled in a simple and convenient way. This book attempts to familiarize the reader with the standard tools in nonlinear dynamical systems theory. The usage of these tools is demonstrated with simple examples from different fields like business cycle theory, optimal control theory, growth theory, and population dynamics. The presentation encompasses a short reminder of linear dynamical systems and traditional themes in nonlinear systems like the existence and uniqueness of limit cycles and closed orbits in predator-prey systems. The main part of the book deals with chaotic motion in economic systems. It is demonstrated, that irregular dynamical behavior can be generated in deterministic economic systems with relative ease, once the linear approach in modelling dynamic phenomena is abandoned. One-dimensional and higher-dimensional difference equations as well as higher-dimensional differential equation systems are discussed and illustrated with a variety of economic examples. Recently emerging techniques for discriminating stochastic and deterministic dynamical systems are presented, as well as the question of whether the nature of actual economic time series is chaotic or stochastic.

Stochastic bifurcation

Stochastic bifurcation

The dynamic behavior of nonlinear systems can be concluded as chaos, periodicity, and the motion between chaos and periodicity; therefore, the key to study the nonlinear system is identifying dynamic behavior considering the different values of the system parameters. For the uncertainty of high‐dimensional nonlinear dynamical systems, the methods for identifying the dynamics of nonlinear nonautonomous and autonomous systems are treated. In addition, the numerical methods are employed to determine the dynamic behavior and periodicity ratio of a typical hull system and Rössler dynamic system, respectively. The research findings will develop the evaluation method of dynamic characteristics for the high‐dimensional nonlinear system.

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

The present research intends to investigate the characteristics of the periodicity ratio and its implementation in analyzing the nonlinear behavior of dynamic systems governed by second-order differential equations. Numerical analyses on the nonlinear dynamic systems with employment of the periodicity ratio for diagnosing chaotic, regular, and irregular behaviors of dynamic systems are performed. To characterize the approach with periodicity ratio in distinguishing different behaviors of the nonlinear dynamic systems, a comparison of periodicity ratio with the widely used Lyapunov exponent in numerically assessing the responses of nonlinear dynamic systems is presented.

This paper deals with the modeling and the prediction of the dynamic behavior of uncertain nonlinear systems. An efficient method is proposed to treat these problems. It is based on the Wiener–Haar chaos concept resulting from the polynomial chaos theory and it generalizes the use of the multiresolution analysis well known in the signal processing theory. The method provides a powerful tool to describe stochastic processes as series of orthonormal piecewise functions whose weighting coefficients are identified using the Mallat pyramidal algorithm. This paper shows that the Wiener–Haar model allows an efficient description and prediction of the dynamic behavior of nonlinear systems with probabilistic uncertainty in parameters. Its contribution, compared to the representation using the generalized polynomial chaos model, is illustrated by evaluating the two models via their application to the problems of the modeling and the prediction of the dynamic behavior of a self-excited uncertain nonlinear system.

A Visual Introduction to Nonlinear Systems: Chaos, Fractals, and the Limits of Prediction

Nearly all nontrivial real-world systems are nonlinear dynamical systems. Chaos describes certain nonlinear dynamical systems that have a very sensitive dependence on initial conditions. Chaotic systems are always deterministic and may be very simple, yet they produce completely unpredictable and divergent behavior. Systems of nonlinear equations are difficult to solve analytically, and scientists have relied heavily on visual and qualitative approaches to discover and analyze the dynamics of nonlinearity. Indeed, few fields have drawn as heavily from visualization methods for their seminal innovations: from strange attractors, to bifurcation diagrams, to cobweb plots, to phase diagrams and embedding. Although the social sciences are increasingly studying these types of systems, seminal concepts remain murky or loosely adopted. This article has three aims. First, it argues for several visualization methods to critically analyze and understand the behavior of nonlinear dynamical systems. Second, it uses these visualizations to introduce the foundations of nonlinear dynamics, chaos, fractals, self-similarity and the limits of prediction. Finally, it presents Pynamical, an open-source Python package to easily visualize and explore nonlinear dynamical systems’ behavior.

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.

This paper presents a semi-analytical method for periodic flows in continuous nonlinear dynamical systems. For the semi-analytical approach, differential equations of nonlinear dynamical systems are discretized to obtain implicit maps, and a mapping structure based on the implicit maps is employed for a periodic flow. From mapping structures, periodic flows in nonlinear dynamical systems are predicted analytically and the corresponding stability and bifurcations of the periodic flows are determined through the eigenvalue analysis. The periodic flows predicted by the single-step implicit maps are discussed first, and the periodic flows predicted by the multistep implicit maps are also presented. Periodic flows in time-delay nonlinear dynamical systems are discussed by the single-step and multistep implicit maps. The time-delay nodes in discretization of time-delay nonlinear systems were treated by both an interpolation and a direct integration. Based on the discrete nodes of periodic flows in nonlinear dynamical systems with/without time-delay, the discrete Fourier series responses of periodic flows are presented. To demonstrate the methodology, the bifurcation tree of period-1 motion to chaos in a Duffing oscillator is presented as a sampled problem. The method presented in this paper can be applied to nonlinear dynamical systems, which cannot be solved directly by analytical methods.

Extending the perturbation technique to the modal representation of nonlinear systems

Chapter 11 - Elements of Nonlinear Digital Control Systems