Nonlinear Dynamical Systems Research Articles

Neural downscaling for complex systems: from large-scale to small-scale by neural operator

Researchers have long been working on interpreting and predicting the dynamics of complex systems in various fields. Conventional methods including full-scale simulations and reduced-order models are used to solve related problems, but often yield unsatisfactory results in terms of precision and computational efficiency. This is primarily due to the challenges posed by simulating small-scale dynamics. An alternative consideration is to leverage well-represented and readily accessible large-scale dynamics to assist small-scale modelling. This paper proposes a novel methodology, named as Neural Downscaling (ND), that effectively captures small-scale dynamics within a complementary subspace from corresponding large-scale dynamics well-represented in a low-dimensional space. The framework of ND integrates neural operator techniques with the principles of inertial manifold and nonlinear Galerkin theory. The effectiveness and efficiency of ND are demonstrated in the complex systems governed by Kuramoto–Sivashinsky and Navier–Stokes equations. Being an unprecedentedly deterministic model targeted at general small-scale dynamics, ND sheds light on the intricate spatiotemporal nonlinear dynamics of complex systems and reveals how small-scale dynamics are interacting with large-scale dynamics.

Deep learning architectures for data-driven damage detection in nonlinear dynamic systems under random vibrations

The primary goal of structural health monitoring is to detect damage at its onset before it reaches a critical level. In the present work an in-depth investigation addresses deep learning applied to data-driven damage detection in nonlinear dynamic systems. In particular, autoencoders and generative adversarial networks are implemented leveraging on 1D convolutional neural networks. The onset of damage is detected in the investigated nonlinear dynamic systems by exciting random vibrations of varying intensity, without prior knowledge of the system or the excitation and in unsupervised manner. The comprehensive numerical study is conducted on dynamic systems exhibiting different types of nonlinear behavior. An experimental application related to a magneto-elastic nonlinear system is also presented to corroborate the conclusions.

Development and Analysis of novel Integrable Nonlinear Dynamical Systems on Quasi-One-Dimensional Lattices. Parametrically Driven Nonlinear System of Pseudo-Excitations on a Two-Leg Ladder Lattice

Following the main principles of developing the evolutionary nonlinear integrable systems on quasi-one-dimensional lattices, we suggest a novel nonlinear integrable system of parametrically driven pseudo-excitations on a regular two-leg ladder lattice. The initial (prototype) form of the system is derivable in the framework of semi-discrete zero-curvature equation with the spectral and evolution operators specified by the properly organized 3 × 3 square matrices. Although the lowest conserved local densities found via the direct recursive method do not prompt us the algebraic structure of system’s Hamiltonian function, but the heuristically substantiated search for the suitable two-stage transformation of prototype field functions to the physically motivated ones has allowed to disclose the physically meaningful nonlinear integrable system with time-dependent longitudinal and transverse inter-site coupling parameters. The time dependencies of inter-site coupling parameters in the transformed system are consistently defined in terms of the accompanying parametric driver formalized by the set of four homogeneous ordinary linear differential equations with the time-dependent coefficients. The physically meaningful parametrically driven nonlinear system permits its concise Hamiltonian formulation with the two pairs of field functions serving as the two pairs of canonically conjugated field amplitudes. The explicit example of oscillatory parametric drive is described in full mathematical details.

Practical fixed-time non-singular sliding mode control of second order nonlinear dynamic systems with chattering and overshooting avoidance

This paper introduces a Practical Fixed-Time Stabilization technique (PFxT) designed for a specific class of nonlinear second-order systems. The closed-loop systems, when appropriately parameterized, exhibit convergence within a predetermined time frame to a confined region near the origin. This outcome is achieved by amalgamating a nonlinear sliding mode approach with a practical fixed-time tracking virtual trajectory. The control algorithmś design incorporates considerations for overshoot reduction and chattering cancellation. Furthermore, the Lyapunov method is employed to establish the practical fixed-time stability of the proposed solution, providing insights into the limits within which the algorithm can be effectively deployed. To complete the algorithm specification, a parameter selection approach is introduced, enabling customization of the desired settling time. Comprehensive simulations are conducted to validate the effectiveness and viability of the proposed PFxT technique.

Nonlinear Dynamics Induced by Coil Heat in the PMDC Motor and Control

In this paper, the interesting dynamics of chaos induced by the effect of the variation of internal average heat during operation in the DC motor control by the full bridge drive are analyzed. By using simple powerful tools of analyzing nonlinear dynamical systems like phase portraits, time traces and frequency spectrum in the MATLAB-SIMULINK environment, we showed that under certain conditions, the PMDC motor develops different behaviors so as periodic limited, and chaotic attractors, when the motor drive different form of external load torque and the windings resistance variation. This paper presents the first studies on the variation of the average heat of the motor and the amplitude of the triangular load torque to produce the strange phenomena like chaos as far as our knowledge go. A chaos control of the unstable regime is proposed to stabilize the PM DC motor in a desire regime. This contribution is very important in industry because some unexplained dynamical behaviors of the DC motor driven by a full bridge now can be avoided.

Analysis of a nonlinear dynamic absorption system

Analysis of a nonlinear dynamic absorption system

Distributed fault estimation of nonlinear dynamical systems over sensor networks: a non-fragile approach

PurposeThe purpose of this paper is to investigate the fault estimation issue of nonlinear dynamical systems via distributed sensor networks. Furthermore, based on the communication topology of sensor networks, the nonfragile design strategy considering the gain fluctuation is also adopted for distributed fault estimators.Design/methodology/approachBy means of intensive dynamical model transformation, sufficient conditions with disturbance attenuation performance are established to design desired fault estimator gains with the help of convex optimization.FindingsA novel distributed fault estimation framework for a class of nonlinear dynamical systems is established over a set of distributed sensor networks, where sampled data of sensor nodes via local information exchanges can be used for more efficiency.Originality/valueThe proposed distributed fault estimator gain fluctuations are taken into account for the nonfragile strategy, such that the distributed fault estimators are more applicable for practical sensor networks implementations. In addition, an illustrative example with simulation results are provided to validate the effectiveness and applicableness of the developed distributed fault estimation technique.

Towards optimal spatio‐temporal decomposition of control‐related sum‐of‐squares programs

AbstractThis paper presents a method for calculating the Region of Attraction (ROA) of nonlinear dynamical systems, both with and without control. The ROA is determined by solving a hierarchy of semidefinite programs (SDPs) defined on a splitting of the time and state space. Previous works demonstrated that this splitting could significantly enhance approximation accuracy, although the improvement was highly dependent on the ad‐hoc selection of split locations. In this work, we eliminate the need for this ad‐hoc selection by introducing an optimization‐based method that performs the splits through conic differentiation of the underlying semidefinite programming problem. We provide the differentiability conditions for the split ROA problem, prove the absence of a duality gap, and demonstrate the effectiveness of our method through numerical examples.

Bifurcation Diagrams of Nonlinear Oscillatory Dynamical Systems: A Brief Review in 1D, 2D and 3D.

We discuss 1D, 2D and 3D bifurcation diagrams of two nonlinear dynamical systems: an electric arc system having both chaotic and periodic steady-state responses and a cytosolic calcium system with both periodic/chaotic and constant steady-state outputs. The diagrams are mostly obtained by using the 0-1 test for chaos, but other types of diagrams are also mentioned; for example, typical 1D diagrams with local maxiumum values of oscillatory responses (periodic and chaotic), the entropy method and the largest Lyapunov exponent approach. Important features and properties of each of the three classes of diagrams with one, two and three varying parameters in the 1D, 2D and 3D cases, respectively, are presented and illustrated via certain diagrams of the K values, -1≤K≤1, from the 0-1 test and the sample entropy values SaEn>0. The K values close to 0 indicate periodic and quasi-periodic responses, while those close to 1 are for chaotic ones. The sample entropy 3D diagrams for an electric arc system are also provided to illustrate the variety of possible bifurcation diagrams available. We also provide a comparative study of the diagrams obtained using different methods with the goal of obtaining diagrams that appear similar (or close to each other) for the same dynamical system. Three examples of such comparisons are provided, each in the 1D, 2D and 3D cases. Additionally, this paper serves as a brief review of the many possible types of diagrams one can employ to identify and classify time-series obtained either as numerical solutions of models of nonlinear dynamical systems or recorded in a laboratory environment when a mathematical model is unknown. In the concluding section, we present a brief overview of the advantages and disadvantages of using the 1D, 2D and 3D diagrams. Several illustrative examples are included.

Variational analysis of sensory feedback mechanisms in powerstroke-recovery systems.

Although the raison d'etre of the brain is the survival of the body, there are relatively few theoretical studies of closed-loop rhythmic motor control systems. In this paper we provide a unified framework, based on variational analysis, for investigating the dual goals of performance and robustness in powerstroke-recovery systems. To demonstrate our variational method, we augment two previously published closed-loop motor control models by equipping each model with a performance measure based on the rate of progress of the system relative to a spatially extended external substrate-such as a long strip of seaweed for a feeding task, or progress relative to the ground for a locomotor task. The sensitivity measure quantifies the ability of the system to maintain performance in response to external perturbations, such as an applied load. Motivated by a search for optimal design principles for feedback control achieving the complementary requirements of efficiency and robustness, we discuss the performance-sensitivity patterns of the systems featuring different sensory feedback architectures. In a paradigmatic half-center oscillator-motor system, we observe that the excitation-inhibition property of feedback mechanisms determines the sensitivity pattern while the activation-inactivation property determines the performance pattern. Moreover, we show that the nonlinearity of the sigmoid activation of feedback signals allows the existence of optimal combinations of performance and sensitivity. In a detailed hindlimb locomotor system, we find that a force-dependent feedback can simultaneously optimize both performance and robustness, while length-dependent feedback variations result in significant performance-versus-sensitivity tradeoffs. Thus, this work provides an analytical framework for studying feedback control of oscillations in nonlinear dynamical systems, leading to several insights that have the potential to inform the design of control or rehabilitation systems.

Engineering approach to construct robust filter for mismatched nonlinear dynamic systems

AbstractThis article proposes a novel approach to design a robust estimator that is able to keep its consistency in system state estimation when system process model mismatch occurs. To successfully develop such an estimator, not only the estimation strategy proposed but also the designer's knowledge and experience about the system behavior are crucial and determining. To assess the performance of the resultant estimator, its performance is compared with that of three well‐known estimators, that is, the unscented Kalman filter, the cubature Kalman filter, and the extended Kalman filter on the IEEE 5‐generator 14‐bus system. The results indicate that the proposed method has led to an estimator outperforming its rivals under the presence of model errors.

The Impact of the Nonlinear Integral Positive Position Feedback (NIPPF) Controller on the Forced and Self-Excited Nonlinear Beam Flutter Phenomenon

This article presents a novel approach to impact regulation of nonlinear vibrational responses in a beam flutter system subjected to harmonic excitation. This study introduces the use of a Nonlinear Integral Positive Position Feedback (NIPPF) controller for this purpose. This technique models the system as a three-degree-of-freedom nonlinear system representing the beam flutter, coupled with a first-order and a second-order filter representing the NIPPF controller. By applying perturbation analysis to the linearized system model, the authors obtain analytical solutions for the autonomous system with the controller. This study aims to reduce vibration amplitudes in a nonlinear dynamic system, specifically when 1:1 internal resonance occurs. The Routh–Hurwitz criterion is utilized to evaluate the system’s stability. Furthermore, the frequency–response curves (FRCs) exhibit symmetry across a range of parameter values. The findings highlight that the effectiveness of vibration suppression is directly related to the product of the NIPPF control signal after comparing with different controllers. Numerical simulations, conducted using the fourth-order Runge–Kutta method, validate the analytical solutions and demonstrate the system’s amplitude response. The strong correlation between the analytical and numerical results highlights the accuracy and dependability of the proposed method.

Finding the rhythm: Humans exploit nonlinear intrinsic dynamics of compliant systems in periodic interaction tasks.

Activities like ball bouncing and trampoline jumping showcase the human ability to intuitively tune to system dynamics and excite motions that the system prefers intrinsically. This human sensitivity to resonance has been experimentally supported for interactions with simple linear systems but remains a challenge to validate in more complex scenarios where nonlinear dynamics cannot be predicted analytically. However, it has been found that many nonlinear systems exhibit periodic orbits similar to the eigenmodes of linear systems. These nonlinear normal modes (NNM) are computable with a recently developed numerical mode tool. Using this tool, the present resarch compared the motions that humans excite in nonlinear systems with the predicted NNM of the energy-conservative systems. In a user study consisting of three experiment parts, participants commanded differently configured virtual double pendula with joint compliance through a haptic joystick. The task was to alternately hit two targets, which were either aligned with the NNM (Experiments 1 and 2) or purposefully arranged offset (Experiment 3). In all tested experiment variations, participants intuitively applied a control strategy that excited the resonance and stabilized an orbit close to the ideal NNM of the conservative systems. Even for increased task accuracy (Experiment 2) and targets located away from the NNM (Experiment 3), participants could successfully accomplish the task, likely by adjusting their arm stiffness to alter the system dynamics to better align the resonant motions to the task. Consequently, our experiments extend the existing research on human resonance sensitivity with data-based evidence to nonlinear systems. Our findings emphasize the human capabilities to apply control strategies to excite and exploit resonant motions in dynamic object interactions, including possibly shaping the dynamics through changes in muscle stiffness.

Data-Driven Fault Detection and Isolation for Multirotor System Using Koopman Operator

This paper presents a data-driven fault detection and isolation (FDI) for a multirotor system using Koopman operator and Luenberger observer. Koopman operator is an infinite-dimensional linear operator that can transform nonlinear dynamical systems into linear ones. Using this transformation, our aim is to apply the linear fault detection method to the nonlinear system. Initially, a Koopman operator-based linear model is derived to represent the multirotor system, considering factors like non-diagonal inertial tensor, center of gravity variations, aerodynamic effects, and actuator dynamics. Various candidate lifting functions are evaluated for prediction performance and compared using the root mean square error to identify the most suitable one. Subsequently, a Koopman operator-based Luenberger observer is proposed using the lifted linear model to generate residuals for identifying faulty actuators. Simulation and experimental results demonstrate the effectiveness of the proposed observer in detecting actuator faults such as bias and loss of effectiveness, without the need for an explicitly defined fault dataset.

Chaos and resonance of nonlinear FG shallow shells reinforced by oblique stiffeners

This paper presents an analysis of the nonlinear dynamics of functionally graded (FG) shallow shells that are simply supported, reinforced with oblique stiffeners, and subjected to external excitation. The properties of these oblique stiffened functionally graded (OSFG) shallow shells vary across their thickness, following a power law distribution based on the volume fractions. Employing the first-order shear deformation theory (FSDT) coupled with the von Kármán nonlinear strain-displacement relations, the derivation of the nonlinear equations of motion is accomplished through the application of Hamilton's principle. The model of FG shallow shells reinforced with oblique two-series stiffeners is developed, allowing for the angles of the two stiffeners to be similar or different from each other. In this context, the employment of the first-order shear deformation theory (FSDT) results in the enhancement of the complexity in modeling the oblique stiffeners. Galerkin's method is applied to discretize the given equations, converting them into a nonlinear dynamical system with two degrees of freedom. This system incorporates both quadratic and cubic nonlinearities and considers the effects of external excitation. The analysis of this research particularly focuses on the resonant scenario involving principal resonance and 1:1 internal resonance. Through the asymptotic perturbation method, a four-dimensional nonlinear averaged equation is obtained. For the first time, a numerical investigation of the research reveals critical insights into the behavior of the system, such as waveforms, phase diagrams, and Poincaré maps, highlighting the effects of different angles of the stiffeners and variations of material parameters on the nonlinear dynamics of these shells.

Investigating amplitude amplification in optimization-based control for a continuous stirred tank reactor

Quantum computers, which utilize quantum states called ‘qubits’ to process information, are becoming of increased interest in a variety of fields because they have the potential to outperform classical computers in certain situations. However, many challenges and hurdles remain, including the development of quantum algorithms that offer a speedup and can be applied to practical problems (some quantum algorithms that offer a speedup may only be applicable in specific and sometimes contrived circumstances). Our previous works have studied the use of quantum computers and quantum algorithms for process control applications. Our prior work evaluated the use of a gate-based quantum amplitude amplification algorithm when applied to a model predictive control optimization problem, specifically evaluating a linear systems example. The results suggested that the algorithm may be suited for the specific linear problem studied (meaning that there is a high probability of obtaining the desired result when measuring the quantum state after the algorithm has been applied). However, the results cannot be generalized to a wider class of systems or to systems containing nonlinearities. To begin to gain an understanding of how the amplitude amplification algorithm might be generalized, this work evaluates the use of the algorithm for the optimization-based control of a nonlinear dynamic system (specifically, a continuous stirred tank reactor). We seek to understand aspects of the problem, such as the development of a solution space, the effects of changing process and control parameters, and the operation of the amplitude amplification algorithm itself. The results demonstrate the challenges that still exist, and demonstrate a method involving inverse sampling transformations to potentially aid in addressing these issues.

Detecting causalities between strongly coupled dynamical systems

A new version of the convergent cross mapping is proposed to detect causalities from records for strongly coupled nonlinear dynamical systems, where the mutual entropy is used to measure nonlinear correlations, and the time delay stability is adopted to filter out false identifications. Calculations on various deterministic dynamic systems show that it is applicable not only to strongly coupled systems but also to non-interacting systems influenced by a common environment. Compared with the original version of convergent cross mapping, under strong couplings our proposed method has significantly higher accuracy, and is more robust to coupling strength. As a typical example, it is used to detect the causal effects between arterial blood pressure (ABP) and intracranial pressure (ICP) of patients diagnosed with traumatic brain injury (TBI). A mono-directional causality from ICP to ABP is identified.

Investigating the nonlinear dynamics of photosensitive Belousov-Zhabotinsky gels via bifurcation analyses.

Controlling the dynamics of active stimuli-responsive smart materials is essential to replicate the biomimetic functionalities at different length scales for a variety of biological systems-based applications. Photosensitive Belousov-Zhabotinsky (BZ) gels, powered by a nonlinear chemical oscillator, called a BZ reaction are one of the stimuli-responsive smart materials in demand due to their ability to continuously transduce chemical oscillations into mechanical deformations. The chemical oscillations in a BZ reaction and subsequent mechanical oscillations in photosensitive BZ gels occur due to the redox cycle of photosensitive ruthenium complex-based catalysts. In this work, our objective is to identify how the behavior of photosensitive BZ gels can be tuned and used for biomimetic applications by investigating its dynamical characteristics using bifurcation analyses. Specifically, we use the normal form approach and perform linear and nonlinear stability analyses to identify high-order bifurcations by computing higher-order Lyapunov and frequency coefficients. We revealed the existence of domains that encompass coexisting stable and unstable limit cycles (LCs), which merge to form a semi-stable LC at the limit point of cycle (LPC). Their existence shows how a slight variation in the BZ gel recipe can significantly alter its dynamics. Subsequently, we quantify the amplitude and frequency of oscillations in different domains under the effect of variation of BZ reaction formulations. We believe that the outcomes of our work serve as an efficient template for the design and control of BZ gel-based applications. The usage of a normal form and a systematic representation of nonlinear dynamics allow our framework to be extended for other nonlinear dynamical systems.

Existence of positive periodic solutions for nonlinear Lotka-Volterra delay dynamic systems on time scales

Existence of positive periodic solutions for nonlinear Lotka-Volterra delay dynamic systems on time scales

Active particle motion in Poiseuille flow through rectangular channels.

We investigate the dynamics of a pointlike active particle suspended in fluid flow through a straight channel. For this particle-fluid system, we derive a constant of motion for a general unidirectional fluid flow and apply it to an approximation of Poiseuille flow through channels with rectangular cross- sections. We obtain a 4D nonlinear conservative dynamical system with one constant of motion and a dimensionless parameter describing the ratio of maximum flow speed to intrinsic active particle speed. Applied to square channels, we observe a diverse set of active particle trajectories with variations in system parameters and initial conditions which we classify into different types of swinging, trapping, tumbling, and wandering motion. Regular (periodic and quasiperiodic) motion as well as chaotic active particle motion are observed for these trajectories and quantified using largest Lyapunov exponents. We explore the transition to chaotic motion using Poincaré maps and show "sticky" chaotic tumbling trajectories that have long transients near a periodic state. We briefly illustrate how these results extend to rectangular cross-sectionswith a width-to-height ratio larger than one. Outcomes of this paper may have implications for dynamics of natural and artificial microswimmers in experimental microfluidic channels that typically have rectangular cross sections.