Nonlinear Second-order Dynamical Systems Research Articles

Few- and single-molecule reservoir computing experimentally demonstrated with surface-enhanced Raman scattering and ion gating.

Molecule-based reservoir computing (RC) is promising for achieving low power consumption neuromorphic computing, although the information-processing capability of small numbers of molecules is not clear. Here, we report a few- and single-molecule RC that uses the molecular vibration dynamics in the para-mercaptobenzoic acid (pMBA) detected by surface-enhanced Raman scattering (SERS) with tungsten oxide nanorod/silver nanoparticles. The Raman signals of the pMBA molecules, adsorbed at the SERS active site of the nanorod, were reversibly perturbated by the application of voltage-induced local pH changes near the molecules, and then used to perform time-series analysis tasks. Despite the small number of molecules used, our system achieved good performance, including >95% accuracy in various nonlinear waveform transformations, 94.3% accuracy in solving a second-order nonlinear dynamic system, and a prediction error of 25.0 milligrams per deciliter in a 15-minute-ahead blood glucose level prediction. Our work provides a concept of few-molecular computing with practical computation capabilities.

Data-Driven Operator Theoretic Methods for Phase Space Learning and Analysis

This paper uses data-driven operator theoretic approaches to explore the global phase space of a dynamical system. We defined conditions for discovering new invariant subsets in the state space of a dynamical system starting from an invariant subset based on the spectral properties of the Koopman operator. When the system evolution is known locally in several invariant subsets in the state space of a dynamical system, a phase space stitching result is derived that yields the global Koopman operator. Additionally, in the case of equivariant systems, a phase space stitching result is developed to identify the global Koopman operator using the symmetry properties between the invariant subsets of the dynamical system and time-series data from any one of the invariant subsets. Finally, these results are extended to topologically conjugate dynamical systems; in particular, the relation between the Koopman tuple of topologically conjugate systems is established. The proposed results are demonstrated on several second-order nonlinear dynamical systems including a bistable toggle switch. Our method elucidates a strategy for designing discovery experiments: experiment execution can be done in many steps, and models from different invariant subsets can be combined to approximate the global Koopman operator.

Swarm intelligence based approach for efficient training of regressive neural networks

This work proposes an efficient approach to solve the problem of training a regressive neural network efficiently. Regressive networks are characterized by delay lines possibly in both the input and the output feedback. Each delay line is connected to the network with synaptic weights and thus increases the number of parameters that must be optimized by the training algorithm. Training algorithms such as the Levenberg–Marquardt, normally used to train neural networks, are prone to local minima entrapment, and for this reason, a strategy to initialize the training procedure correctly is needed. To solve this problem, the continuous flock of starling optimization algorithm, a highly explorative optimizer based on swarm intelligence, is used. The proposed approach is tested and validated on an experimental benchmark featuring a second-order nonlinear dynamic system.

Physical reservoir computing using magnetic skyrmion memristor and spin torque nano-oscillator

Spintronic nanodevices have ultrafast nonlinear dynamic and recurrence behaviors on a nanosecond scale that promises to enable a high-performance spintronic reservoir computing (RC) system. Here, two physical RC systems based on one single magnetic skyrmion memristor (MSM) and 24 spin-torque nano-oscillators (STNOs) are numerically modeled to process image classification task and nonlinear dynamic system prediction, respectively. Based on the nonlinear responses of the MSM and STNO with current pulse stimulation, our results demonstrate that the MSM-based RC system exhibits excellent performance on image classification, while the STNO-based RC system does well in solving the complex unknown nonlinear dynamic problems, e.g., a second-order nonlinear dynamic system and NARMA10. Our result and analysis of the current-dependent nonlinear dynamic properties of the MSM and STNO provide the strategy to optimize the experimental parameters in building the better spintronic-based brainlike devices for machine learning based computing.

The Vehicle-to-Vehicle Link Duration Scheme Using Platoon-Optimized Clustering Algorithm

In the scenario of the vehicle network, the unstable vehicle to vehicle (V2V) connectivity caused by a high-speed vehicle movement is improved through efficient clustering algorithm to some extent. However, in view of the insufficient consideration of the variation of cluster stability in existing clustering algorithms, the V2V connectivity needs to be further enhanced. To this end, in this paper, a V2V link duration scheme using platoon-optimized clustering algorithm is proposed, which consists of three aspects: system model, a platoon-optimized clustering algorithm, and the analysis of link duration. Specifically, the second-order nonlinear dynamic system for platoon is analyzed to initialize the network scene. Then, a platoon-optimized clustering algorithm is performed, which combines the platoon leader (PL) selection based on motion consistency and the updated algorithm based on the dynamically stable cluster to reduce the randomness caused by the dynamic change of network topology. In addition, the V2V link duration based on the proposed scheme is quantitatively implemented through specific mathematical formulas. Finally, the extensive simulation results show that the proposed scheme can effectively guarantee the cluster stability and prolong the V2V link duration.

Finite-Time Distributed Average Tracking for Second-Order Nonlinear Systems.

This paper studies the distributed average tracking (DAT) problem for multiple reference signals described by the second-order nonlinear dynamical systems. Leveraging the state-dependent gain design and the adaptive control approaches, a couple of DAT algorithms are developed in this paper, which are named finite-time and adaptive-gain DAT algorithms. Based on the finite-time one, the states of the physical agents in this paper can track the average of the time-varying reference signals within a finite settling time. Furthermore, the finite settling time is also estimated by considering a well-designed Lyapunov function in this paper. Compared with asymptotical DAT algorithms, the proposed finite-time algorithm not only solve finite-time DAT problems but also ensure states of physical agents to achieve an accurate average of the multiple signals. Then, an adaptive-gain DAT algorithm is designed. Based on the adaptive-gain one, the DAT problem is solved without global information. Thus, it is fully distributed. Finally, numerical simulations show the effectiveness of the theoretical results.

Exponential synchronization for second-order nonlinear systems in complex dynamical networks with time-varying inner coupling via distributed event-triggered transmission strategy

This paper focuses on the exponential synchronization problem of complex dynamical networks (CDNs) with time-varying inner coupling via distributed event-triggered transmission strategy. Information update is driven by properly defined event, which depends on the measurement error. Each node is described as a second-order nonlinear dynamic system and only exchanges information with its neighboring nodes over a directed network. Suppose that the network communication topology contains a directed spanning tree. A sufficient condition for achieving exponential synchronization of second-order nonlinear systems in CDNs with time-varying inner coupling is derived. Detailed theoretical analysis on exponential synchronization is performed by the virtues of algebraic graph theory, distributed event-triggered transmission strategy, matrix inequality and the special Lyapunov stability analysis method. Moreover, the Zeno behavior is excluded as well by the strictly positive sampling intervals based on the upper right-hand Dini derivative. It is noted that the amount of communication among network nodes and network congestion have been significantly reduced so as to avoid the waste of network resources. Finally, a simulation example is given to show the effectiveness of the proposed exponential synchronization criteria.

Numerical scheme for period-m motion of second-order nonlinear dynamical systems based on generalized harmonic balance method

Prediction of periodic motion plays key roles in identifying bifurcations and chaos for nonlinear dynamical systems. In this paper, a semi-analytical and semi-numerical scheme is developed as a combination of the analytical generalized harmonic balance method and the Newton–Raphson iteration for period-m solution of second-order nonlinear systems. The nonlinear external loading is approximated by the Taylor’s expansion of displacement and velocity, and is expressed as summations of many orders of Fourier harmonics pairs. A set of nonlinear algebraic equations are solved iteratively for the coefficients of harmonic pairs until the convergence of solution is achieved. The periodic solutions for period-2 motion in a periodically forced Duffing oscillator and period-3 motion in a buckled, nonlinear Jeffcott rotor system are obtained from the present scheme, and the corresponding stability and bifurcation are evaluated through eigenvalue analysis. The results from the present scheme are found in good agreement with the existent analytical solutions. The present scheme can be used as a general purpose numerical realization of the generalized harmonic balance method in evaluating periodical nonlinear dynamical systems since it is not involved with analytical derivation of Fourier expansion of external loading.

Output Feedback Fractional-Order Nonsingular Terminal Sliding Mode Control of Underwater Remotely Operated Vehicles

For the 4-DOF (degrees of freedom) trajectory tracking control problem of underwater remotely operated vehicles (ROVs) in the presence of model uncertainties and external disturbances, a novel output feedback fractional-order nonsingular terminal sliding mode control (FO-NTSMC) technique is introduced in light of the equivalent output injection sliding mode observer (SMO) and TSMC principle and fractional calculus technology. The equivalent output injection SMO is applied to reconstruct the full states in finite time. Meanwhile, the FO-NTSMC algorithm, based on a new proposed fractional-order switching manifold, is designed to stabilize the tracking error to equilibrium points in finite time. The corresponding stability analysis of the closed-loop system is presented using the fractional-order version of the Lyapunov stability theory. Comparative numerical simulation results are presented and analyzed to demonstrate the effectiveness of the proposed method. Finally, it is noteworthy that the proposed output feedback FO-NTSMC technique can be used to control a broad range of nonlinear second-order dynamical systems in finite time.

Some features of the deceleration of an inhomogeneous ball in the air

A model of the deceleration of an inhomogeneous ball acted upon by the drag of the air is discussed, taking into account the interaction of the translational and rotational motions. The problem is reduced to analysing a non-linear second-order dynamical system. The steady motions of the ball, including self-oscillatory and self-rotation motions, are obtained. The bifurcation values of the parameters defining these motions are determined. The corresponding phase portraits are constructed and an interesting interpretation of them is given.

SYNCHRONIZATION OF A CLASS OF SECOND-ORDER NONLINEAR SYSTEMS

The asymptotic behavior of a class of coupled second-order nonlinear dynamical systems is studied in this paper. Using very mild assumptions on the vector-field, conditions on the coupling parameters that guarantee synchronization are provided. The proposed result does not require solutions to be ultimately bounded in order to prove synchronization, therefore it can be used to study coupled systems that do not globally synchronize, including synchronization of unbounded solutions. In this case, estimates of the synchronization region are obtained. Synchronization of two-coupled nonlinear pendulums and two-coupled Duffing systems are studied to illustrate the application of the proposed theory.

Groups and nonlinear dynamical systems dynamics on the SU (2) group

An abstract Newton-like equation on a general Lie algebra is introduced such that orbits of the Lie-group action are attracting set. This equation generates the nonlinear dynamical system satisfied by the group parameters having an attractor coinciding with the orbit. The periodic solutions of the abstract equation on a Lie algebra are discussed. The particular case of the SU (2) group is investigated. The resulting nonlinear second-order dynamical system in R 3 as well as its constrained version referring to the generalized spherical pendulum are shown to exhibit global Hopf bifurcation.

Stochastic linearization method with random parameters for SDOF nonlinear dynamical systems: prediction and identification procedures

This paper describes firstly, the calculation of the Power Spectral Density Function (PSDF) for the stationary response of SDOF nonlinear second-order dynamical systems excited by a white or a broad-band Gaussian noise, and secondly, the identification of a single-degree-of-freedom (SDOF) nonlinear dynamical second-order dynamical system driven by a broad-band or a colored Gaussian noise. The two aspects are based on the use of a stochastic linearization method with random parameters which is an efficient way of approximating the PSDF. The gain obtained by this method is shown on a SDOF nonlinear dynamical system. In addition, it is shown that the stochastic linearization method with random parameters is an efficient approach for identifying a SDOF nonlinear dynamical system.

Method of Modeling, Parameter and State Estimation of Nonlinear Systems

THE problem of obtaining the simultaneous weighted least squares (WLS) estimate of the parameters p and states r(p,t) of a given general first- or second-order nonlinear dynamic system is addressed, respectively, via the use of a quadratic (QSS) or cubic spline series (CSS) time representation of r(p,t). The measured states or functions of the states are assumed to be the true values corrupted by additive zero-mean uncorrelated random noise. The CSS is described in Eq. (6) below.! Unknown state functional dependence in the dynamics, e.g., aerodynamic force modeling of an unknown re-entry vehicle, is effectively represented in the time domain with a vector B(t) of unknown time functions. The B(t) are also parameterized via the spline time series.! This has the ability to accurately represent both complex state functional dependence and well behaved solutions of differential equations with a relatively modest number of terms and, hence, unknown coefficients. The initial conditions and unknown system parameters are WLS estimated from a given set of discrete noisy measurements on the system output at observation times tj(/=!,/) whence r(p,t) and its time derivatives at any t, follow immediately as a linear function of p. A variety of analytical spline algorithms are described that include iterative batch processing for maximum accuracy and sequential recursive processing for real time operation. Sensitivity differential equations, as in standard maximum likelihood parameter estimation, and error covariance propagation equations, as with the extended Kalman filter, are eliminated. Precision, generality, flexibility, and computational efficiency are achieved. These are illustrated in the full paper2'7 with a numerical example based upon the tracking of Van der Pol's equation. The solution is attained via hand calculation for comparison with familiar methods. Additionally, analytical solution expressions are obtained for a practical naval combat weapon systems problem.