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Chaos is considered to be essential for life, since it can bring diversity to the world. However, it is difficult to detect the pattern of chaos due to its apparent randomness. In this paper, a nonlinear system based on a Hénon-like map is studied to show the semi-ordered structure of its chaotic sequences. The periodic sequences of such a map are solved analytically, and the stability is analyzed. Multiple coexisting unstable sequences with different periods and the chaotic attractor are obtained. An evidence is found that chaos is not purely apparently random, and it is weakly attracted by different unstable periodic sequences. Due to the weak attraction, it transits from one unstable periodic sequence to another at random instances, and it appears unpredictable.

Optimal control of nonlinear vibration requires precise knowledge of the system and the solution to Hamilton–Jacobi–Bellman (HJB) equation. However, in practical engineering applications, acquiring precise system parameters poses challenges, and the analytical solutions for the HJB equation are difficult to obtain. In this paper, two reinforcement learning algorithms, Twin Delayed Deep Deterministic Policy Gradient (TD3) algorithm and Soft Actor-Critic (SAC) algorithm, are employed to train neural network-based optimal controllers for the van der Pol vibration system in the presence of unknown system parameters. To validate their performance, the controllers undergo testing in a series of experiments, including assessments of free vibration, frequency sweep excitation, and Gaussian noise excitation. The results indicate that both the TD3-trained and SAC-trained neural network-based controller are capable of proficiently suppress the vibration of the van der Pol oscillator. Additionally, these two model-free controllers can approximate the optimal control law which solved based on the dynamic model of the nonlinear system.

A neural-model-based control design for some nonlinear systems is addressed. The design approach is to approximate the nonlinear systems with neural networks of which the activation functions satisfy the sector conditions. A novel neural network model termed standard neural network model (SNNM) is advanced for describing this class of approximating neural networks. Full-order dynamic output feedback control laws are then designed for the SNNMs with inputs and outputs to stabilize the closed-loop systems. The control design equations are shown to be a set of linear matrix inequalities (LMIs) which can be easily solved by various convex optimization algorithms to determine the control signals. It is shown that most neural-network-based nonlinear systems can be transformed into input-output SNNMs to be stabilization synthesized in a unified way. Finally, some application examples are presented to illustrate the control design procedures.

A new approach for nonlinear system identification and control based on modular neural networks (MNN) is proposed in this paper. The computational complexity of neural identification can be greatly reduced if the whole system is decomposed into several subsystems. This is obtained using a partitioning algorithm. Each local nonlinear model is associated with a nonlinear controller. These are also implemented by neural networks. The switching between the neural controllers is done by a dynamical switcher, also implemented by neural networks, that tracks the different operating points. The proposed multiple modelling and control strategy has been successfully tested on simulated laboratory scale liquid-level system.

Recurrent wavelet neural network (RWNN) has the advantages such as fast learning property, good generalization capability and information storing ability. With these advantages, this paper proposes an RWNN-based adaptive control (RBAC) system for multi-input multi-output (MIMO) uncertain nonlinear systems. The RBAC system is composed of a neural controller and a bounding compensator. The neural controller uses an RWNN to online mimic an ideal controller, and the bounding compensator can provide smooth and chattering-free stability compensation. From the Lyapunov stability analysis, it is shown that all signals in the closed-loop RBAC system are uniformly ultimately bounded. Finally, the proposed RBAC system is applied to the MIMO uncertain nonlinear systems such as a mass-spring-damper mechanical system and a two-link robotic manipulator system. Simulation results verify that the proposed RBAC system can achieve favorable tracking performance with desired robustness without any chattering phenomenon in the control effort.

We study the influence of randomly distributed phase directions of external force in an array of coupled pendula, instead of studying the influence of continuous phase. We find that with the increase of the absolute value of the phase, the chaotic behaviors of the coupled arrays may be controlled and different synchronized patterns can be induced. These results demonstrate that by introducing the randomness of the phase directions, rather than the continuous value of the phase, it can lead to a synchronization in nonlinear systems. This finding may provide a new insight for understanding the mechanism of disorder induced synchronization.

We introduce new quantum states of light field, called the nonlinear even and odd displaced number states, by introducing the nonlinearity. The statistical properties investigated presents several interesting quantum effects.

The nonlinear dynamics of a vibration-controlled magnetic system are studied via a three-mode discretization of the governing partial differential equations. The analysis focuses specifically on the effects of modal coupling through the nonlinear terms of the system equation. A bifurcation analysis of the system is performed using sophisticated nonlinear theories, including the center manifold theory and the normal form theorem. The results show that when the first mode and the higher modes are excited simultaneously by the control forces, the three-mode approximation method predicts the existence of a triple zero degeneracy accompanied by complicated bifurcation phenomena. Comparing the dynamics structure predicted by the three-mode approximation model with that obtained from a single-mode approach, it is found that if the higher modes are excited by the control forces, the effects of modal coupling should be taken into consideration since a complicated dynamics structure may exist as a result.

In this paper, we investigate a nonlinear system, which describes the marginally unstable baroclinic wave packets in the geophysical fluid. Based on the symbolic computation and Hirota method, bright one- and two-soliton solutions for such a system are derived. Propagation and collisions of the solitons are graphically shown and discussed with $\beta$, which reflects the collision between the wave packet and mean flow, $\alpha$, which measures the state of the basic flow, and group velocity $\gamma$. $\gamma$ is observed to affect the amplitudes of the solitons, and $\alpha$ can influence the solitons’ traveling directions. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions are derived. Properties of the first- and second-order rogue waves are graphically presented and analyzed: The first-order rogue waves are shown in the figures. $\alpha$ has no effects on A, which is the amplitude of the wave packet, but with the increase of $\alpha$, amplitude of B, which is a quantity measuring the correction of the basic flow, decreases. When $\beta$ is chosen differently, A and B do not keep their shapes invariant. With the value of $\gamma$ increasing, amplitudes of A and B become larger. The second-order rogue wave is presented, from which we observe that with $\alpha$ increasing, amplitude of B decreases, but $\alpha$ has no effects on A. Collision features of A and B alter with the value of $\beta$ changing. When we make the value of $\gamma$ larger, amplitudes of A and B increase.

In this paper, we present a new method for the modeling and characterization of oscillator circuit with a Van Der Pol (VDP) model using parameter identification. We also discussed and investigated the problem of estimation in nonlinear system based on time domain data. The approach is based on an appropriate state space representation of Van der Pol oscillator that allows an optimal parameter estimation. Using sampled output voltage signal, model parameters are obtained by an iterative identification algorithm based on Output Error method. Normalization issues are fixed by an appropriate transformation allowing a quickly global minimum search. Finally, the proposed estimation method is tested and validated using simulation data from a 1 GHz oscillator circuit in GaAs technology.

This paper proposes the design of a precoder in MIMO systems dedicated to JPEG Wireless (JPWL) (ISO/IEC 15444-11) image transmission which jointly takes into account the channel noise and the nonlinear distortions due to the high power amplifier (HPA) used in the transmitter. MIMO-OFDM (Orthogonal Frequency Division Multiplexing) systems dedicated to image transmission rely only on the Channel State Information (CSI) and the image content to design their precoding solutions. However, the nonlinear behavior of the HPA used at the front-end is not taken into account when designing the current precoders. In order to ensure the QoS of the transmitted image, a large level back-off is usually applied to avoid the nonlinear distortions generated by the HPA. In addition, the Input Back-off (IBO) value is statically chosen and does not depend on the other parameters of the link such as the instantaneous MIMO channel status, the modulation order and the scalability of the JPWL content. This may reduce both the global energy efficiency and the transmission quality of the Radio-Frequency (RF) link. Therefore, this work proposes a new precoding scheme to allocate the total available power while dynamically adjusting the optimal power IBO value according to the HPA parameters, the instantaneous channel status and the image content to be transmitted. Numerical results using a commercial HPA model under the IEEE 802.11n standard show that the proposed power allocation algorithm improves the visual quality of the received JPWL images compared to the state-of-the-art over a realistic radio channel model.

In this paper, fractional-order chaotic systems in an analog-based platform are realized using field programmable analog arrays (FPAA) hardware. With the help of this work, we aim to increase the complexity of chaotic systems. Approximated transfer functions in frequency domain are obtained by analyzing different values of fractional-order integrator with the Charef approximation method. In this study, fractional-order numerical calculation of Rssler and Sprott type-H chaotic systems is carried out. MATLAB Simulink model for chaotic systems that satisfy the conditions of chaos in the boundaries of fractional order value is schematically presented. Moreover, CAM designs and analysis that facilitate the realization of fractional-order transfer functions in FPAA platforms are introduced. The analog-based FPAA experimental and numerical outcomes for fractional order chaotic systems are demonstrated. The comparison of the results obtained in the numerical analysis and simulation study with the experimental results is given. This study confirms that the unpredictability of the chaos carrier signals realized by digital-based can be increased with analog-based FPAA hardware and fractional-order structures so as to provide safer transfer of information signals.

Based on the variable structure model reference adaptive control (VS-MRAC) theory, a new control system for the control of chaos in Lorenz system, using only the measured output variable, is designed. For the derivation of the control law, it is assumed that the parameters of the model are unknown. Moreover, it is assumed that a disturbance input is present in the system. It is shown that in the closed-loop system, the output variable tracks a given reference trajectory, and the state vector converges to the equilibrium state. Digital simulation results show that the closed-loop system has good transient behavior and robustness to the uncertainties and disturbance input.

In this Letter a feedback linearizing adaptive control system for the control of Chua's circuits is presented. It is assumed that all the parameters of the system are unknown. Using a backstepping design procedure, an adaptive control system is designed which accomplishes trajectory control of the chosen node voltage by an independent voltage source. Simulation results are presented which show precise trajectory tracking and regulation of the state vector to the desired terminal state.

Nonlinear dynamical models are frequently used to approximate and predict observed physical, biological and economic systems. Such models will be subject to errors both in the model dynamics, and the observations of the underlying system. In order to improve models, it is necessary to understand the causes of error growth. A complication with chaotic models is that small errors may be amplified by the model dynamics. This paper proposes a technique for estimating levels of both dynamical and observational noise, based on the model drift. The method is demonstrated for a number of models, for cases with both stochastic and nonstochastic dynamical errors. The effect of smoothing or treating the observations is also considered. It is shown that use of variational smoothing techniques in the presence of dynamical model errors can lead to potentially deceptive patterns of error growth.

In this paper, we examine the localization problem of compact invariant sets of systems with the differentiable right-side. The localization procedure consists in applying the iterative algorithm based on the first order extremum condition originally proposed by one of authors for periodic orbits. Analysis of a location of compact invariant sets of the Lanford system is realized for all values of its bifurcational parameter.

In this paper we study the localization problem of compact invariant sets of the optically injected laser system by applying the first order extremum conditions and cylindrical coordinates. Our main results consist in finding localization sets of simple forms which can be easily computed. Special attention is focussed on the case where we obtain a compact localization set.

Intermittent instability is commonly observed in switching power supplies during the design and development phase. It manifests as symmetrical period-doubling bifurcation in the time domain with long intermittent periods. Such intermittent operation is considered undesirable in practice and is usually avoided by appropriate adjustments of circuit parameters. This paper explores the mechanism and conditions for the emergence of intermittency in a common voltage-mode controlled buck converter. It is found that interference at frequencies near the switching frequency or its rational multiples will induce intermittent operation. The strengths and frequencies of the interfering signals determine the type and period of intermittency. The problem is analyzed by transforming the conventional parameter-bifurcation analysis to a time-bifurcation analysis. Analytical results are verified by simulations and experimental measurements.

In this paper we study the localization problem of all compact invariant sets of the five-dimensional coupled laser system by applying the first-order extremum conditions. Our main results consist in finding a number of localization sets formed by frusta, a circular cylinder, two parabolic cylinders and some other quadratic surfaces. Parameters of these localization sets are computed explicitly.

In this paper we examine the localization problem of compact invariant sets of nonlinear time-varying systems with the differentiable right-side. We extend our results respecting the localization problem obtained earlier for time-invariant systems and apply them to a damped driven pendulum and the Vallis model with regard to the seasonal cycle.