Nonlinear Nonautonomous Systems Research Articles

The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

In this work, we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded random hyperbolic solution for the associated nonlinear nonautonomous random dynamical systems. Moreover, we show that these random hyperbolic solutions converge to the autonomous equilibrium. As an application, we consider a semilinear differential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation.

Application of the Method of Non-Classical Approach to Certain Problems of the Theory of Anisotropic Space Plazma in Fluid Description

In this paper, the new well-posed boundary value problem to the system of singular nonlinear differential equations is considered, which describes the stationary radial outflow of anisotropic plasma from the Sun (solar wind). These equations are obtained on the basis of 16-moment MHD (magnetohydrodynamic) transport equations for a collisionless magnetized plasma, which takes into account the temperature anisotropy relative to the direction of the magnetic field and the heat flux carried by the wind. This is a generalization of the classical isotropic Parker model taking into account the effects of anisotropy. In this paper, the equations under study are characterized as a non-autonomous nonlinear system of ordinary differential equations the coefficients in which degenerate and simultaneously have singularities. These equations are related to an unsolved problem in the general theory of ODEs (Ordinary Differential Equations). At first, according to the conditions of the coefficients of the equations, a non-classical boundary value problem is set, and the solvability is established for the same non-autonomous and nonlinear system of equations under consideration. The found analytical solution reconstructs numerical solutions, which are simultaneously automatically established by classical formulation of boundary value problem. Parker’s solutions are also partially included in this obtained class of solutions, which is presented with strictly proves. Further, by means of the methods of “ -regularization” and “fixed point” the theorem of solvability for the considered differential equations is obtained. After constructed nonsingular system equations with well-posed boundary value problem, the analytical solutions are founded. Using the sketch of graph of these solutions their family is established.

Successive estimations of bilateral bounds and trapping/stability regions of solution to some nonlinear nonautonomous systems

Estimation of the degree of stability and the bounds of solutions to nonautonomous nonlinear systems present major concerns in numerous applied problems. Yet, current techniques frequently yield overconservative conditions which are unable to effectively gage these characteristics in time-varying nonlinear systems. This paper develops a novel methodology providing successive approximations to solutions that are stemmed from the trapping/stability regions of these systems and estimate the errors of such approximations. In turn, this leads to successive approximations of both the bilateral bounds of solutions and the boundaries of trapping/stability regions of the underlying systems. Along these lines we formulate enhanced stability/boundedness criteria and contrast our inferences with inclusive simulations which reveal dependence of the trapping/stability regions upon the structure of time-dependent components and initial time moment.

Bending-torsion coupling bursting oscillation of a sandwich conical panel under parametric excitation

Regarding the dynamic behavior of the functionally graded materials (FGM) conical panels under the static and harmonic in-plane loading, most studies focus on parametric resonances and dynamic instability which determine the stability regions. Studies on the bursting oscillation of the conical panels under the lower frequency excitation are quite rare. In this paper, we present a novel nonautonomous system of the cantilever functionally graded materials sandwich conical panels with a slowly varying parametric excitation and study the bending-torsion coupling bursting oscillations. The face sheets and core layer are the metal ceramic composite layer and homogeneous metal layer, respectively. In two surfaces layers, the component materials are gradually changed in the transverse direction and the temperature dependent properties show the form of the power law distribution. The excitations in the meridional direction include the static preload and lower frequency harmonic excitation. The mode functions of the first-order bending mode and the first-order torsion mode are obtained in terms of Chebyshev polynomials with the help of energy principle. The bending-torsion coupling nonautonomous nonlinear oscillation system is derived by Lagrange equations. Under the slowly varying in-plane load, the equilibrium solutions of the system are obtained by treating them as a generalized autonomous system and they will be no longer stable through the pitchfork bifurcation with the symmetry breaking. The bursting oscillations of the novel dynamic system are analyzed by numerical methods. It is interesting to find that the mechanisms of bursting oscillation are associated with the pitchfork bifurcation with the symmetry breaking.

New results on stability for non‐linear Markov switched stochastic functional differential systems

For Markov switched stochastic functional differential systems (SFDSs), asymptotic property is one of the most desired issues. Recently, a new class of delay-dependent asymptotic stability for non-linear Markov switched SFDSs was investigated. However, the existing references do not take the convergent speed and non-autonomous factor into consideration. Therefore, by means of multiple Lyapunov–Krasovskii functionals, this study is devoted to examine the exponential stability for highly non-linear autonomous Markov switched SFDSs and the exponential stability, polynomial stability and polynomial growth at most for highly non-linear non-autonomous systems, where all the criteria rely on the intervals lengths of continuous delays. In addition, the properties of boundedness and asymptotic stability are as well explored.

Multistability in a Periodically Forced Brusselator

In this paper, we report on multistability in a periodically forced Brusselator, which is modeled by a nonlinear nonautonomous system of two first-order ordinary differential equations. Multistability regions are detected in a cross section of the four-dimensional parameter space of the model, namely the (ω, F) parameter plane, where ω and F are respectively angular frequency and amplitude of an external forcing. Lyapunov exponents spectra are used to characterize the dynamical behavior of each point in the abovementioned parameter plane. Moreover, basins of attraction, bifurcation diagrams, and phase-space portraits are used to illustrate the coexistence of periodic and chaotic behaviors.

On inhomogeneous nonholonomic Bilimovich system

We consider a simple motivating example of a non-Hamiltonian dynamical system with time-dependent constraints obtained by imposing rheonomic non-integrable Bilimovich’s constraint on a freely rotating rigid body. Dynamics of this low-dimensional nonlinear nonautonomous dynamic system involves different kinds of stable and unstable attractors, quasi and strange attractors, compact and noncompact invariant attractive curves, etc. To study this cautionary example we apply the Poincaré map method to disambiguate and discover multiscale temporal dynamics, specifically the coarse-grained dynamics resulting from fast-scale nonlinear control via nonholonomic Bilimovich’s constraint.

A GUAS Joint Position Tracking Controller of Torque-Driven Robot Manipulators Influenced by Dynamic Dahl Friction: Theory and Experiments

This article presents a controller for global joint position tracking control of torque-driven robot manipulators involving dynamic Dahl friction. The control law has been designed using a novel theoretical Hamiltonian inspired framework, which allows global uniform asymptotic stability (GUAS) of the nonautonomous nonlinear closed-loop system. An observer is provided to estimate the nonmeasurable state of internal friction of the Dahl model, preserving the GUAS attribute in the closed-loop system. The performance of the proposed control scheme is illustrated through experiments on a two-degree-of-freedom direct-drive robot arm.

Nonlinear Responses of An Unbalanced Overhung Rotor-Short Journal Bearing System with Some Bifurcation Results

Both autonomous and non-autonomous nonlinear systems display complex responses including the transient and long run responses. An overhung rotor-short journal bearing system (ORSJB) is the excellent paradigm exhibiting nonlinear responses and of practical importance. The unbalanced overhung rotor-journal bearing system is formulated by using modified Laval-Jeffcott rotor model. The nonlinear fluid forces model of the short journal bearing is adopted and rearranged in a suitable form for numerical analysis. This paper investigates numerically a bifurcation point, transient and long run responses of the ORSJB system. The rotational speed is preferable bifurcation parameter herein. The ORSJB system without unbalanced eccentricity for the selected parameters yields the bifurcation value of Ω = 3.0091. Limits cycles in the long-run responses with eU =0.002 m and without unbalanced eccentricity at long run are presented in the bifurcation regime at the speed of Ω = 6.6869 and Ω = 0.01671 respectively. The transient response with eU =0.002 m at the speed of Ω = 4.7142is elucidated. The evident results in this paper give only a partial view of bifurcation behaviours. Bifurcation analysis is a useful tool for design and operation overhung rotor dynamic systems.

Coexistence of bistable multi-pulse chaotic motions with large amplitude vibrations in buckled sandwich plate under transverse and in-plane excitations.

This paper researches the dynamic snap-through phenomena and the coexistence of the multi-pulse jumping chaotic dynamics in bistable equilibrium positions and the large amplitude nonlinear vibrations for a simply supported buckled 3D-kagome truss core sandwich rectangular plate under combined transverse and in-plane excitations based on the extended high-dimensional Melnikov method. According to the two-degree-of-freedom non-autonomous nonlinear dynamical system of the buckled truss core sandwich rectangular plate and the theory of normal form, it is found that some nonlinear terms have less effects on the nonlinear dynamic responses than other nonlinear terms. The extended high-dimensional Melnikov method is employed to investigate the dynamic snap-through phenomena and the bistable multi-pulse jumping chaotic vibrations around the upper-mode and the lower-mode for two different cases of buckling in the truss core sandwich rectangular plate. A numerical method is used to detect the bistable multi-pulse jumping chaotic vibrations around two stable equilibrium positions for the non-autonomous buckled truss core sandwich rectangular plate. The obtained theoretical and numerical results indicate that there exists the coexistence of the bistable multi-pulse jumping chaotic vibrations and the large amplitude nonlinear vibrations in the buckled truss core sandwich rectangular plate under combined transverse and in-plane excitations.

A New Class of Digital Circuits for the Design of Entropy Sources in Programmable Logic

We propose a novel class of Digital Nonlinear Oscillators (DNOs) supporting complex dynamics, including chaos, suitable for the definition of high-performance and low-complexity entropy sources in Programmable Logic Devices (PLDs). We derive our proposal from the analysis of simplified models, investigated as non-autonomous nonlinear dynamical systems under different excitation conditions. The study lead the authors to the design of a fully digital entropy source consuming only two slices of a Xilinx FPGA, including post-processing, sufficient to define a class of TRNGs capable to pass the NIST standard tests for randomness in any worst case experimentally tested by the authors (6 chips, 96 generators). The solution has been compared with others published in the literature, confirming the validity of the proposal.

Robust stabilization of non-linear non-autonomous control systems with periodic linear approximation

Abstract The paper deals with the problem of stabilizing the equilibrium states of a family of non-linear non-autonomous systems. It is assumed that the nominal system is a linear controlled system with periodic coefficients. For the nominal controlled system, a new method for constructing a Lyapunov function in the quadratic form with a variable matrix is proposed. This matrix is defined as an approximate solution of the Lyapunov matrix differential equation in the form of a piecewise exponential function based on partial sums of a W. Magnus series. A stabilizing control in the form of a linear feedback with a piecewise constant periodic matrix is constructed. This control simultaneously stabilizes the considered family of systems. The estimates of the domain of attraction of an asymptotically stable equilibrium state of a closed-loop system that are common for all systems are obtained. A numerical example is given.

On stability of nonlinear nonautonomous discrete fractional Caputo systems

Conditions to establish Mittag-Leffler stability of solutions for nonlinear nonautonomous discrete Caputo-like fractional systems just from the linear associated system is shown. Mittag-Leffler stability for linear systems is tackled pointing out properties the matrix must satisfy. Additionally features on solutions for linear systems are included in order to establish the equivalence between uniform asymptotical stability and exponential stability. Converse-like-Lyapunov Theorem is developed and a particular result to show Mittag-Leffler stability from a condition on a linear part of the full system is worked.

Lax pair, conservation laws, Darboux transformation, breathers and rogue waves for the coupled nonautonomous nonlinear Schrödinger system in an inhomogeneous plasma

Abstract Plasmas are believed to be possibly “the most abundant form of ordinary matter in the Universe”. In this paper, a coupled nonautonomous nonlinear Schrodinger system is investigated, which describes the propagation of two envelope solitons in a weakly inhomogeneous plasma with the t-dependent linear and parabolic density profiles and nonconstant collisional damping. Lax pair with the nonisospectral parameter and infinitely-many conservation laws are derived. Based on the Lax pair, the Nth-step Darboux transformation is constructed. Utilizing the Nth-step Darboux transformation, we obtain the breather and rogue wave solutions, and find that the amplitude of the nonzero background is nonconstant and dependent on the inhomogeneous coefficients in the system under investigation. Characteristics of the breathers and rogue waves are discussed, and effects of the inhomogeneous coefficients on the breathers and rogue waves are analyzed. Breathers and rogue waves with the dark or bright soliton together are also constructed and their characteristics are discussed. We find that the dark and bright solitons can coexist and generate the breather-like waves.

Differentiator-Based Output-Feedback Controller for Uncertain Nonautonomous Nonlinear Systems With Unknown Relative Degree

A novel output-feedback controller for uncertain nonautonomous nonlinear systems with unknown relative-degree is proposed in this study. With the assumption that the upper bound of the relative degree is known, the proposed control scheme is designed based on the input filter and higher-order switching differentiator(HOSD) with over dimension. Using the overestimated time-derivatives of tracking error by HOSD, the proposed controller compensates for the effects of uncertainties including an unknown relative degree in the controlled system. Assuming that the internal zero dynamics, if they exist, are stable, the asymptotic stability of the closed-loop system is guaranteed.

Periodic pulse control of Hopf bifurcation in a fractional-order delay predator\u2013prey model incorporating a prey refuge

This paper is concerned with periodic pulse control of Hopf bifurcation for a fractional-order delay predator–prey model incorporating a prey refuge. The existence and uniqueness of a solution for such system is studied. Taking the time delay as the bifurcation parameter, critical values of the time delay for the emergence of Hopf bifurcation are determined. A novel periodic pulse delay feedback controller is introduced into the first equation of an uncontrolled system to successfully control the delay-deduced Hopf bifurcation of such a system. Since the stability theory is not well-developed for nonlinear fractional-order non-autonomous systems with delays, we investigate the periodic pulse control problem of the original system by a semi-analytical and semi-numerical method. Specifically, the stability of the linearized averaging system of the controlled system is first investigated, and then it is shown by numerical simulations that the controlled system has the same stability characteristics as its linearized averaging system. The proposed periodic pulse delay feedback controller has more flexibility than a classical linear delay feedback controller guaranteeing the control effect, due to the fact that the pulse width in each control period can be flexibly selected.

On mappability of control systems to linear systems with analytic matrices

The paper deals with the problem of analytic linearizability for nonlinear non-autonomous control systems, i.e., the problem of mapping of a nonlinear non-autonomous control system to a linear system with analytic matrices by a transformation of coordinates. We introduce a concept of a driftless form of a non-autonomous system, which allows simplifying our analysis. We give linearizability conditions and find invariants for nonlinear systems that are mappable to a preassigned linear system.

Poincaré maps design for the stabilization of limit cycles in non-autonomous nonlinear systems via time-piecewise-constant feedback controllers with application to the chaotic Duffing oscillator

Abstract In this paper, a design of Poincare maps and time–piecewise–constant state–feedback control laws for the stabilization of limit cycles in periodically–forced, non–autonomous, nonlinear dynamical systems is achieved. Our methodology is based mainly on the linearization of the nonlinear dynamics around a desired period–m unstable limit cycle. Thus, this strategy permits to construct an explicit mathematical expression of a controlled Poincare map from the structure of several local maps. An expression of the generalized controlled Poincare map is also developed. To make comparisons, we design three different time–piecewise–constant control laws: an mT–piecewise–constant control law, a T–piecewise–constant control law and a T n –piecewise–constant control law. As an illustrative application, we adopt the chaotic Duffing oscillator. By applying the designed piecewise–constant control laws to the Duffing oscillator, the system is stabilized on its desired period–m limit cycle and hence the chaotic motion is controlled.

Approximation-Free Output-Feedback Non-Backstepping Controller for Uncertain SISO Nonautonomous Nonlinear Pure-Feedback Systems

A novel differentiator-based approximation-free output-feedback controller for uncertain nonautonomous nonlinear pure-feedback systems is proposed. Using high-order sliding mode observer, which is a finite-time exact differentiator, the time-derivatives of the signal generated using tracking error and filtered input are directly estimated. As a result, the proposed non-backstepping control law and stability analysis are drastically simple. The tracking error vector is guaranteed to be exponentially stable in finite time regardless of the nonautonomous property in the considered system. It does not require neural networks or fuzzy logic systems, which are typically adopted to capture unstructured uncertainties intrinsic in the controlled system. As far as the authors know, there are no research results on the output-feedback controller for the uncertain nonautonomous pure-feedback nonlinear systems. The results of the simulation show clearly the performance and compactness of the control scheme proposed.

Chaotic dynamics of a non-autonomous nonlinear system for a smart composite shell subjected to the hygro-thermal environment

In this research, nonlinear dynamic behaviors of multiscale composites doubly curved shells have been investigated by employing multiple scales Perturbation Method. Three-phase composites shells with polymer/Carbon nanotube/fiber (PCF) according to Halpin–Tsai model have been assumed. The displacement- strain of nonlinear vibration of multiscale laminated doubly curved shells via higher order shear deformation (HSDT) theory and using Green–Lagrange nonlinear shell theory is obtained. The governing equations of composite doubly curved shell have been derived by implementing Hamilton’s principle and shell considered to be simply supported. For investigating correctness and accuracy, this paper is validated by other previous researches. Finally, bifurcation diagram, phase portraits and Poincare maps are investigated. The results indicate different dimensionless force; curvature ratio and kind of distribution pattern have strong influence on nonlinear vibration control of the composite multiscale doubly curved shell.