Nonlinear Autonomous Systems Research Articles

Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems

In this paper, for any positive integer $n$, we study the Maslov-type index theory of $i_{L_0}$, $i_{L_1}$ and $i_{\sqrt{-1}}^{L_0}$ with $L_0=\{0\}\times \R^n\subset \R^{2n}$ and $L_1=\R^n\times \{0\} \subset \R^{2n}$. As applications we study the minimal period problems for brake orbits of nonlinear autonomous reversible Hamiltonian systems. For first order nonlinear autonomous reversible Hamiltonian systems in $\R^{2n}$, which are semipositive, and superquadratic at zero and infinity, we prove that for any $T>0$, the considered Hamiltonian systems possesses a nonconstant $T$ periodic brake orbit $X_T$ with minimal period no less than $\frac{T}{2n+2}$. Furthermore if $\int_0^T H"_{22}(x_T(t))dt$ is positive definite, then the minimal period of $x_T$ belongs to $\{T,\;\frac{T}{2}\}$. Moreover, if the Hamiltonian system is even, we prove that for any $T>0$, the considered even semipositive Hamiltonian systems possesses a nonconstant symmetric brake orbit with minimal period belonging to $\{T,\;\frac{T}{3}\}$

Existence and analytical approximations of limit cycles in a three-dimensional nonlinear autonomous feedback control system

This paper is concerned with the existence and the analytical approximations of limit cycles in a three-dimensional nonlinear autonomous feedback control system. Based on three-dimensional Hopf bifurcation theorem, the existence of limit cycles is first proved. Then the homotopy analysis method (HAM) is applied to obtain the analytical approximations of the limit cycle and its frequency. In deriving the higher-order approximations, the authors utilized the idea of a perturbation procedure proposed for limit cycles’ approximation in van der Pol equation. By comparing with the numerical integration solutions, it is shown that the accuracy of the analytical results obtained in this paper is very high, even when the amplitude of the limit cycle is large.

Optimal scheduling for reference tracking or state regulation using reinforcement learning

The problem of optimal control of autonomous nonlinear switching systems with infinite-horizon cost functions, for the purpose of tracking a family of reference signals or regulation of the states, is investigated. A reinforcement learning scheme is presented which learns the solution and provides scheduling between the modes in a feedback form without enforcing a mode sequence or a number of switching. This is done through a value iteration based approach. The convergence of the iterative learning scheme to the optimal solution is proved. After answering different analytical questions about the solution, the learning algorithm is presented. Finally, numerical analyses are provided to evaluate the performance of the developed technique in practice.

A novel model predictive control scheme based on bees algorithm in a class of nonlinear systems: Application to a three tank system

This paper proposes a novel algorithm for utilizing bees algorithm in a model predictive control (MPC) in order to control a class of nonlinear systems. The bees algorithm is utilized in order to solve the open loop optimization problem (OOP), and it is based on the foraging behavior of honey bees. The proposed algorithm makes use of the bees algorithm for minimizing a predefined cost function in order to find the best input signals subject to constraints and a model of the system. The class of systems considered in this paper includes autonomous nonlinear systems without delay and with continuous and discrete inputs. The proposed algorithm is validated by simulating a three tank system as a case study. A comparison between the proposed novel MPC with different predictive horizons and a conventional MPC demonstrates the potential advantages of the proposed method such as reduction in computation time, good convergence toward desired values and ability of control management. Simulations also show the simplicity of applying and efficiency of the proposed algorithm for designing an MPC based on the bees algorithm.

Asymmetric vibration characteristics of two-cylinder four-stroke single-piston hydraulic free piston engine

The structure and working principle of a two-cylinder four-stroke single-piston hydraulic free piston engine (HFPE) were introduced. The basic vibration equation of free piston assembly (FPA) was established based upon the energy conversion between the injected fuel and the friction together with the load. Both the theoretical and numerical results show that the vibration system of FPA is a nonlinear conservative autonomous system in one cycle. The FPA vibration is symmetric with constant amplitude when FPA is only driven by the compression pressure in the compression accumulator and that in the combustion chamber. When considering the friction and load, FPA could still achieve a stable vibration after a few cycles’ adjustment whether the input energy is equal to the consumed energy or not. The vibration characteristics are different when FPA vibrates in the compression stroke and the expansion stroke, which is the unique feature of the single-piston HFPE.

A Method of Nonlinear Modal Superposition for Weakly Nonlinear Autonomous Systems

For a weakly nonlinear autonomous systems, the nonlinear coordinate transformation is provided in the form of power series expansions and the mathematical transformation from the physics system coordinate to the modal coordinate is presented. And then, according to the relations between physics system coordinate and modal coordinate, a method of nonlinear modal superposition is obtained. Using this method, the general solution of the vibration under any initial condition can be answered. In order to explain the applications of the method, numerical examples are presented in this paper.

A wave interaction approach to studying non-modal homogeneous and stratified shear instabilities

AbstractHomboe (Geophys. Publ., vol. 24, 1962, pp. 67–112) postulated that resonant interaction between two or more progressive, linear interfacial waves produces exponentially growing instabilities in idealized (broken-line profiles), homogeneous or density-stratified, inviscid shear layers. Here we have generalized Holmboe’s mechanistic picture of linear shear instabilities by (i) not initially specifying the wave type, and (ii) providing the option for non-normal growth. We have demonstrated the mechanism behind linear shear instabilities by proposing a purely kinematic model consisting of two linear, Doppler-shifted, progressive interfacial waves moving in opposite directions. Moreover, we have found a necessary and sufficient (N&S) condition for the existence of exponentially growing instabilities in idealized shear flows. The two interfacial waves, starting from arbitrary initial conditions, eventually phase-lock and resonate (grow exponentially), provided the N&S condition is satisfied. The theoretical underpinning of our wave interaction model is analogous to that of synchronization between two coupled harmonic oscillators. We have re-framed our model into a nonlinear autonomous dynamical system, the steady-state configuration of which corresponds to the resonant configuration of the wave interaction model. When interpreted in terms of the canonical normal-mode theory, the steady-state/resonant configuration corresponds to the growing normal mode of the discrete spectrum. The instability mechanism occurring prior to reaching steady state is non-modal, favouring rapid transient growth. Depending on the wavenumber and initial phase-shift, non-modal gain can exceed the corresponding modal gain by many orders of magnitude. Instability is also observed in the parameter space which is deemed stable by the normal-mode theory. Using our model we have derived the discrete spectrum non-modal stability equations for three classical examples of shear instabilities: Rayleigh/Kelvin–Helmholtz, Holmboe and Taylor–Caulfield. We have shown that the N&S condition provides a range of unstable wavenumbers for each instability type, and this range matches the predictions of the normal-mode theory.

Nonuniform dichotomy spectrum and normal forms for nonautonomous differential systems

The aim of this paper is to study the normal forms of nonautonomous differential systems. For doing so, we first investigate the nonuniform dichotomy spectrum of the linear evolution operators that admit a nonuniform exponential dichotomy, where the linear evolution operators are defined by nonautonomous differential equations x˙=A(t)x in Rn. Using the nonuniform dichotomy spectrum we obtain the normal forms of the nonautonomous linear differential equations. Finally we establish the finite jet normal forms of the nonlinear differential systems x˙=A(t)x+f(t,x) in Rn, which is based on the nonuniform dichotomy spectrum and the normal forms of the nonautonomous linear systems.

Quantitative Studies on Asymptotic Growth Behaviors of Trajectories of Nonlinear Discrete Dynamical Systems

This technical note studies quantitatively asymptotic growth behaviors of trajectories (AGBT) of nonlinear autonomous discrete dynamical system that has unbounded domain, non-Lipschitz continuous nonlinear operator, and stable or unstable equilibrium point. We explain how trajectory motion speed is quantitatively determined in the system, and study exact computation and sharp estimation of the smallest exponential bound of trajectories. We characterize exponential stability and asymptotic stability of the system from a new point of view, and provide a simple condition to distinguish them from each other. These results extend existing results that were obtained in some special cases of the system, and are helpful for quantitative analysis and understanding of AGBT of the system.

Dynamics of a nonlinear mathematical model for three interacting populations

In this paper, we derive and analyze both analytically and numerically a mathematical model for three interacting populations. These take the form of a herbivore, a plant, and a pollinator. The full model is a nonlinear reaction–diffusion–advection system, which is derived on the basis of a series of plausible and widely supported ecological hypothesis. The study we present here deals with the conditions for the coexistence of the three interacting species. The analysis is carried out in two stages. For the homogeneous case, the mathematical model reduces to a nonlinear three-dimensional autonomous ODE system, which, as the ecological parameters change, exhibits different dynamical behaviors, namely a limit cycle and a positive attractor. The non-homogeneous case is studied mainly by means of numerical simulations of the full model defined on a rectangular region and considering appropriate initial and boundary conditions. Our results strongly suggest the stabilizing role played by the herbivore population which, in turn means that the introduction of this population into the mutualistic pollinator–plant interaction, favors the coexistence of the three interacting species.

Stability of limit cycles in autonomous nonlinear systems

Periodical solutions or limit cycles (LC) comprise a significant family among the response types of nonlinear autonomous systems. Their identification and stability assessment is of a great importance during the analysis of an unknown system. A new analytical/iterative method of LC identification and portrait investigation was presented recently. The current study proposes a novel technique for their stability assessment. This strategy facilitates the distinction of stable and unstable LCs, thereby allowing the definition of attractive and repulsive response fields. A narrow toroidal domain is constructed around the LC, which is arithmetized by an orthogonal system that is positioned by tangential and normal vectors to the LC. The stability of the LC is investigated using the transformed differential system of the normal components of the response, which are functions of the coordinate along the LC trajectory. Exponential LC stability criteria are also proposed, which are based on the first degree of the perturbation procedure. Theoretical considerations are illustrated using single and two degree of freedom systems including demonstrations with specific systems. The strengths, future steps, and shortcomings of this method are evaluated.

Remarks on integrable systems

The problem of integrability conditions for systems of differential equations is discussed. Darboux’s classical results on the integrability of linear non-autonomous systems with an incomplete set of particular solutions are generalized. Special attention is paid to linear Hamiltonian systems. The paper discusses the general problem of integrability of the systems of autonomous differential equations in an n-dimensional space, which admit the algebra of symmetry fields of dimension ⩽ n. Using a method due to Liouville, this problem is reduced to investigating the integrability conditions for Hamiltonian systems with Hamiltonians linear in the momenta in phase space of dimension that is twice as large. In conclusion, the integrability of an autonomous system in three-dimensional space with two independent non-trivial symmetry fields is proved. It should be emphasized that no additional conditions are imposed on these fields.

Liapunov's stability on autonomous nuclear reactor dynamical systems

Point kinetics model for nuclear reactor physics leads to a stiff system of ordinary differential equations. Stability conditions for linear and nonlinear nuclear reactor point kinetics equations are determined without explicitly solving these systems. Three different techniques, linearization, Liapunov's direct, and variable gradient methods, are applied to investigate the stability of the equilibrium solutions of the proposed systems. Linear system is solved analytically to check the results obtained by Liapunov's direct and variable gradient methods. For the autonomous nonlinear system with Newtonian feedback, linearization technique is used to discuss the stability in a small vicinity of the equilibrium solution, while variable gradient method is applied to establish asymptotic stability. The nonlinear system is close to the linear system, so that it has been proved to be almost linear. Generalized power series method (GPWS) with variable step sizes is used as a numerical method to solve the nonlinear system. Results of numerical calculations are presented in the form of diagrams that confirm the validity of stability conditions obtained by Liapunov's methods. Our results indicate that no trajectories go to infinity as the time tends to infinity. Also, it is proved that the system is conservative by constructing a Liapunov's function, which is valid everywhere without any restrictions, associated with zero Liapunov's derivative function.

Criterions for the positive definiteness of real supersymmetric tensors

The positive definiteness of an even-degree homogeneous polynomial form f ( x ) plays an important role in the stability study of nonlinear autonomous systems via Lyapunov's direct method in automa...

Exponential Stability Criteria for Nonautonomous Difference Systems

The aim of this paper is to characterize the exponential stability of linear systems of difference equations with slowly varying coefficients. Our approach is based on the generalization of the freezing method for difference equations combined with new estimates for the norm of bounded linear operators. The main novelty of this work is that we use estimates for the absolute values of entries of a matrix-valued function, instead of bounds on its eigenvalues. By this method, new explicit stability criteria for linear nonautonomous systems are derived.

An explicit criterion for finite-time stability of linear nonautonomous systems with delays

In this paper, the problem of finite-time stability of linear nonautonomous systems with time-varying delays is considered. Using a novel approach based on some techniques developed for linear positive systems, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. These conditions are shown to be relaxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate the effectiveness of the obtained result.

STABILITY BOUNDARY CHARACTERIZATION OF NONLINEAR AUTONOMOUS DYNAMICAL SYSTEMS IN THE PRESENCE OF A SUPERCRITICAL HOPF EQUILIBRIUM POINT

A complete characterization of the boundary of the stability region (or area of attraction) of nonlinear autonomous dynamical systems is developed admitting the existence of a particular type of nonhyperbolic equilibrium point on the stability boundary, the supercritical Hopf equilibrium point. Under a condition of transversality, it is shown that the stability boundary is comprised of all stable manifolds of the hyperbolic equilibrium points lying on the stability boundary union with the center-stable and\or center manifolds of the type-k, k ≥ 1, supercritical Hopf equilibrium points on the stability boundary.

Existence and construction of a function with sign semidefinite derivative for dynamic systems

The problem of finding a function with sign semidefinite derivative along the solutions to the system of ordinary differential equations is considered. It is shown that a sufficient existence condition is the existence of trajectories that leave a neighborhood of the equilibrium point both in forward or backward time. Two approaches to the construction of the desired function in the autonomous and nonautonomous cases are proposed. The results are illustrated by the example of a nonlinear autonomous system (Artstein's circle).

Stability and dynamics of nonautonomous systems with pulsed nonlinearity

We study the dynamics of a class of nonautonomous systems with pulsed nonlinearity that consist of a periodic sequence of linear and nonlinear autonomous systems, each one acting alone in a different time or space interval. We focus on the investigation of control capabilities of such systems in terms of altering their fundamental dynamical properties by appropriate parameter selections. For the case of single oscillators, the stability of the zero solution as well as the phase space topology is shown to drastically depend on parameters such as the frequency of the linear oscillations and the durations of the linear and nonlinear intervals. In cases of chain of coupled oscillators with pulsed onsite nonlinearity, it is shown that appropriate parameter selections can stabilize an otherwise unstable zero background allowing for the existence of dynamically robust localized excitations, whose evolution properties can now be explicitly determined and controlled.

Semistability of Nonlinear Control Systems

The problem of semistability for nonlinear autonomous systems is investigated. We develop semistability and unsemistability theorems for nonlinear autonomous systems which is stable. These results of this paper are more tractable and practical for semistability of nonlinear autonomous systems than the existing results in the literature. Several examples are included to show that the proposed method is effective.