Nonlinear Quasi-periodic System Research Articles

Reducibility for a Class of Nonlinear Quasi-Periodic Systems under Brjuno-Russmann’s Non-resonance Conditions

Reducibility for a Class of Nonlinear Quasi-Periodic Systems under Brjuno-Russmann’s Non-resonance Conditions

Order Reduction of Nonlinear Quasi-Periodic Systems Using Lyapunov–Perron Transformation

Abstract In this paper, multiple order reduction techniques for parametrically excited nonlinear quasi-periodic systems are presented. The linear time-varying part of the quasi-periodic system is transformed into a linear time-invariant form via the Lyapunov–Perron (L–P) transformation. The analytical computation of such a transformation is performed using an intuitive state augmentation and the normal forms technique. This L–P transformation is further utilized in analyzing the nonlinear part of the original quasi-periodic system. Using the L–P transformation, three-order reduction techniques are detailed in this work. First, a Guyan linear reduction method is applied to reduce the order. The second method is to determine a nonlinear projection based on the singular perturbation method. In the third technique, the method of Invariant Manifold is applied to identify a relationship between the dominant and nondominant system states. Furthermore, in this work, all three order reduction techniques are demonstrated on the class of commutative and noncommutative/Hills-type nonlinear quasi-periodic systems. The behavior of the reduced system states of the resulting solution is compared with the numerical integration results and their performance is studied using the error plots for each technique.

Order reduction of nonlinear quasi-periodic systems subjected to external excitations

This paper presents order reduction techniques for nonlinear quasi-periodic systems subjected to external excitations. The order reduction techniques presented here are based on the Lyapunov–Perron (L–P) Transformation. For a class of non-resonant quasi-periodic systems, the L-P Transformation can convert a linear quasi-periodic system into a linear time-invariant one. This Linear Time-Invariant (LTI) system retains the dynamics of the original quasi-periodic system. Once this LTI system is obtained, the tools and techniques available for analysis of LTI systems can be used, and the results could be obtained for the original quasi-periodic system via the L–P Transformation. This approach is similar to the Lyapunov–Floquet (L–F) transformation to convert a linear time-periodic system into an LTI system and perform analysis and control.Order reduction is a systematic way of constructing dynamical system models with relatively smaller states that accurately retain large-scale systems’ essential dynamics. This work presents reduced-order modeling techniques for nonlinear quasi-periodic systems subjected to external excitations. The methods proposed here use the L–P Transformation that makes the linear part of transformed equations time-invariant. In this work, two order reduction techniques are suggested. The first method is simply an application of the well-known Guyan like reduction method to nonlinear systems. The second technique is based on the concept of an invariant manifold for quasi-periodic systems.The ‘quasi-periodic invariant manifold’ based technique yields ‘reducibility conditions’. These conditions (referred to in the perturbation literature as resonance conditions) help us understand the system’s various types of resonant interactions. These resonances indicate energy interactions between the system states, nonlinearity, and external excitation. To retain the essential dynamical characteristics, one must preserve all these ‘resonant’ states in the reduced-order model. Thus, if the ‘reducibility conditions’ are satisfied then only, a nonlinear order reduction based on the quasi-periodic invariant manifold approach is possible. It is found that the invariant manifold approach yields good results. These methodologies are general and can be used for parametric study, sensitivity analysis, and controller design.

A new semi-analytical approach for quasi-periodic vibrations of nonlinear systems

This paper presents a new semi-analytical approach, namely the enhanced time-domain minimum residual method to solve the semi-analytical quasi-periodic solution for nonlinear system. This approach does not require multiple numerical integration and can be applied to strongly nonlinear systems. The approach is mainly three-fold. Firstly, the semi-analytical solution of the nonlinear quasi-periodic system is expanded into a set of trigonometric series with unknown coefficients, i.e., x(t)≈∑k=1N[bkcos(ωkt)+cksin(ωkt)]. Then, the problem of solving quasi-periodic solution can be expressed as: determining the coefficients of the trigonometric series so that the residual objective function R=Mx¨+Cx˙+Kx+N(x¨,x˙,x,t)−F(t) is minimum over a period, i.e., mina∈A∫0TR(a,t)TR(a,t)dt. Finally, the nonlinear minimum optimization problem is solved iteratively through the enhanced response sensitivity approach. Moreover, the “adaptive zero-setting curve” is introduced to accelerate the convergence. Two numerical examples, a van der Pol-Duffing system and a nonlinear energy sink system are adopted to verify the feasibility of the proposed approach.

Response solutions of 3-dimensional degenerate quasi-periodic systems with small parameter

This paper considers a class of 3-dimensional real analytic nonlinear quasi-periodic systems with a small perturbation parameter, whose unperturbed part has a degenerate equilibrium point. Based on the Leray-Schauder Continuation Theorem [18] , [20] and using technique of outer parameter, we prove that the system has a small response solution for many sufficiently small parameters.

Approximate Lyapunov–Perron Transformations: Computation and Applications to Quasi-Periodic Systems

Abstract A new technique for the analysis of dynamical equations with quasi-periodic coefficients (so-called quasi-periodic systems) is presented. The technique utilizes Lyapunov–Perron (L–P) transformation to reduce the linear part of a quasi-periodic system into the time-invariant form. A general approach for the construction of L–P transformations in the approximate form is suggested. First, the linear part of a quasi-periodic system is replaced by a periodic system with a “suitable” large principal period. Then, the state transition matrix (STM) of the periodic system is computed in the symbolic form using Floquet theory. Finally, Lyapunov–Floquet theorem is used to compute approximate L–P transformations. A two-frequency quasi-periodic system is studied and transformations are generated for stable, unstable, and critical cases. The effectiveness of these transformations is demonstrated by investigating three distinct quasi-periodic systems. They are applied to a forced linear quasi-periodic system to generate analytical solutions. It is found that the closeness of the analytical solutions to the exact solutions depends on the principal period of the periodic system. A general approach to obtain the stability bounds on linear quasi-periodic systems with stochastic perturbations is also discussed. Finally, the usefulness of approximate L–P transformations is presented by analyzing a nonlinear quasi-periodic system with cubic nonlinearity using time-dependent normal form (TDNF) theory. The closed-form solution generated is found to be in good agreement with the exact solution.

Lyapunov–Perron Transformation for Quasi-Periodic Systems and Its Applications

Abstract This paper depicts the application of symbolically computed Lyapunov–Perron (L–P) transformation to solve linear and nonlinear quasi-periodic systems. The L–P transformation converts a linear quasi-periodic system into a time-invariant one. State augmentation and the method of normal forms are used to compute the L–P transformation analytically. The state augmentation approach converts a linear quasi-periodic system into a nonlinear time-invariant system as the quasi-periodic parametric excitation terms are replaced by “fictitious” states. This nonlinear system can be reduced to a linear system via normal forms in the absence of resonances. In this process, one obtains near identity transformation that contains fictitious states. Once the quasi-periodic terms replace the fictitious states they represent, the near identity transformation is converted to the L–P transformation. The L–P transformation can be used to solve linear quasi-periodic systems with external excitation and nonlinear quasi-periodic systems. Two examples are included in this work, a commutative quasi-periodic system and a non-commutative Mathieu–Hill type quasi-periodic system. The results obtained via the L–P transformation approach match very well with the numerical integration and analytical results.

Reducibility of a class of nonlinear quasi-periodic systems with Liouvillean basic frequencies

In this paper we consider the following nonlinear quasi-periodic system:$$\begin{eqnarray}{\dot{x}}=(A+\unicode[STIX]{x1D716}P(t,\unicode[STIX]{x1D716}))x+\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})+h(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $A$ is a $d\times d$ constant matrix of elliptic type, $\unicode[STIX]{x1D716}g(t,\unicode[STIX]{x1D716})$ is a small perturbation with $\unicode[STIX]{x1D716}$ as a small parameter, $h(x,t,\unicode[STIX]{x1D716})=O(x^{2})$ as $x\rightarrow 0$, and $P,g$ and $h$ are all analytic quasi-periodic in $t$ with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$, where $\unicode[STIX]{x1D6FC}$ is irrational. It is proved that for most sufficiently small $\unicode[STIX]{x1D716}$, the system is reducible to the following form: $$\begin{eqnarray}{\dot{x}}=(A+B_{\ast }(t))x+h_{\ast }(x,t,\unicode[STIX]{x1D716}),\quad x\in \mathbb{R}^{d},\end{eqnarray}$$ where $h_{\ast }(x,t,\unicode[STIX]{x1D716})=O(x^{2})~(x\rightarrow 0)$ is a high-order term. Therefore, the system has a quasi-periodic solution with basic frequencies $\unicode[STIX]{x1D714}=(1,\unicode[STIX]{x1D6FC})$, such that it goes to zero when $\unicode[STIX]{x1D716}$ does.

Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings

A major challenge in the study of dynamical systems is that of model discovery: turning data into models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. This problem is made more difficult by the fact that many systems of interest exhibit diverse behaviors across multiple time scales. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SINDy to scale efficiently to problems with multiple time scales. Specifically, we can discover distinct governing equations at slow and fast scales. For systems with incomplete observations, we show that the Hankel alternative view of Koopman (HAVOK) method, based on time-delay embedding coordinates, can be used to obtain a linear model and Koopman invariant measurement system that nearly perfectly captures the dynamics of nonlinear quasiperiodic systems. We introduce two strategies for using HAVOK on systems with multiple time scales. Together, our approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.

On reducibility for a class of n-dimensional quasi-periodic systems with a small parameter

In this paper we consider the reducibility of a class of n-dimensional real analytic quasi-periodic systems with a small parameter:x˙=(A+ϵQ(t,ϵ))x,x∈Rn.We prove that if the basic frequencies of Q and the eigenvalues of A satisfy some non-resonance conditions, then for most of the sufficiently small parameters in the sense of Lebesgue measure, the system is reducible without any non-degeneracy assumption with respect to the parameter. Moreover, under some assumptions, we obtain a similar result for nonlinear quasi-periodic systems.

On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point

This paper focuses on quasi-periodic perturbation of four dimensional nonlinear quasi-periodic system. Using the KAM method, the perturbed system can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation. Hence, the perturbed system has a quasi-periodic solution near the equilibrium point.

On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point

In this paper we consider small quasi-periodic perturbation of two-dimensional nonlinear quasi-periodic system with hyperbolic-type degenerate equilibrium point. By KAM method we prove that it can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation. Hence, the perturbed system has a quasi-periodic solution near the equilibrium point.

Hofstadter butterflies in nonlinear Harper lattices, and their optical realizations

The ubiquitous Hofstadter butterfly describes a variety of systems characterized by incommensurable periodicities, ranging from Bloch electrons in magnetic fields and the quantum Hall effect to cold atoms in optical lattices and more. Here, we introduce nonlinearity into the underlying (Harper) model and study the nonlinear spectra and the corresponding extended eigenmodes of nonlinear quasiperiodic systems. We show that the spectra of the nonlinear eigenmodes form deformed versions of the Hofstadter butterfly and demonstrate that the modes can be classified into two families: nonlinear modes that are a ‘continuation’ of the linear modes of the system and new nonlinear modes that have no counterparts in the linear spectrum. Finally, we propose an optical realization of the linear and nonlinear Harper models in transversely modulated waveguide arrays, where these Hofstadter butterflies can be observed. This work is relevant to a variety of other branches of physics beyond optics, such as disorder-induced localization in ultracold bosonic gases, localization transition processes in disordered lattices, and more.

On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point

This work focuses on the reducibility of the following real nonlinear analytical quasiperiodic system: x ̇ = A x + f ( t , x , ϵ ) , x ∈ R 2 where A is a real 2×2 constant matrix, and f ( t , 0 , ϵ ) = O ( ϵ ) and ∂ x f ( t , 0 , ϵ ) = O ( ϵ ) as ϵ → 0 . With some non-resonant conditions of the frequencies with the eigenvalues of A and without any nondegeneracy condition with respect to ϵ , by an affine analytic quasiperiodic transformation we change the system to a suitable norm form at the zero equilibrium for most of the sufficiently small perturbation parameter ϵ .

Lagrangian stability of a nonlinear quasi-periodic system

In this paper, we prove the Lagrangian stability of the quasi-periodic system d 2 x/ dt 2+ G x ( x, t)=0, where G is quasi-periodic in both x and t, respectively.

Optical multistability in a nonlinear Fibonacci multilayer.

The transmission properties of a multilayered medium consisting of $N$ nonlinear slabs are studied. A general characteristic matrix formalism is applied to obtain the power dependence of the transmission coefficient. As an application, a nonlinear Fibonacci multilayer with as many as 55 and 233 nonlinear slabs is considered. The nonlinear quasiperiodic system is shown to offer a wide variety of bistable and multistable operations.