Normal Forms for High Co-Dimension Bifurcations of Nonlinear Time-Periodic Systems with Nonsemisimple Eigenvalues

The normal forms for time-periodic nonlinear variational equations witharbitrary linear Jordan form undergoing bifurcations of highco-dimension are found. First, the equations are transformed via theLyapunov–Floquet (L–F) transformation into an equivalent form in whichthe linear matrix is constant with degenerate nonsemisimple lineareigenvalues while the nonlinear monomials have periodic coefficients. Byconsidering the resulting coupling of the bases of the near identitytransformation, the solvability condition for an arbitrary Jordan matrixis then derived. It is shown that time-independent and/or time-dependentnonlinear resonance terms remain in the normal form for various Jordanmatrices. Specifically, the normal forms for quadratic and cubicnonlinearities with the following linear Jordan forms are explicitlyderived: double zero eigenvalues (co-dimension two bifurcation), triplezero eigenvalues (co-dimension three bifurcation), and two repeatedpairs of purely imaginary eigenvalues (co-dimension two bifurcation). Acommutative system with cubic nonlinearities and a double inverted pendulum with a periodicfollower force are used as illustrative examples.     

Normal Forms and the Structure of Resonance Sets in Nonlinear Time-Periodic Systems

The structure of time-dependent resonances arising in themethod of time-dependent normal forms (TDNF) for one andtwo-degrees-of-freedom nonlinear systems with time-periodic coefficientsis investigated. For this purpose, the Liapunov–Floquet (L–F)transformation is employed to transform the periodic variationalequations into an equivalent form in which the linear system matrix istime-invariant. Both quadratic and cubic nonlinearities are investigatedand the associated normal forms are presented. Also, higher-orderresonances for the single-degree-of-freedom case are discussed. It isdemonstrated that resonances occur when the values of the Floquet multipliers result in MT-periodic (M = 1, 2,...) solutions. The discussion is limited to the Hamiltonian case (which encompasses allpossible resonances for one-degree-of-freedom). Furthermore, it is alsoshown how a recent symbolic algorithm for computing stability andbifurcation boundaries for time-periodic systems may also be employed tocompute the time-dependent resonance sets of zero measure in theparameter space. Unlike classical asymptotic techniques, this method isfree from any small parameter restriction on the time-periodic term inthe computation of the resonance sets. Two illustrative examples (oneand two-degrees-of-freedom) are included.     

A direct analysis of nonlinear systems with external periodic excitations via normal forms

This study presents a direct methodology for a quantitative analysis of nonlinear dynamic systems with external periodic forcing via an application of the theory of normal forms. Rather than introducing a new state variable to reduce the problem to a homogeneous one, a set of time-dependant near-identity transformations is applied to construct the normal forms. In the process, the total response of the system is expressed as superposition of a steady state solution and a transient solution. A steady state solution of the system is obtained by the method of harmonic balance and the transient solution is obtained by solving a set of time periodic homological equations. The proposed method can be applied to time-invariant as well as time varying systems. After discussing the time-invariant case, the methodology is extended to systems with time-periodic coefficients. The case of time periodic systems is handled through an application of the Lyapunov–Floquet (L–F) transformation. Application of the L–F transformation produces a dynamically equivalent system in which the linear part of the system is time-invariant, making the system amenable to near-identity transformations. An example for each type of system, namely, constant coefficients and time-varying coefficients, is included to demonstrate effectiveness of the method. Various resonance conditions are discussed. It is observed that the linear parametric excitation term need not be small as generally assumed in perturbation and averaging techniques. Results obtained by proposed methods are compared with numerical solutions. Close agreements are found in some typical applications.     

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.     

Analysis of time-periodic nonlinear dynamical systems undergoing bifurcations

In this study a new procedure for analysis of nonlinear dynamical systems with periodically varying parameters under critical conditions is presented through an application of the Liapunov-Floquet (L-F) transformation. The L-F transformation is obtained by computing the state transition matrix associated with the linear part of the problem. The elements of the state transition matrix are expressed in terms of Chebyshev polynomials in timet which is suitable for algebraic manipulations. Application of Floquet theory and the eigen-analysis of the state transition matrix at the end of one principal period provides the L-F transformation matrix in terms of the Chebyshev polynomials. Since this is a periodic matrix, the L-F transformation matrix has a Fourier representation. It is well known that such a transformation converts a linear periodic system into a linear time-invariant one. When applied to quasi-linear equations with periodic coefficients, a dynamically similar system is obtained whose linear part is time-invariant and the nonlinear part consists of coefficients which are periodic. Due to this property of the L-F transformation, a periodic orbit in original coordinates will have a fixed point representation in the transformed coordinates. In this study, the bifurcation analysis of the transformed equations, obtained after the application of the L-F transformation, is conducted by employingtime-dependent center manifold reduction andtime-dependent normal form theory. The above procedures are analogous to existing methods that are employed in the study of bifurcations of autonomous systems. For the two physical examples considered, the three generic codimension one bifurcations namely, Hopf, flip and fold bifurcations are analyzed. In the first example, the primary bifurcations of a parametrically excited single degree of freedom pendulum is studied. As a second example, a double inverted pendulum subjected to a periodic loading which undergoes Hopf or flip bifurcation is analyzed. The methodology is semi-analytic in nature and provides quantitative measure of stability when compared to point mappings method. Furthermore, the technique is applicable also to those systems where the periodic term of the linear part does not contain a small parameter which is certainly not the case with perturbation or averaging methods. The conclusions of the study are substantiated by numerical simulations. It is believed that analysis of this nature has been reported for the first time for this class of systems.     

Analysis of periodic-quasiperiodic nonlinear systems via Lyapunov-Floquet transformation and normal forms

In this paper a general technique for the analysis of nonlinear dynamical systems with periodic-quasiperiodic coefficients is developed. For such systems the coefficients of the linear terms are periodic with frequency ω while the coefficients of the nonlinear terms contain frequencies that are incommensurate with ω. No restrictions are placed on the size of the periodic terms appearing in the linear part of system equation. Application of Lyapunov-Floquet transformation produces a dynamically equivalent system in which the linear part is time-invariant and the time varying coefficients of the nonlinear terms are quasiperiodic. Then a series of quasiperiodic near-identity transformations are applied to reduce the system equation to a normal form. In the process a quasiperiodic homological equation and the corresponding ‘solvability condition’ are obtained. Various resonance conditions are discussed and examples are included to show practical significance of the method. Results obtained from the quasiperiodic time-dependent normal form theory are compared with the numerical solutions. A close agreement is found.     

Versal Deformation and Local Bifurcation Analysis of Time-Periodic Nonlinear Systems

In this study a local semi-analytical method of quantitativebifurcation analysis for time-periodic nonlinear systems is presented.In the neighborhood of a local bifurcation point the system equationsare simplified via Lyapunov–Floquet transformation whichtransforms the linear part of the equation into a dynamically equivalenttime-invariant form. Then the time-periodic center manifoldreduction is used to separate the `critical' states and reduce thedimension of the system to a possible minimum. The center manifoldequations can be simplified further via time-dependent normal formtheory. For most codimension one cases these nonlinear normal forms arecompletely time-invariant. Versal deformation of thesetime-invariant normal forms can be found and the bifurcation phenomenoncan be studied in the neighborhood of the critical point. However, ingeneral, it is not a trivial task to find a quantitatively correctversal deformation for time-periodic systems. In order to do so, onemust find a relationship between the bifurcation parameter of theoriginal time-periodic system and the versal deformation parameter ofthe time-invariant normal form. Essentially one needs to find theeigenvalues of the fundamental solution matrix of the time-periodicproblem in terms of the system parameters, which, in general, cannot bedone due to computational difficulties. In this work two ideas areproposed to achieve this goal. The eigenvalues of the fundamentalsolution matrix can be related to the versal deformation parameter bysensitivity analysis and an approximation of any desired order can beobtained. This idea requires a symbolic computational procedure whichcan be very time consuming in some cases. An alternative method issuggested for faster results in which a second or higher order curvefitting technique is used to find the relationship. Once thisrelationship is established, closed form post-bifurcation steady-statesolutions can be obtained for flip, symmetry breaking, transcritical andsecondary Hopf bifurcations. Unlike averaging and perturbation methods,the proposed technique is applicable at any bifurcation point in theparameter space. As physical examples, a simple and a double pendulumsubjected to periodic parametric excitation are considered. A simple twodegrees of freedom model is also studied and the results are comparedwith those obtained from the traditional averaging method. All resultsare verified by numerical integration. It is observed that the proposedtechnique yields results which are very close to the numericalsolutions, unlike the averaging method.     

Van der Pol type self-excited micro-cantilever probe of atomic force microscopy

A technique for dimensional reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. The DDEs considered here have a canonical form with at most cubic nonlinearities and periodic coefficients. The nonlinear terms are multiplied by a perturbation parameter. Perturbation expansion converts the nonlinear response problem into solutions of a series of nonhomogeneous linear ordinary differential equations (ODEs) with time-periodic coefficients. One set of linear nonhomogeneous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. The linear part of the map is the monodromy operator whose eigenvalues characterize stability. Dimensional reduction on the map is then carried out. In the case of critical eigenvalues, this corresponds to center manifold reduction, while for the noncritical case resonance conditions are derived. The accuracy of the nonlinear Chebyshev collocation map is demonstrated by finding the solution of a nonlinear delayed Mathieu equation and then a milling model via the method of steps. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation. The method of this paper is shown by example to be applicable to systems with strong parametric excitations.     

一种构造高维Nilpotent正规型的方法

正规型方法是一种有效的简化一类非线方程的方法。今提出了一种简便的代数方法去构造高维非线性系统的Nilpotent范式。通过引入一系列简单的变换和代数运算,而无需求解任何偏微分方程,即可得到高维非线性系统的Nilpotent范式。以四维非线性系统为例介绍这个方法。该方法完全适用于分析高于四维的非线性系统的Nilpotent范式。     

Invariant linearization criteria for a three-dimensional dynamical system of second-order ordinary differential equations and applications

Second-order dynamical systems are of paramount importance as they arise in mechanics and many applications. It is essential to have workable explicit criteria in terms of the coefficients of the equations to effect reduction and solutions for such types of equations. One important aspect is linearization by invertible point transformations which enables one to reduce a non-linear system to a linear system. The solution of the linear system allows one to solve the non-linear system by use of the inverse of the point transformation. It was proved that the n-dimensional system of second-order ordinary differential equations obtained by projecting down the system of geodesics of a flat (n+1)-dimensional space can be converted to linear form by a point transformation. This is a generalization of the Lie linearization criteria for a scalar second-order equation. In this case it is of the maximally symmetric class for a system and the linearizing transformation as well as the solution can be directly written down. This was explicitly used for two-dimensional dynamical systems. The criteria were written down in terms of the coefficients and the linearizing transformation allowed for the general solution of the original system. Here the work is extended to a three-dimensional dynamical system and we find explicit criteria, including the linearization test given in terms of the coefficients of the cubic in the first derivatives of the system and the construction of the transformations, that result in linearization. Applications to equations of classical mechanics and relativity are given to illustrate our results.     

Control techniques for impacting flexible systems

Summary In this article, a comparative study of the control for the repetitive impacting elastic link with parametrically excited base in rotational motion is considered. First, a sliding mode control strategy based on linearized inverse model is designed and employed to suppress the vibrations of the elastic beam after the impact. The control concept involves the usage of an adaptive plant inverse model as controller in feedforward configurations. Next, a linear controller is designed via Lyapunov-Floquet transformation. In this approach, the time-periodic equations of motion are transformed into a time-invariant form, which is suitable for the application of standard time-invariant controller-design techniques. Finally, a fuzzy logic controller is applied for the nonlinear model of the impacting system. The momentum balance method and an empirical coefficient of restitution is used in the collision. Received 27 January 1999; accepted for publication 3 June 1999     

A stability in the large investigation for a non-linear second order two-degree-of-freedom system with periodic excitation

An extension to an algorithm due to Simpson has been developed for the analysis of a non-linear second order two-degree-of-freedom system with external periodic excitation. The form of equations considered arises from the study of mechanical systems with a single concentrated weak non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. The system considered is such that in the absence of external excitation, it possesses a stable equilibrium point and an unstable limit cycle arising from a sub-critical Hopf bifurcation. When forcing is applied, the stable equilibrium point may then be replaced by a stable periodic attractor, and the limit cycle by an unstable multi-periodic attractor. The method has been applied to the problem of locating these attractors, and if they exist, of finding the stable attractor's basin of attraction in terms of initial conditions. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution.The method was applied to three non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in those cases where the non-linearity was weak. However, the method would not be expected to give such accurate results if the non-linear effect was more significant. This was illustrated for a case involving the freeplay non-linearity.     

Equivalence of Dynamics for Nonholonomic Systems with Transverse Constraints

This paper is concerned with the dynamics of a mechanical system subject to nonintegrable constraints. In the first part, we prove the equivalence between the classical nonholonomic equations and those derived from the nonholonomic variational formulation, proposed by Kozlov in [10–12], for a class of constrained systems with constraints transverse to a foliation. This result extends the equivalence between the two formulations, proved for holonomic constraints, to a class of linear nonintegrable ones. In the second part, we derive the nonholonomic variational reduced equations for a constrained system with symmetry and constraint transverse to a principal bundle fibration, using a reduction procedure similar to the one developed in [5]. The resulting equations are compared with the nonholonomic reduced ones through mechanical examples.     

Application of the Krylov–Bogoliubov–Mitropolsky method to weakly damped strongly non-linear planar Hamiltonian systems

In this paper, an analytical approximation of damped oscillations of some strongly non-linear, planar Hamiltonian systems is considered. To apply the Krylov–Bogoliubov–Mitropolsky method in this strongly non-linear case, we mainly provide the formal and exact solutions of the homogeneous part of the variational equations with periodic coefficients resulting from the Hamiltonian systems. It is shown that these are simply expressed in terms of the partial derivatives of the solutions, written in action-angle variables, of the Hamiltonian systems. Two examples, including a non-linear harmonic oscillator and the Morse oscillator, are presented to illustrate this extension of the method. The approximate first order solution obtained in each case is observed to be quite satisfactory.     

Thermal buckling and post-buckling of pinned–fixed Euler–Bernoulli beams on an elastic foundation

In this article, both thermal buckling and post-buckling of pinned–fixed beams resting on an elastic foundation are investigated. Based on the accurate geometrically non-linear theory for Euler–Bernoulli beams, considering both linear and non-linear elastic foundation effects, governing equations for large static deformations of the beam subjected to uniform temperature rise are derived. Due to the large deformation of the beam, the constraint forces of elastic foundation in both longitudinal and transverse directions are taken into account. The boundary value problem for the non-linear ordinary differential equations is solved effectively by using the shooting method. Characteristic curves of critical buckling temperature versus elastic foundation stiffness parameter corresponding to the first, the second, and the third buckling mode shapes are plotted. From the numerical results it can be found that the buckling load-elastic foundation stiffness curves have no intersection when the value of linear foundation stiffness parameter is less than 3000, which is different from the behaviors of symmetrically supported (pinned–pinned and fixed–fixed) beams. As we expect that the non-linear foundation stiffness parameter has no sharp influence on the critical buckling temperature and it has a slight effect on the post-buckling temperature compared with the linear one.     

Generalized variables in boundary-layer theory

Independent variables are widely used in boundary-layer theory to construct efficient methods of solving problems. The Dorodnitsyn variables in Lees' form [1] are the most common and general. This form combines the transformations proposed by Dorodnitsyn [2], Blasius [3], and Mangler-Stepanov [4, 5]. As is well known, transformation of the boundary-layer equations to Dorodnitsyn variables in Lees' form leads to a generalized single system of equations describing plane and axisymmetric gas flows. An analogous generalization of the Mises [6] and Crocco [7] variables is carried out below.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 166–168, September–October, 1976.     

具有系统参数的非线性动力系统周期解方法

给出两种形式的微分方程周期求解方法,这两种方法对称处理奇异的非线性特征值问题有独特的能力,为具有系统参数的非线性动力系统在整个系统参数范围内的动态特性分析提供了有效的方法。     

On order reduction of non-linear equations of mechanics and mathematical physics,new integrable equations and exact solutions

Some classes of non-linear equations of mechanics and mathematical physics are described that admit order reduction through the use of a hydrodynamic-type transformation, where a first-order partial derivative is taken as a new independent variable and a second-order partial derivative is taken as the new dependent variable. The results obtained are used for order reduction of hydrodynamic equations (Navier–Stokes, Euler, and boundary layer) and deriving exact solutions to these equations. Associated Bäcklund transformations are constructed for evolution equations of general form (special cases include Burgers, Korteweg-de Vries, and many other non-linear equations of mathematical physics). A number of new integrable non-linear equations, inclusive of the generalized Calogero equation, are considered.     

Nonholonomic systems with non-linear constraint equations

The development of a form of Lagrange's equations applicable with nonholonomic systems with non-linear constraint equations is presented and discussed. The analysis is based upon, and is an extension of. a method developed by the authors for nonholonomic systems with linear constraint equations in the generalized coordinate derivatives. The method is illustrated with the problem of the “balancing pole”.     

Almost Periodic Solutions and Global Attractors of Non-autonomous Navier–Stokes Equations

The article is devoted to the study of non-autonomous Navier–Stokes equations. First, the authors have proved that such systems admit compact global attractors. This problem is formulated and solved in the terms of general non-autonomous dynamical systems. Second, they have obtained conditions of convergence of non-autonomous Navier–Stokes equations. Third, a criterion for the existence of almost periodic (quasi periodic, almost automorphic, recurrent, pseudo recurrent) solutions of non-autonomous Navier–Stokes equations is given. Finally, the authors have derived a global averaging principle for non-autonomous Navier–Stokes equations.