Algebraic linearization criteria for systems of ordinary differential equations

Algebraic linearization criteria by means of general point transformations for systems of two second-order nonlinear ordinary differential equations (ODEs) are revisited. In previous work due to Wafo Soh and Mahomed (Int. J. Non-Linear Mech. 36:671, 2001) two four-dimensional Lie algebras that result in linearizability in terms of arbitrary point transformation for such systems were studied. Here we consider three more algebras of dimension four that result in linearization. Therefore our results supplement those of Wafo Soh and Mahomed (Int. J. Non-Linear Mech. 36:671, 2001). Moreover, it is shown that these are the only other possibilities for dimension four. Hence we provide the complete algebraic linearization criteria for dimension four algebras. Necessary and sufficient conditions for linearization via invertible maps of a nonlinear to a linear system are given. These are shown to be built up from the Lie algebraic criteria for linearization of scalar second-order ODEs. These results together with very recent work (Bagderina in J. Phys. A, Math. Theor. 43:465201, 2010) give a complete picture on linearizability properties via general point transformations for systems of two second-order ODEs. Furthermore, we provide natural extensions of these algebraic criteria for linearizing arbitrary systems of nonlinear second-order ODEs by means of point transformations. We also obtain algebraic criteria for the reduction of a linear system to the simplest system. Examples from Newtonian mechanics and geodesic equations are presented to illustrate our results.     

Linearization criteria for a system of second-order quadratically semi-linear ordinary differential equations

Conditions are derived for the linearizability via invertible maps of a system of n second-order quadratically semi-linear differential equations that have no lower degree lower order terms in them, i.e., for the symmetry Lie algebra of the system to be sl(n + 2, ℝ). These conditions are stated in terms of the coefficients of the equations and hence provide simple invariant criteria for such systems to admit the maximal symmetry algebra. We provide the explicit procedure for the construction of the linearizing transformation. In the simplest case of a system of two second-order quadratically semi-linear equations without the linear terms in the derivatives, we also provide the construction of the linearizing point transformation using complex variables. Examples are given to illustrate our approach for two- and three-dimensional systems.     

Geometric Proof of Lie's Linearization Theorem

In 1883, S. Lie found the general form of all second-order ordinary differential equations transformable to the linear equation by a change of variables and proved that their solution reduces to integration of a linear third-order ordinary differential equation. He showed that the linearizable equations are at most cubic in the first-order derivative and described a general procedure for constructing linearizing transformations by using an over-determined system of four equations. We present here a simple geometric proof of the theorem, known as Lie's linearization test, stating that the compatibility of Lie's four auxiliary equations furnishes a necessary and sufficient condition for linearization.     

A novel statistical linearization solution for randomly excited coupled bending-torsional beams resting on non-linear supports

A novel statistical linearization technique is developed for computing stationary response statistics of randomly excited coupled bending-torsional beams resting on non-linear elastic supports. The key point of the proposed technique consists in representing the non-linear coupled response in terms of constrained linear modes. The resulting set of non-linear equations governing the modal amplitudes is then replaced by an equivalent linear one via a classical statistical error minimization procedure, which provides algebraic non-linear equations for the second-order statistics of the beam response, readily solved by a simple iterative scheme. Data from Monte Carlo simulations, generated by a pertinent boundary integral method in conjunction with a Newmark numerical integration scheme, are used as benchmark solutions to check accuracy and reliability of the proposed statistical linearization technique.

     

Birkhoff系统的广义正则变换及其基本形式

动力学方程的积分问题是分析力学研究的一个重要方面．由于求解一般的动力学方程往往会遇到很大困难，因此可利用变量变换，使方程变得容易求解．文章研究Birkhoff系统的广义正则变换．首先，建立Birkhoff系统的广义正则变换的充分必要条件；其次，基于该条件，给出Birkhoff系统的广义正则变换的六种基本形式，导出每一种情况下新旧变量之间的变换关系．作为特例，文中给出Hamilton方程的正则变换．文末，给出算例以说明结果的应用．     

Linearizability criteria for systems of two second-order differential equations by complex methods

Lie??s linearizability criteria for scalar second-order ordinary differential equations had been extended to systems of second-order ordinary differential equations by using geometric methods. These methods not only yield the linearizing transformations but also the solutions of the nonlinear equations. Here, complex methods for a scalar ordinary differential equation are used for linearizing systems of two second-order ordinary and partial differential equations, which can use the power of the geometric method for writing the solutions. Illustrative examples of mechanical systems including the Lane?CEmden type equations which have roots in the study of stellar structures are presented and discussed.     

Symmetry classification of first integrals for scalar dynamical equations

We completely classify the first integrals of scalar non-linear second-order ordinary differential equations (ODEs) in terms of their Lie point symmetries. This is performed by first obtaining the classifying relations between point symmetries and first integrals of scalar non-linear second-order equations which admit one, two and three point symmetries. We show that the maximum number of symmetries admitted by any first integral of a scalar second-order non-linear ODE is one which in turn provides reduction to quadratures of the underlying dynamical equation. We provide physical examples of the generalized Emden–Fowler, Lane–Emden and modified Emden equations.     

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.     

Numerical method for unsteady compressible boundary layers driven by compression or expansion wave

A numerical analysis is presented for the unsteady compressible laminar boundary layer driven by a compression or expansion wave. Approximate or series expansion methods have been used for the problems because of the characteristics of the governing equations, such as non-linearity, coupling with the thermal boundary layer equation and initial conditions. Here a transformation of the governing equations and the numerical linearization technique are introduced to deal with the difficulties. First, the governing equations are transformed for the initial conditions by Howarth and semisimilarity variables. These transformations reduce the number of independent variables from three to two and the governing equations from partial to ordinary differential equations at the initial point. Next, the numerical linearization technique is introduced for the non-linearity and the coupling with the thermal boundary layer equation. Because the non-linear terms are linearized without sacrifice of numerical accuracy, the solutions can be obtained without numerical iterations. Therefore the exact numerical solution, not approximate or series expansion, can be obtained. Compared with the approximate or series expansion method, this method is much improved. Results are compared with the series expansion solutions.     

Construction of Dynamically Equivalent Time-Invariant Forms for Time-Periodic Systems

In this study dynamically equivalent time-invariant forms are obtained for linear and non-linear systems with periodically varying coefficients via Lyapunov–Floquet (L–F) transformation. These forms are equivalent in the sense that the local stability and bifurcation characteristics are identical for both systems in the entire parameter space. It is well known that the L–F transformation converts a linear periodic first order system into a time-invariant one. In the first part of this study a set of linear second order periodic equations is converted into an equivalent set of time-independent second order equations through a sequence of linear transformations. Then the transformations are applied to a time-periodic quadratic Hamiltonian to obtain its equivalent time-invariant form. In the second part, time-invariant forms of nonlinear equations are studied. The application of L–F transformation to a quasi-linear periodic equation converts the linear part to a time-invariant form and leaves the non-linear part with time-periodic coefficients. Dynamically equivalent time-invariant forms are obtained via time-periodic center manifold reduction and time-dependent normal form theory. Such forms are constructed for general hyperbolic systems and for some simple critical cases, including that of one zero eigenvalue and a purely imaginary pair. As a physical example of these techniques, a single and a double inverted pendulum subjected to periodic parametric excitation are considered. The results thus obtained are verified by numerical simulation.     

Stochastic averaging of n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations

The averaged generalized Fokker-Planck-Kolmogorov (GFPK) equation for response of n-dimensional (n-d) non-linear dynamical systems to non-Gaussian wide-band stationary random excitation is derived from the standard form of equation of motion. The explicit expressions for coefficients of the fourth-order approximation of the averaged GFPK equation are given in series form. Conditions for convergences of these series are pointed out. The averaged GFPK equation is then reduced to that for 1-d dynamical systems derived by Stratonovich and compared with the closed form of GFPK equation for n-d dynamical systems subject to Poisson white noise derived by Di Paola and Falsone. Finally, this averaged GFPK equation is further reduced to that for quasi linear system subject to non-Gaussian wide-band stationary random excitation. Stationary probability density for quasi linear system subject to filtered Poisson white noise is obtained. Theoretical results for an example are confirmed by using Monte-Carlo simulation for different parameter values.     

An investigation of the stability in the large for an autonomous second-order two-degree-of-freedom system

An extension to an algorithm due to Simpson has been developed for the analysis of a second-order two-degree-of-freedom autonomous system. The form of equations considered arises from the study of mechanical systems with a single concentrated non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. For a system possessing a stable equilibrium point and an unstable limit cycle arising from a subcritical Hopf bifurcation, the method has been applied to the problem of predicting the basin of attraction of the equilibrium point. 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 four weakly 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 the cases considered. However, it would be expected that the method would not give such accurate results if the non-linear effect was more significant. This was illustrated for the case of the cubic hardening non-linearity.     

A PIECEWISE–LINEARIZED METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS: TWO–POINT BOUNDARY–VALUE PROBLEMS

Piecewise-linearized methods for the solution of two-point boundary value problems in ordinary differential equations are presented. These problems are approximated by piecewise linear ones which have analytical solutions and reduced to finding the slope of the solution at the left boundary so that the boundary conditions at the right end of the interval are satisfied. This results in a rather complex system of non-linear algebraic equations which may be reduced to a single non-linear equation whose unknown is the slope of the solution at the left boundary of the interval and whose solution may be obtained by means of the Newton–Raphson method. This is equivalent to solving the boundary value problem as an initial value one using the piecewise-linearized technique and a shooting method. It is shown that for problems characterized by a linear operator a technique based on the superposition principle and the piecewise-linearized method may be employed. For these problems the accuracy of piecewise-linearized methods is of second order. It is also shown that for linear problems the accuracy of the piecewise-linearized method is superior to that of fourth-order-accurate techniques. For the linear singular perturbation problems considered in this paper the accuracy of global piecewise linearizat ion is higher than that of finite difference and finite element methods. For non-linear problems the accuracy of piecewise-linearized methods is in most cases lower than that of fourth-order methods but comparable with that of second-order techniques owing to the linearization of the non-linear terms.     

Treatment of non-linearities in the numerical solution of the incompressible Navier–Stokes equations

The solution of the full non-linear set of discrete fluid flow equations is usually obtained by solving a sequence of linear equations. The type of linearization used can significantly affect the rate of convergence of the sequence to the final solution. The first objective of the present study was to determine the extent to which a full Newton–Raphson linearization of all non-linear terms enhances convergence relative to that obtained using the ‘standard’ incompressible flow linearization. A direct solution procedure was employed in this evaluation. It was found that the full linearization enhances convergence, especially when grid curvature effects are important. The direct solution of the linear set is uneconomical. The second objective of the paper was to show how the equations can be effectively solved by an iterative scheme, based on a coupled-equation line solver, which implicitly retains all the inter-equation couplings. This solution method was found to be competitive with the highly refined segregated solution methods that represent the current state-of-the-art.     

Exact analytic solutions of the porous media and the gas pressure diffusion ODEs in non-linear mechanics

Two kinds of second-order non-linear ordinary differential equations (ODEs) appearing in mathematical physics and non-linear mechanics are analyzed in this paper. The one concerns the Kidder equation in porous media and the second the gas pressure diffusion equation. Both these equations are strongly non-linear including quadratic first-order derivatives (damping terms). By a series of admissible functional transformations we reduce the prescribed equations to Abel's equations of the second kind of the normal form that they do not admit exact analytic solutions in terms of known (tabulated) functions. According to a mathematical methodology recently developed concerning the construction of exact analytic solutions of the above class of Abel's equations, we succeed in performing the exact analytic solutions of both Kidder's and gas pressure diffusion equations. The boundary and initial data being used in the above constructions are in accordance with each specific problem under considerations.     

Higher order linearization in non-linear random vibration

In this paper a higher order linearization method for analyzing non-linear random vibration problems is presented. The non-linear terms of the given equation are replaced by unknown linear terms. These are in turn described by extra non-linear differential equations. The combined system of equations is then linearized to arrive at a higher degree-of-freedom equation for the original system. The method is illustrated by considering the Duffing oscillator under white noise input. The equivalent two d.o.f linear system is derived by the present method. Numerical results on steady state variance and PSD functions are obtained. These are found to be better than the simple linearization results.     

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.     

Symmetry Breaking for a System of Two Linear Second-Order Ordinary Differential Equations

A new canonical form for a system of two linear second-orderordinary differential equations (odes) is obtained. The latter isdecisive in unravelling symmetry structure of a system of two linearsecond-order odes. Namely we establish that the point symmetry Liealgebra of a system of two linear second-order odes can be5-, 6-, 7-, 8- or 15-dimensional. This result enhances both the richness andthe complexity of the symmetry structure of linear systems.     

Evaluation of multi-modes for finite-element models: systems tuned into 1:2 internal resonance

A non-linear multi-mode of vibration arises from the coupling of two or more normal modes of a non-linear system under free-vibration. The ensuing motion takes place on a 2M-dimensional invariant manifold in the phase space of the system, M being the number of coupled linear modes; the manifold contains a stable equilibrium point of interest, and at that point is tangent to the 2M-dimensional eigenspace of the system linearised about that equilibrium point, which characterises the corresponding M linear modes. On this manifold, M pairs of state variables govern the dynamics of the system; that is, the system behaves like an M-degree-of-freedom oscillator. Non-linear multi-modes may therefore come about when the system exhibits non-linear coupling among generalised co-ordinates. That is the case, for instance, of internal resonance of the 1:2 or 1:3 types, for systems with quadratic or cubic non-linearities, respectively, in which a four-dimensional manifold should be determined. Evaluation of non-linear multi-modes poses huge computational challenges, which is the explanation for very limited reports on the subject in the literature so far. The authors developed a procedure to determine the non-linear multi-modes for finite-element models of plane frames, using the method of multiple scales. This paper refers to the case of quadratic non-linearities. The results obtained by the proposed technique are in good agreement with those coming out from direct integration of the equations of motion in the time domain and also with those few available in the literature.     

Multiple scales analysis for double Hopf bifurcation with?1:3?resonance

The dynamical behavior of a general n-dimensional delay differential equation (DDE) around a 1:3 resonant double Hopf bifurcation point is analyzed. The method of multiple scales is used to obtain complex bifurcation equations. By expressing complex amplitudes in a mixed polar-Cartesian representation, the complex bifurcation equations are again obtained in real form. As an illustration, a system of two coupled van der Pol oscillators is considered and a set of parameter values for which a 1:3 resonant double Hopf bifurcation occurs is established. The dynamical behavior around the resonant double Hopf bifurcation point is analyzed in terms of three control parameters. The validity of analytical results is shown by their consistency with numerical simulations.