Solving systems of nonlinear difference equations by the multiple scales perturbation method
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Abstract
In this paper, we apply an improved version of the multiple scales perturbation method to a system of weakly nonlinear, regularly perturbed ordinary difference equations. Such systems arise as a result of the discretization of a system of nonlinear differential equations, or as a result in the stability analysis of nonlinear oscillations. In our procedure, asymptotic approximations of the solutions of the difference equations will be constructed which are valid on long iteration scales.
Keywords
Nonlinear difference equation Multiple scales perturbation method Difference operator1 Introduction
For scientists and engineers, the analysis of nonlinear dynamical systems is an important field of research since the solutions of these systems can exhibit counterintuitive and sometimes unexpected behavior. To obtain useful information from these systems, the multiple scales perturbation method can play an important role.
Nowadays, the multiple time-scales perturbation method for differential equations is well developed, well accepted, and a very popular method to approximate solutions of weakly nonlinear differential equations. For difference equations, this perturbation method is recently improved by Van Horssen [1] such that it can be applied to a large class of problems. In [1], a version of the multiple scales perturbation method is presented in a complete “difference operator” setting. This method can for instance be applied to problems for systems with time-varying masses. Examples of such systems can be found in robotics, rotating crankshafts, conveyor systems, excavators, cranes, biomechanics, and in fluid-structure interaction problems [2, 3]. The oscillations of electric transmission lines and cables of cable-stayed bridges with water rivulets on the surface are also examples of time-varying dynamic systems [4]. For these mechanical constructions, the 1-mode Galerkin approximation of the continuous model will lead to a single degree-of-freedom oscillator (sdofo)-equation. These sdofos are considered to be representative models for testing numerical methods and for studying forces which are acting on the system [5]. In [6] and [7], the forced vibrations of a linear sdofo with a time-varying mass were studied. The forced vibrations are due to small masses which are periodically hitting and leaving the oscillator with different velocities. In [8], the vibrations of a damped, linear sdofo with a time-varying mass were studied, and the stability properties for the free, and for the forced vibrations (due to small masses and an external force (for instance, a windforce)) were presented for various parameter values. A system of two nonlinear ordinary difference equations (OΔEs) is obtained when also windforces are included in the model. To analyze the system of OΔEs, numerical methods were used. In the analysis, a small parameter ε was defined for the relative mass which is added periodically. Then, also a perturbation method can be applied. In this paper, we are going to study similar systems of OΔEs. It will be shown how the improved multiple scales perturbation method can be applied to such systems of OΔEs. Moreover, several bifurcation problems will be studied in detail.
2 Problem definition
3 The multiple scales perturbation method for OΔEs
4 Two complex eigenvalues
Values of M_{k} (for k=1,…,4) in (19) for some representations of the absolute value of the eigenvalues λ_{1} and λ_{2}
r | M_{1}= | M_{2}= | M_{3}= | M_{4}= |
---|---|---|---|---|
0<r<1 | M_{11} | M_{21} | M_{12} | M_{22} |
r=1 | M_{11}+M_{17} | M_{21}+M_{27} | M_{12}+M_{18} | M_{22}+M_{28} |
where λ_{1}=cos(θ)+isin(θ) and \(\lambda _{2}=\overline {\lambda_{1}}\).
5 Conclusions and remarks
In this paper, by applying the multiple scales perturbation method to a general system of two first-order ordinary difference equations, including linear, quadratic, and qubic terms, we obtain approximations of the solutions which are valid on long iteration scales. We considered two cases for which the eigenvalues are complex with nonzero real and imaginary parts, and the modulus is less than or equal to 1. For the case when the modulus is smaller than 1, we found conditions for which the solutions are stable, and for the case when the modulus is equal to 1, and for some special values of the constants, we encounter limit cycles, and a circle of equilibrium points. Our results are in nice agreement with numerical results in [9] when ε is considered to be small. The stable and unstable limit cycles in the phase plane shown by [9] show this agreement.
The methodology that we used in this paper can also be expanded in the same way to the other cases such as two distinct real eigenvalues, and two coinciding real eigenvalues, and based on the values for the constants a_{ij} and b_{ij} we can have the bifurcation diagrams as well. The obtained results help us in a better understanding of the behavior of nonlinear oscillations. In particular, all kinds of bifurcations can be studied in detail.
Notes
Open Access
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References
- 1.Van Horssen, W.T., Ter Brake, M.C.: On the multiple scales perturbation methods for difference equations. Nonlinear Dyn. 55, 401–418 (2009) MATHCrossRefGoogle Scholar
- 2.Irschik, H., Holl, H.J.: Mechanics of variable-mass systems-part 1: balance of mass and linear momentum. Appl. Mech. Rev. 5, 145–160 (2004) CrossRefGoogle Scholar
- 3.Cveticanin, L.: Self-excited vibrations of the variable mass rotor/fluid system. J. Sound Vib. 212, 685–702 (1998) MathSciNetCrossRefGoogle Scholar
- 4.Van der Burgh, A.H.P., Hartono, Abramian, A.K.: A new model for the study of rain-wind-induced vibrations of a simple oscillator. Int. J. Non-Linear Mech. 41, 345–358 (2006) MATHCrossRefGoogle Scholar
- 5.Holl, H.J., Belyaev, A.K., Irschik, H.: Simulation of the Duffing-oscillator with time-varying mass by a BEM in time. Comput. Struct. 73, 177–186 (1999) MATHCrossRefGoogle Scholar
- 6.Van Horssen, W.T., Pischanskyy, O.V., Dubbeldam, J.L.A.: On the stability properties of a periodically forced, time-varying mass system. ASME IDETC/CIE 329, 679–687 (2009) Google Scholar
- 7.Van Horssen, W.T., Pischanskyy, O.V., Dubbeldam, J.L.A.: On the forced vibrations of an oscillator with a periodically time-varying mass. J. Sound Vib. 329, 721–732 (2010) CrossRefGoogle Scholar
- 8.Van Horssen, W.T., Pischanskyy, O.V.: On the stability properties of a damped oscillator with a periodically time-varying mass. J. Sound Vib. 330, 3257–3269 (2011) CrossRefGoogle Scholar
- 9.Pischanskyy, O.V., Van Horssen, W.T.: On the nonlinear dynamics of a single degree of freedom oscillator with a time-varying mass. J. Sound Vib. 331, 1887–1897 (2012) CrossRefGoogle Scholar