Acta Mech
DOI 10.1007/s00707-016-1582-9
ORIGINAL PAPER
L. Cveticanin · M. Zukovic · Gy. Mester · I. Biro · J. Sarosi
Oscillators with symmetric and asymmetric quadratic
nonlinearity
Received: 22 July 2015 / Revised: 22 December 2015
© Springer-Verlag Wien 2016
Abstract In this paper, oscillators with asymmetric and symmetric quadratic nonlinearity are compared. Both
oscillators are modeled as ordinary second-order differential equations with strong quadratic nonlinearities:
one with positive quadratic term and the second with a quadratic term which changes the sign. Solutions for
both equations are obtained in the form of Jacobi elliptic functions. For the asymmetric oscillator, conditions for
the periodic motion are determined, while for the symmetric oscillator a new approximate solution procedure
based on averaging is developed. Obtained results are tested on an optomechanical system where the motion
of a cantilever in the intracavity field is oscillatory. Two types of quadratic nonlinearities in the system are
investigated: symmetric and asymmetric. The advantage and disadvantage of both models is discussed. The
analytical procedure suggested in the paper is applied. The obtained solution agrees well with a numerical one.
1 Introduction
In this paper, two types of oscillators with quadratic nonlinearity are investigated: one is the so-called symmetric,
and the other the asymmetric oscillator. The mathematical model of the asymmetric oscillator is assumed to
be
¨ x + c
1
x + c
2
x
2
= 0, (1)
and the mathematical model of the symmetric oscillator is then assumed to be
¨ x + c
1
x + c
2
x |x |= 0, (2)
where c
1
and c
2
are positive parameters of the linear and of the quadratic term, respectively. The meaning of
‘symmetry’ and ‘asymmetry’ in the oscillator is linked with the form of the equation of motion and also the
solution of the equation of motion. For the symmetric quadratic oscillator (2), the function F (x ) = c
1
x +c
2
x |x |
is odd, that is F (−x ) =−F (x ). The same relation is satisfied for the harmonic linear oscillator, which is named
‘symmetric oscillator,’ too. In the symmetric oscillator, the amplitude of vibration in the positive direction is
equal to the value in the negative direction of x . In (1), the quadratic term is nonnegative independent of the
sign of x and the condition of symmetry is not satisfied, that is F (−x ) =−F (x ). As a result, the amplitude
L. Cveticanin (B ) · M. Zukovic
University of Novi Sad, Trg D. Obradovica 6, Novi Sad 21000, Serbia
E-mail: cveticanin@uns.ac.rs
L. Cveticanin · Gy. Mester
Obuda University, Nepszinhaz u.8, Budapest 1081, Hungary
I. Biro · J. Sarosi
University of Szeged, Mars tér 7, Szeged 6724, Hungary