Coupled Van der Pol oscillator with non-integer order connection

• System of two coupled oscillators with pure nonlinear connection of any positive rational order is introduced.
• Analytical method for approximate solutions in the form of the Ateb function is proposed and verified on numerical examples.
• The procedure is applied for solving the two-mass Van der Pol oscillator system.
• Solution dependence on the order of nonlinearity is discussed.

In this paper the dynamics of a system of two oscillators with strong nonlinear connection is considered. The two mass system connected with a spring with pure nonlinear force of any positive rational order (integer or noninteger), on which some additional small nonlinear forces act, is analyzed. The mathematical model of the system contains two coupled second order differential equations of oscillatory type with strong pure nonlinearity and small additional terms. In the paper an analytical solving procedure which introduces the periodical Ateb function is developed. The averaging solution method is adopted to this special function and gives the new type of averaged differential equations.The special attention is given to the steady-state motion of a two-degree-of-freedom Van der Pol oscillator system of positive rational order of nonlinearity. The influence of the order of nonlinearity on the motion of the system is analyzed. using the suggested approximate method three numerical examples are solved. The obtained results are much more accurate than those obtained by the already published methods based on the trigonometric functions.