OSCILLATORS NEAR HOPF BIFURCATION OSCILLATORS NEAR HOPF BIFURCATION

1,2,3 be the differential transformations of functions g, g i , i = 1,2,3, then the following correspondences hold In this paper the differential transformation method (DTM) is employed to solve a system of linear differential equations derived for energy optimal control theory and nonlinear differential equations and their systems for oscillators near Hopf bifurcation. They are given by differential equations in a complex plane. Different types of forced terms are considered. The approximate solutions are given in the form of series with required accuracy. Some numerical examples are provided.


Introduction
The collective behaviour of oscillators and coupled oscillators represents the complex dynamic system. It demonstrates interest in experimental as well as numerical research. Examples of such applications may be found in mechanical oscillators, limit cycle oscillators, semiconductor lasers, superconducting Josephson junction or pacemakers. Synchronization, where individual oscillators oscillate at a common phase and frequency, depends on coupling force. A weak coupling leads to the character of non-coupled oscillators, for stronger coupling the considering of both frequency and amplitude is required. An approach used in the investigation is based on the study of the stability -the system is linearized near zero [1]. The solving of differential equations or their systems by DTM avoids such a restriction. This paper presents two main goals. Firstly, we apply the differential transformation method for oscillators and coupled oscillators. Secondly, we reduce a system of linear equations in an engineering application to the system of linear equations to energy optimal control of motors.

DTM -differential transformation method
In this section we bring a brief overview of the differential transformation method that proposes a technique which does not evaluate the derivatives symbolically; the differential transformation DT of the n th derivative of a function y(t) is defined by where y(t) is an original and Y(t) is a transformed function.
If t 0 0 = , then the representation of the function reduces to y t Y n t n n 0 The properties of the DT with their proofs are published in [2 -4]; the used ones of them are introduced below.
Let functions G, G i , i = 1,2,3 be the differential transformations of functions g, g i , i = 1,2,3, then the following correspondences hold g t g t g t G n G n G n

OSCILLATORS NEAR HOPF BIFURCATION OSCILLATORS NEAR HOPF BIFURCATION Helena Samajova -Tongxing Li *
In this paper the differential transformation method (DTM) is employed to solve a system of linear differential equations derived for energy optimal control theory and nonlinear differential equations and their systems for oscillators near Hopf bifurcation. They are given by differential equations in a complex plane. Different types of forced terms are considered. The approximate solutions are given in the form of series with required accuracy. Some numerical examples are provided.
Keywords: Oscillators, systems of ordinary differential equations, differential transformation method.
for the same conditions as in R1. and , k l R3. If the forcing term is of the exponential type f t e je = and for identical initial condition we obtain following solutions In the special case Remark: From the theory of Taylor series and relations (7), (8) and (13), it is evident that the influence of the power type right hand side (RHS) manifests from the term of the lower degree of k,l unlike the RHS of exponential type changes all the terms of solutions.

Differential equation for an oscillator with a complex parameter ~.
Consider the equation (11) with j C

Differential equation for an oscillator with a real parameter ~.
We consider nonlinear differential equation used for oscillators [1] near Hopf bifurcation [5] with different types of forcing terms y t y t j y t f t 1 where j Using the properties (4)     The difference of two frequencies 1 2 T~= and a coupling constant K introduced in the system of differential equations (23) have a decisive influence on the behaviour of such systems of coupled oscillators. It could be one of the parameters which may be the reason of quenching of oscillators´ amplitudes.

System of differential equations for coupled oscillators with real parameters
The numerical solutions of (21) are given for the initial conditions  = + h is studied in [6]. Parameters for this model of an engine are given in [8]

Conclusion
We used the differential transformation method DTM for two different problems -coupled nonlinear oscillators and energy optimal speed and position control of PMSM drive. Solutions to linear and nonlinear systems of ODE are calculated for systems with power and exponential types of RHS.