Chapter 6
Time-Delayed Nonlinear Oscillators
In this chapter, analytical solutions for period-m motions in a time-delayed, nonlinear oscillator will be presented through the Fourier series, and the stability and bifurcation analyses of the corresponding periodic motions are presented through the eigenvalue analysis. Analytical bifurcation trees of periodic motions to chaos will be presented through the frequency-amplitude curves. Trajectories and amplitude spectrums of periodic motions in such a time-delayed nonlinear system are illustrated numerically for a better understanding of time-delayed nonlinear dynamical systems.
6.1 Analytical Solutions
In this section, the analytical solutions of periodic motions in time-delayed, nonlinear systems will be developed through finite Fourier series. Consider a periodically forced, time-delayed, nonlinear oscillator as
where 
and 
. Coefficients in Equation (6.1) are 
and 
for linear damping, 
and 
for linear springs and delay, 
and 
for quadratic nonlinearity and delay, 
and 
for the cubic nonlinearity and delay, 
and 
for excitation amplitude and frequency, respectively. In Luo (2012a), the standard form of Equation (6.1) can be written as
where
The analytical solution of period-m motion for the above equation is
where 
and 
. The coefficients 
vary with time, and the derivatives of the foregoing equations are
Substitution of Equations (6.4)–(6.6) into Equation (6.1) and application of the virtual work principle for a basis of constant, 
and 
as a set of virtual displacements gives
where
Therefore, the coefficients of constant, 
and 
for the function of 
can be obtained. The constant term is given by
The constants caused by quadratic nonlinearity are
The constants caused by cubic nonlinearity are
with
The time-delay related constants, caused by cubic nonlinearity, are
with
where
The cosine term is given by
The cosine terms, caused by the quadratic nonlinear terms, are
with
where
The cosine terms, caused by the cubic nonlinearity, are given by
with
where
The time-delayed cosine terms, caused by the cubic nonlinearity, are given by
with
The sine term is given by
The sine term, caused by the quadratic nonlinearity, is given by
with
where
The sine term, caused by the cubic nonlinearity, is given by
with
where
The time-delayed sine term, caused by the cubic nonlinearity, is given by
with
Define
Equation (6.7) can be expressed in the form of vector field as
where
and
Introducing
Equation (6.38) becomes
The steady-state solutions for periodic motion in Equation (6.1) can be obtained by setting 
and 
, that is,
The 
nonlinear equations in Equation (6.43) are solved by the Newton-Raphson method. In Luo (2012a), the linearized equation at equilibrium 
and 
is given by
where
The Jacobian matrices are
and
for 
with
for 
The corresponding components for constants are
where for 
with
The corresponding components for cosine terms are
where
with
and
with
The corresponding components for sine terms are
where
with
and
with
The components relative to time-delay for constants are for 
where
and
with
The components relative to time-delay for cosine terms are
where
with
and
with
The components relative to time-delay for sine terms are
where
with
and