<s>
The	O
Duffing	B-Algorithm
equation	I-Algorithm
(	O
or	O
Duffing	B-Algorithm
oscillator	I-Algorithm
)	O
,	O
named	O
after	O
Georg	O
Duffing	B-Algorithm
(	O
1861	O
–	O
1944	O
)	O
,	O
is	O
a	O
non-linear	O
second-order	O
differential	O
equation	O
used	O
to	O
model	O
certain	O
damped	O
and	O
driven	O
oscillators	O
.	O
</s>
<s>
where	O
the	O
(	O
unknown	O
)	O
function	O
is	O
the	O
displacement	O
at	O
time	O
is	O
the	O
first	B-Algorithm
derivative	I-Algorithm
of	O
with	O
respect	O
to	O
time	O
,	O
i.e.	O
</s>
<s>
velocity	O
,	O
and	O
is	O
the	O
second	O
time-derivative	O
of	O
i.e.	O
</s>
<s>
The	O
equation	O
describes	O
the	O
motion	O
of	O
a	O
damped	O
oscillator	O
with	O
a	O
more	O
complex	O
potential	O
than	O
in	O
simple	O
harmonic	O
motion	O
(	O
which	O
corresponds	O
to	O
the	O
case	O
)	O
;	O
in	O
physical	O
terms	O
,	O
it	O
models	O
,	O
for	O
example	O
,	O
an	O
elastic	O
pendulum	O
whose	O
spring	O
's	O
stiffness	B-Algorithm
does	O
not	O
exactly	O
obey	O
Hooke	O
's	O
law	O
.	O
</s>
<s>
The	O
Duffing	B-Algorithm
equation	I-Algorithm
is	O
an	O
example	O
of	O
a	O
dynamical	O
system	O
that	O
exhibits	O
chaotic	O
behavior	O
.	O
</s>
<s>
Moreover	O
,	O
the	O
Duffing	B-Algorithm
system	O
presents	O
in	O
the	O
frequency	O
response	O
the	O
jump	O
resonance	O
phenomenon	O
that	O
is	O
a	O
sort	O
of	O
frequency	O
hysteresis	O
behaviour	O
.	O
</s>
<s>
controls	O
the	O
linear	O
stiffness	B-Algorithm
,	O
</s>
<s>
controls	O
the	O
amount	O
of	O
non-linearity	O
in	O
the	O
restoring	O
force	O
;	O
if	O
the	O
Duffing	B-Algorithm
equation	I-Algorithm
describes	O
a	O
damped	O
and	O
driven	O
simple	O
harmonic	O
oscillator	O
,	O
</s>
<s>
The	O
Duffing	B-Algorithm
equation	I-Algorithm
can	O
be	O
seen	O
as	O
describing	O
the	O
oscillations	O
of	O
a	O
mass	O
attached	O
to	O
a	O
nonlinear	O
spring	O
and	O
a	O
linear	O
damper	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
adjectives	O
hardening	O
and	O
softening	O
are	O
used	O
with	O
respect	O
to	O
the	O
Duffing	B-Algorithm
equation	I-Algorithm
in	O
general	O
,	O
dependent	O
on	O
the	O
values	O
of	O
(	O
and	O
)	O
.	O
</s>
<s>
The	O
number	O
of	O
parameters	O
in	O
the	O
Duffing	B-Algorithm
equation	I-Algorithm
can	O
be	O
reduced	O
by	O
two	O
through	O
scaling	O
(	O
in	O
accord	O
with	O
the	O
Buckingham	O
π	O
theorem	O
)	O
,	O
e.g.	O
</s>
<s>
The	O
dots	O
denote	O
differentiation	O
of	O
with	O
respect	O
to	O
This	O
shows	O
that	O
the	O
solutions	O
to	O
the	O
forced	O
and	O
damped	O
Duffing	B-Algorithm
equation	I-Algorithm
can	O
be	O
described	O
in	O
terms	O
of	O
the	O
three	O
parameters	O
(	O
and	O
)	O
and	O
two	O
initial	B-Algorithm
conditions	I-Algorithm
(	O
i.e.	O
</s>
<s>
In	O
general	O
,	O
the	O
Duffing	B-Algorithm
equation	I-Algorithm
does	O
not	O
admit	O
an	O
exact	O
symbolic	O
solution	O
.	O
</s>
<s>
The	O
term	O
,	O
also	O
called	O
the	O
Duffing	B-Algorithm
term	O
,	O
can	O
be	O
approximated	O
as	O
small	O
and	O
the	O
system	O
treated	O
as	O
a	O
perturbed	O
simple	O
harmonic	O
oscillator	O
.	O
</s>
<s>
Any	O
of	O
the	O
various	O
numeric	B-General_Concept
methods	I-General_Concept
such	O
as	O
Euler	B-Algorithm
's	I-Algorithm
method	I-Algorithm
and	O
Runge	B-Algorithm
–	I-Algorithm
Kutta	I-Algorithm
methods	I-Algorithm
can	O
be	O
used	O
.	O
</s>
<s>
The	O
homotopy	O
analysis	O
method	O
(	O
HAM	O
)	O
has	O
also	O
been	O
reported	O
for	O
obtaining	O
approximate	O
solutions	O
of	O
the	O
Duffing	B-Algorithm
equation	I-Algorithm
,	O
also	O
for	O
strong	O
nonlinearity	O
.	O
</s>
<s>
In	O
the	O
special	O
case	O
of	O
the	O
undamped	O
(	O
)	O
and	O
undriven	O
(	O
)	O
Duffing	B-Algorithm
equation	I-Algorithm
,	O
an	O
exact	O
solution	O
can	O
be	O
obtained	O
using	O
Jacobi	O
's	O
elliptic	O
functions	O
.	O
</s>
<s>
Multiplication	O
of	O
the	O
undamped	O
and	O
unforced	O
Duffing	B-Algorithm
equation	I-Algorithm
,	O
with	O
gives	O
:	O
</s>
<s>
Without	O
forcing	O
the	O
damped	O
Duffing	B-Algorithm
oscillator	I-Algorithm
will	O
end	O
up	O
at	O
(	O
one	O
of	O
)	O
its	O
stable	O
equilibrium	O
point(s )	O
.	O
</s>
<s>
The	O
forced	O
Duffing	B-Algorithm
oscillator	I-Algorithm
with	O
cubic	O
nonlinearity	O
is	O
described	O
by	O
the	O
following	O
ordinary	O
differential	O
equation	O
:	O
</s>
<s>
Depending	O
on	O
the	O
type	O
of	O
nonlinearity	O
,	O
the	O
Duffing	B-Algorithm
oscillator	I-Algorithm
can	O
show	O
hardening	O
,	O
softening	O
or	O
mixed	O
hardening	O
–	O
softening	O
frequency	O
response	O
.	O
</s>
<s>
For	O
the	O
parameters	O
of	O
the	O
Duffing	B-Algorithm
equation	I-Algorithm
,	O
the	O
above	O
algebraic	O
equation	O
gives	O
the	O
steady	O
state	O
oscillation	O
amplitude	B-Application
at	O
a	O
given	O
excitation	O
frequency	O
.	O
</s>
<s>
Derivation	B-Algorithm
of	O
the	O
frequency	O
responseUsing	O
the	O
method	O
of	O
harmonic	O
balance	O
,	O
an	O
approximate	O
solution	O
to	O
the	O
Duffing	B-Algorithm
equation	I-Algorithm
is	O
sought	O
of	O
the	O
form	O
:	O
</s>
<s>
Application	O
in	O
the	O
Duffing	B-Algorithm
equation	I-Algorithm
leads	O
to	O
:	O
</s>
<s>
Squaring	O
both	O
equations	O
and	O
adding	O
leads	O
to	O
the	O
amplitude	B-Application
frequency	O
response	O
:	O
</s>
<s>
For	O
certain	O
ranges	O
of	O
the	O
parameters	O
in	O
the	O
Duffing	B-Algorithm
equation	I-Algorithm
,	O
the	O
frequency	O
response	O
may	O
no	O
longer	O
be	O
a	O
single-valued	B-Algorithm
function	I-Algorithm
of	O
forcing	O
frequency	O
For	O
a	O
hardening	O
spring	O
oscillator	O
(	O
and	O
large	O
enough	O
positive	O
)	O
the	O
frequency	O
response	O
overhangs	O
to	O
the	O
high-frequency	O
side	O
,	O
and	O
to	O
the	O
low-frequency	O
side	O
for	O
the	O
softening	O
spring	O
oscillator	O
(	O
and	O
)	O
.	O
</s>
<s>
when	O
the	O
angular	O
frequency	O
is	O
slowly	O
increased	O
(	O
with	O
other	O
parameters	O
fixed	O
)	O
,	O
the	O
response	O
amplitude	B-Application
drops	O
at	O
A	O
suddenly	O
to	O
B	O
,	O
</s>
<s>
if	O
the	O
frequency	O
is	O
slowly	O
decreased	O
,	O
then	O
at	O
C	O
the	O
amplitude	B-Application
jumps	O
up	O
to	O
D	O
,	O
thereafter	O
following	O
the	O
upper	O
branch	O
of	O
the	O
frequency	O
response	O
.	O
</s>
<s>
Some	O
typical	O
examples	O
of	O
the	O
time	O
series	O
and	O
phase	B-Application
portraits	I-Application
of	O
the	O
Duffing	B-Algorithm
equation	I-Algorithm
,	O
showing	O
the	O
appearance	O
of	O
subharmonics	O
through	O
period-doubling	O
bifurcation	O
–	O
as	O
well	O
chaotic	O
behavior	O
–	O
are	O
shown	O
in	O
the	O
figures	O
below	O
.	O
</s>
