Nonlinear oscillator with velocity-dependent frequency
$begingroup$
In a physical problem I need to investigate the following nonlinear differential equation
$$ddot x+omega^2left (1+frac{m^2dot x^2}{p^2}right)x=0,$$
where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found the following:
- Ronald E. Mickens, Generalized harmonic oscillators, Journal of Sound and Vibration, Volume 236, Issue 4, Pages 730-732, 28 September 2000.
mp.mathematical-physics differential-equations
edited Jan 21 at 21:39
Rodrigo de Azevedo
1,8422719
asked Jan 21 at 12:13
Zurab SilagadzeZurab Silagadze
10.9k2669
$endgroup$
add a comment |
$begingroup$
In a physical problem I need to investigate the following nonlinear differential equation
$$ddot x+omega^2left (1+frac{m^2dot x^2}{p^2}right)x=0,$$
where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found the following:
- Ronald E. Mickens, Generalized harmonic oscillators, Journal of Sound and Vibration, Volume 236, Issue 4, Pages 730-732, 28 September 2000.
mp.mathematical-physics differential-equations
edited Jan 21 at 21:39
Rodrigo de Azevedo
1,8422719
asked Jan 21 at 12:13
Zurab SilagadzeZurab Silagadze
10.9k2669
$endgroup$
add a comment |
$begingroup$
In a physical problem I need to investigate the following nonlinear differential equation
$$ddot x+omega^2left (1+frac{m^2dot x^2}{p^2}right)x=0,$$
where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found the following:
- Ronald E. Mickens, Generalized harmonic oscillators, Journal of Sound and Vibration, Volume 236, Issue 4, Pages 730-732, 28 September 2000.
mp.mathematical-physics differential-equations
edited Jan 21 at 21:39
Rodrigo de Azevedo
1,8422719
asked Jan 21 at 12:13
Zurab SilagadzeZurab Silagadze
10.9k2669
$endgroup$
In a physical problem I need to investigate the following nonlinear differential equation
$$ddot x+omega^2left (1+frac{m^2dot x^2}{p^2}right)x=0,$$
where $p$ is some constant with a dimension of momentum. I will be grateful for references about such type of oscillators. So far I only found the following:
- Ronald E. Mickens, Generalized harmonic oscillators, Journal of Sound and Vibration, Volume 236, Issue 4, Pages 730-732, 28 September 2000.
mp.mathematical-physics differential-equations
mp.mathematical-physics differential-equations
edited Jan 21 at 21:39
Rodrigo de Azevedo
1,8422719
asked Jan 21 at 12:13
Zurab SilagadzeZurab Silagadze
10.9k2669
edited Jan 21 at 21:39
Rodrigo de Azevedo
1,8422719
asked Jan 21 at 12:13
Zurab SilagadzeZurab Silagadze
10.9k2669
edited Jan 21 at 21:39
Rodrigo de Azevedo
1,8422719
edited Jan 21 at 21:39
Rodrigo de Azevedo
1,8422719
edited Jan 21 at 21:39
Rodrigo de Azevedo
1,8422719
1,8422719
asked Jan 21 at 12:13
Zurab SilagadzeZurab Silagadze
10.9k2669
asked Jan 21 at 12:13
Zurab SilagadzeZurab Silagadze
10.9k2669
asked Jan 21 at 12:13
Zurab SilagadzeZurab Silagadze
10.9k2669
10.9k2669
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This type of ODE,
$$ddot{x}+f(x)dot{x}^2+g(x)=0$$
is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).
A particular case with nice properties is
$$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
see Design of nonlinear isochronous oscillators.
answered Jan 21 at 12:45
Carlo BeenakkerCarlo Beenakker
75.4k9171280
$endgroup$
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This type of ODE,
$$ddot{x}+f(x)dot{x}^2+g(x)=0$$
is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).
A particular case with nice properties is
$$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
see Design of nonlinear isochronous oscillators.
answered Jan 21 at 12:45
Carlo BeenakkerCarlo Beenakker
75.4k9171280
$endgroup$
add a comment |
$begingroup$
This type of ODE,
$$ddot{x}+f(x)dot{x}^2+g(x)=0$$
is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).
A particular case with nice properties is
$$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
see Design of nonlinear isochronous oscillators.
answered Jan 21 at 12:45
Carlo BeenakkerCarlo Beenakker
75.4k9171280
$endgroup$
add a comment |
$begingroup$
This type of ODE,
$$ddot{x}+f(x)dot{x}^2+g(x)=0$$
is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).
A particular case with nice properties is
$$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
see Design of nonlinear isochronous oscillators.
answered Jan 21 at 12:45
Carlo BeenakkerCarlo Beenakker
75.4k9171280
$endgroup$
This type of ODE,
$$ddot{x}+f(x)dot{x}^2+g(x)=0$$
is known as a Liénard equation of the second kind. It has been studied for example in Monotonicity of the period function of the Liénard equation of second kind (2016).
A particular case with nice properties is
$$ddot{x}-frac{f'(x)}{f(x)}dot{x}^2+f(x)int_0^xfrac{1}{f(u)}du=0,$$
see Design of nonlinear isochronous oscillators.
answered Jan 21 at 12:45
Carlo BeenakkerCarlo Beenakker
75.4k9171280
answered Jan 21 at 12:45
Carlo BeenakkerCarlo Beenakker
75.4k9171280
answered Jan 21 at 12:45
Carlo BeenakkerCarlo Beenakker
75.4k9171280
answered Jan 21 at 12:45
Carlo BeenakkerCarlo Beenakker
75.4k9171280
75.4k9171280
add a comment |
add a comment |
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