Nonlinear oscillator with velocity dependent frequency
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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 https://www.sciencedirect.com/science/article/pii/S0022460X00929894
mp.mathematical-physics differential-equations
asked 4 hours ago
Zurab SilagadzeZurab Silagadze
10.8k2569
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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 https://www.sciencedirect.com/science/article/pii/S0022460X00929894
mp.mathematical-physics differential-equations
asked 4 hours ago
Zurab SilagadzeZurab Silagadze
10.8k2569
$endgroup$
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 https://www.sciencedirect.com/science/article/pii/S0022460X00929894
mp.mathematical-physics differential-equations
asked 4 hours ago
Zurab SilagadzeZurab Silagadze
10.8k2569
$endgroup$
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 https://www.sciencedirect.com/science/article/pii/S0022460X00929894
mp.mathematical-physics differential-equations
mp.mathematical-physics differential-equations
asked 4 hours ago
Zurab SilagadzeZurab Silagadze
10.8k2569
asked 4 hours ago
Zurab SilagadzeZurab Silagadze
10.8k2569
asked 4 hours ago
Zurab SilagadzeZurab Silagadze
10.8k2569
asked 4 hours ago
Zurab SilagadzeZurab Silagadze
10.8k2569
asked 4 hours ago
Zurab SilagadzeZurab Silagadze
10.8k2569
10.8k2569
add a comment |
add a comment |
1 Answer
1
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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 3 hours ago
Carlo BeenakkerCarlo Beenakker
74.9k9170277
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1 Answer
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1 Answer
1
active
oldest
votes
active
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active
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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 3 hours ago
Carlo BeenakkerCarlo Beenakker
74.9k9170277
$endgroup$
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 3 hours ago
Carlo BeenakkerCarlo Beenakker
74.9k9170277
$endgroup$
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 3 hours ago
Carlo BeenakkerCarlo Beenakker
74.9k9170277
$endgroup$
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 3 hours ago
Carlo BeenakkerCarlo Beenakker
74.9k9170277
answered 3 hours ago
Carlo BeenakkerCarlo Beenakker
74.9k9170277
answered 3 hours ago
Carlo BeenakkerCarlo Beenakker
74.9k9170277
answered 3 hours ago
Carlo BeenakkerCarlo Beenakker
74.9k9170277
74.9k9170277
add a comment |
add a comment |
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