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Resonance in Mathieu's Equation
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Consider the Mathieu's Equation:
$$fracd^2udt^2+[omega^2 + 2epsilon cos(2t)]u=0$$ with $u(0)=1$ and $u'(0)=0$
What I have done is, assume $u(t)=u_0(t)+epsilon u_1(t)+cdots$, substitute into the DE, and we get
$$u_0(t)=cos(omega t)$$ and $$u_1(t)=frac(1-omega)cos[(2+omega)t]-2cos(omega t)+(1+omega)cos[(2-omega)t]4-4omega^2$$.
The question is, what modes are resonant at order $epsilon$? and what frequencies are resonant at the next order (i think it means order $epsilon^2$)?
What are the definition of modes and resonance in this problem? I have no idea how to proceed because I do not know the definition. Can anyone tell me what does it mean by "modes" and "resonant"? Moreover, What is "frequency" here? The problem is just an ODE, where does the word "frequency" come from?
differential-equations stability-in-odes perturbation-theory
asked Jul 21 at 3:12
bbw
31517
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up vote
0
down vote
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Consider the Mathieu's Equation:
$$fracd^2udt^2+[omega^2 + 2epsilon cos(2t)]u=0$$ with $u(0)=1$ and $u'(0)=0$
What I have done is, assume $u(t)=u_0(t)+epsilon u_1(t)+cdots$, substitute into the DE, and we get
$$u_0(t)=cos(omega t)$$ and $$u_1(t)=frac(1-omega)cos[(2+omega)t]-2cos(omega t)+(1+omega)cos[(2-omega)t]4-4omega^2$$.
The question is, what modes are resonant at order $epsilon$? and what frequencies are resonant at the next order (i think it means order $epsilon^2$)?
What are the definition of modes and resonance in this problem? I have no idea how to proceed because I do not know the definition. Can anyone tell me what does it mean by "modes" and "resonant"? Moreover, What is "frequency" here? The problem is just an ODE, where does the word "frequency" come from?
differential-equations stability-in-odes perturbation-theory
asked Jul 21 at 3:12
bbw
31517
At resonance the value of the function goes to infinity. The modes are typically orthogonal functions whose linear combinations add up to $u(t)$. Note sure whether $u_0$ and $u_1$ are orthogonal in this case.
– Biswajit Banerjee
Jul 21 at 4:12
add a comment |Â
up vote
0
down vote
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up vote
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down vote
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Consider the Mathieu's Equation:
$$fracd^2udt^2+[omega^2 + 2epsilon cos(2t)]u=0$$ with $u(0)=1$ and $u'(0)=0$
What I have done is, assume $u(t)=u_0(t)+epsilon u_1(t)+cdots$, substitute into the DE, and we get
$$u_0(t)=cos(omega t)$$ and $$u_1(t)=frac(1-omega)cos[(2+omega)t]-2cos(omega t)+(1+omega)cos[(2-omega)t]4-4omega^2$$.
The question is, what modes are resonant at order $epsilon$? and what frequencies are resonant at the next order (i think it means order $epsilon^2$)?
What are the definition of modes and resonance in this problem? I have no idea how to proceed because I do not know the definition. Can anyone tell me what does it mean by "modes" and "resonant"? Moreover, What is "frequency" here? The problem is just an ODE, where does the word "frequency" come from?
differential-equations stability-in-odes perturbation-theory
asked Jul 21 at 3:12
bbw
31517
Consider the Mathieu's Equation:
$$fracd^2udt^2+[omega^2 + 2epsilon cos(2t)]u=0$$ with $u(0)=1$ and $u'(0)=0$
What I have done is, assume $u(t)=u_0(t)+epsilon u_1(t)+cdots$, substitute into the DE, and we get
$$u_0(t)=cos(omega t)$$ and $$u_1(t)=frac(1-omega)cos[(2+omega)t]-2cos(omega t)+(1+omega)cos[(2-omega)t]4-4omega^2$$.
The question is, what modes are resonant at order $epsilon$? and what frequencies are resonant at the next order (i think it means order $epsilon^2$)?
What are the definition of modes and resonance in this problem? I have no idea how to proceed because I do not know the definition. Can anyone tell me what does it mean by "modes" and "resonant"? Moreover, What is "frequency" here? The problem is just an ODE, where does the word "frequency" come from?
differential-equations stability-in-odes perturbation-theory
asked Jul 21 at 3:12
bbw
31517
asked Jul 21 at 3:12
bbw
31517
asked Jul 21 at 3:12
bbw
31517
asked Jul 21 at 3:12
bbw
31517
31517
At resonance the value of the function goes to infinity. The modes are typically orthogonal functions whose linear combinations add up to $u(t)$. Note sure whether $u_0$ and $u_1$ are orthogonal in this case.
– Biswajit Banerjee
Jul 21 at 4:12
add a comment |Â
At resonance the value of the function goes to infinity. The modes are typically orthogonal functions whose linear combinations add up to $u(t)$. Note sure whether $u_0$ and $u_1$ are orthogonal in this case.
– Biswajit Banerjee
Jul 21 at 4:12
At resonance the value of the function goes to infinity. The modes are typically orthogonal functions whose linear combinations add up to $u(t)$. Note sure whether $u_0$ and $u_1$ are orthogonal in this case.
– Biswajit Banerjee
Jul 21 at 4:12
At resonance the value of the function goes to infinity. The modes are typically orthogonal functions whose linear combinations add up to $u(t)$. Note sure whether $u_0$ and $u_1$ are orthogonal in this case.
– Biswajit Banerjee
Jul 21 at 4:12
add a comment |Â
1 Answer
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You can rewrite the last formula as
$$
u_1(t)=fracsin(Ét)2left(fracsin((1-É)t)1-É-fracsin((1+É)t)1+Éright)
$$
which under $Éto1$ has the limit
$$
u_1(t)=fracsin(t)2left(t-fracsin(2t)2right)
$$
which is growing without bound.
As $u_1$ has frequencies $2+É$, $É$ and $|2-É|$, the multiplication with a frequency $2$ term will lead to terms with frequencies $4+É$, $2+É$, $É$, $|2-É|$ and $|4-É|$. Resonance happens when two of these frequencies fall together, which is at $É=2$ and again at $É=1$.
answered Jul 21 at 7:41
LutzL
49.8k31849
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
You can rewrite the last formula as
$$
u_1(t)=fracsin(Ét)2left(fracsin((1-É)t)1-É-fracsin((1+É)t)1+Éright)
$$
which under $Éto1$ has the limit
$$
u_1(t)=fracsin(t)2left(t-fracsin(2t)2right)
$$
which is growing without bound.
As $u_1$ has frequencies $2+É$, $É$ and $|2-É|$, the multiplication with a frequency $2$ term will lead to terms with frequencies $4+É$, $2+É$, $É$, $|2-É|$ and $|4-É|$. Resonance happens when two of these frequencies fall together, which is at $É=2$ and again at $É=1$.
answered Jul 21 at 7:41
LutzL
49.8k31849
add a comment |Â
up vote
0
down vote
You can rewrite the last formula as
$$
u_1(t)=fracsin(Ét)2left(fracsin((1-É)t)1-É-fracsin((1+É)t)1+Éright)
$$
which under $Éto1$ has the limit
$$
u_1(t)=fracsin(t)2left(t-fracsin(2t)2right)
$$
which is growing without bound.
As $u_1$ has frequencies $2+É$, $É$ and $|2-É|$, the multiplication with a frequency $2$ term will lead to terms with frequencies $4+É$, $2+É$, $É$, $|2-É|$ and $|4-É|$. Resonance happens when two of these frequencies fall together, which is at $É=2$ and again at $É=1$.
answered Jul 21 at 7:41
LutzL
49.8k31849
add a comment |Â
up vote
0
down vote
up vote
0
down vote
You can rewrite the last formula as
$$
u_1(t)=fracsin(Ét)2left(fracsin((1-É)t)1-É-fracsin((1+É)t)1+Éright)
$$
which under $Éto1$ has the limit
$$
u_1(t)=fracsin(t)2left(t-fracsin(2t)2right)
$$
which is growing without bound.
As $u_1$ has frequencies $2+É$, $É$ and $|2-É|$, the multiplication with a frequency $2$ term will lead to terms with frequencies $4+É$, $2+É$, $É$, $|2-É|$ and $|4-É|$. Resonance happens when two of these frequencies fall together, which is at $É=2$ and again at $É=1$.
answered Jul 21 at 7:41
LutzL
49.8k31849
You can rewrite the last formula as
$$
u_1(t)=fracsin(Ét)2left(fracsin((1-É)t)1-É-fracsin((1+É)t)1+Éright)
$$
which under $Éto1$ has the limit
$$
u_1(t)=fracsin(t)2left(t-fracsin(2t)2right)
$$
which is growing without bound.
As $u_1$ has frequencies $2+É$, $É$ and $|2-É|$, the multiplication with a frequency $2$ term will lead to terms with frequencies $4+É$, $2+É$, $É$, $|2-É|$ and $|4-É|$. Resonance happens when two of these frequencies fall together, which is at $É=2$ and again at $É=1$.
answered Jul 21 at 7:41
LutzL
49.8k31849
answered Jul 21 at 7:41
LutzL
49.8k31849
answered Jul 21 at 7:41
LutzL
49.8k31849
answered Jul 21 at 7:41
LutzL
49.8k31849
49.8k31849
add a comment |Â
add a comment |Â
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At resonance the value of the function goes to infinity. The modes are typically orthogonal functions whose linear combinations add up to $u(t)$. Note sure whether $u_0$ and $u_1$ are orthogonal in this case.
– Biswajit Banerjee
Jul 21 at 4:12