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yourquantummechanicguynextdoor
yourquantummechanicguynextdoor
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Consider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions X = (x_1, x_2, ..., x_N)$X = (x_1, x_2, ..., x_N)$, we have

X'' = A*X$X'' = A*X$

How can this system become chaotic by introducing an extra term? For instance, would it become chaotic by adding a higher-order term like f_{(x_1)}$f_{(x_1)}$ to the equation of the first oscillator, such as x_1^2$x_1^2$ or x_1^4$x_1^4$? Or, do I need to add nonlinear coupling?

Consider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions X = (x_1, x_2, ..., x_N), we have

X'' = A*X

How can this system become chaotic by introducing an extra term? For instance, would it become chaotic by adding a higher-order term like f_{(x_1)} to the equation of the first oscillator, such as x_1^2 or x_1^4? Or, do I need to add nonlinear coupling?

Consider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions $X = (x_1, x_2, ..., x_N)$, we have

$X'' = A*X$

How can this system become chaotic by introducing an extra term? For instance, would it become chaotic by adding a higher-order term like $f_{(x_1)}$ to the equation of the first oscillator, such as $x_1^2$ or $x_1^4$? Or, do I need to add nonlinear coupling?

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yourquantummechanicguynextdoor
yourquantummechanicguynextdoor
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AssumeConsider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions X = (x_1, x_2, ..., x_N), we have

X'' = A*X

How can I make this system become chaotic by introducing an extra term? For instance, would it become chaotic by adding a higher-order term like f_{(x_1)} to the equation of the first oscillator, such as x_1^2 or x_1^4? Or, do I need to add nonlinear coupling?

Assume a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions X = (x_1, x_2, ..., x_N), we have

X'' = A*X

How can I make this system chaotic by introducing an extra term? For instance, would it become chaotic by adding a term like f_{(x_1)} to the equation of the first oscillator, such as x_1^2 or x_1^4? Or, do I need to add nonlinear coupling?

Consider a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions X = (x_1, x_2, ..., x_N), we have

X'' = A*X

How can this system become chaotic by introducing an extra term? For instance, would it become chaotic by adding a higher-order term like f_{(x_1)} to the equation of the first oscillator, such as x_1^2 or x_1^4? Or, do I need to add nonlinear coupling?

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yourquantummechanicguynextdoor
yourquantummechanicguynextdoor
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Making a system of second-order ODEs chaotic

Assume a system of N linear 2nd-order OEDs, describing a system of coupled one-dimensional harmonic oscillators, with couplings given by matrix A and positions X = (x_1, x_2, ..., x_N), we have

X'' = A*X

How can I make this system chaotic by introducing an extra term? For instance, would it become chaotic by adding a term like f_{(x_1)} to the equation of the first oscillator, such as x_1^2 or x_1^4? Or, do I need to add nonlinear coupling?