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Complex Eigenvalues of system of DE
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- Complex Differential equations
In summary, complex eigenvalues occur in systems of differential equations when the characteristic equation has non-real roots. These eigenvalues typically indicate oscillatory behavior in the system's solutions. When eigenvalues are complex conjugates, the corresponding eigenvectors lead to solutions that exhibit sinusoidal functions, often resulting in a combination of exponential growth or decay with oscillations. Understanding complex eigenvalues is crucial for analyzing stability and dynamic behavior in various applications, including engineering and physics.
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Just perform the multiplications and collect real and imaginary terms in their own matrices.
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renormalize
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The yellow expression contains a sign error; the second term should be:$$ChiralSuperfields said:
i\left(\begin{array}{c}
-e^{-2t}\sin3t\\
+e^{-2t}\cos3t
\end{array}\right)
$$
- #4
member 731016
Thank you for your replies @Orodruin and @renormalize ! Yeah that was part of my confusion is that there is a typo
It makes sense now that I know it is a typo!
Thanks!
It makes sense now that I know it is a typo!
Thanks!
FAQ: Complex Eigenvalues of system of DE
What are complex eigenvalues in the context of differential equations?
Complex eigenvalues arise when analyzing systems of differential equations, particularly when the characteristic polynomial has roots that are not real numbers. These eigenvalues are typically associated with oscillatory behavior in the solutions of the system, leading to solutions that can exhibit exponential growth or decay combined with sinusoidal oscillations.
How do complex eigenvalues affect the stability of a system?
The presence of complex eigenvalues indicates that the system may exhibit oscillatory behavior. If the real part of the complex eigenvalues is positive, the system is unstable, as solutions will grow unbounded over time. Conversely, if the real part is negative, the system is stable, and solutions will decay to zero while oscillating. If the real part is zero, the solutions will oscillate indefinitely without growing or decaying.
Can a system of differential equations have only complex eigenvalues?
Yes, a system of differential equations can have only complex eigenvalues, particularly when the characteristic polynomial has no real roots. This situation often occurs in systems with certain symmetries or specific parameters that lead to purely imaginary eigenvalues, resulting in solutions that oscillate without exponential growth or decay.
How do you find the eigenvalues of a system of differential equations?
To find the eigenvalues of a system of differential equations, you typically start by writing the system in matrix form. You then compute the characteristic polynomial by taking the determinant of the matrix minus a scalar multiple of the identity matrix. Setting this determinant equal to zero allows you to solve for the eigenvalues, which may be real or complex numbers.
What is the significance of complex eigenvectors in relation to complex eigenvalues?
Complex eigenvectors correspond to complex eigenvalues and provide insight into the behavior of the system's solutions. Each complex eigenvalue typically has a complex conjugate eigenvalue, and their associated eigenvectors also come in conjugate pairs. These eigenvectors help to construct the general solution of the system, which will typically involve exponential and oscillatory terms derived from the eigenvalues and eigenvectors.