It would appear that you have a double mode, or a weakly coupled mode pair.

This occurs when two parts of a resonant structure or circuit are weakly coupled, and each happens to support a mode at the same time.

Then there are two different solutions to the same equations for the resonant structure, causing two singular values to go to zero, and giving two different singular vectors (or eigenfunctions),

Recall that the singular vectors -- which are the mode functions for your two modes -- must be orthogonal to one another. The theory of SVD requires this. This is why your modes should have a phase reversal somewhere. It makes the two mode functions orthogonal to each other.

Weakly coupled mode pairs can have interesting properties. And in principle you could have any number of coincident modes in weakly coupled structures, but it would be unusual to have more than two.

I came across mode pairs with two singular values zeroed in seismic modes in layered media. See Section 5.3.3 in:

Thesis Scattering of Elastic Waves in Layered Media: A boundary int...

Ronald

Ronald Kessel Thank you very much for the answer! I will read you thesis about it. But for the artificial test signal matrix that I generated in the code for benchmarking. Only one mode with fixed frequency and phase are set. I don't understand how could my test signal contributes to 2 coupled modes.

sig(t, i) = cos(2*pi*f*t + n*2*pi*i/N)

Where i is the index of the artificial probes, N is the total number of the probes. n is the toroidal/poloidal mode number.

And simulation results suggest that only the mode with n=0 will get a single singular value, the other modes with n>0 will get 2 singular values.

If I have well understood, you applied the SVD to a simulated signal in order to find the indipendent components (the technique is something similar to SSA https://en.wikipedia.org/wiki/Singular_spectrum_analysis). Your signal is a combination of multiple cosines with different time frequency (f) and spatial frequency (n, the number of mode) and noise. The "i" simulates the equally spaced positions of each probe.

For each mode with spatial frequency (n>0) you have 2 independent modes with 90/n degrees spatial phase difference (but the same time frequency). To represent phase difference you have to consider the immaginary part of the complex solution.

From a physical point of view the solutions with same frequency have to be summed.

This technique like a Fourier transform produce 2 values (amplitude and phase or the amplitude of sine and cosine) for each indipendent mode (in both cases this mode are sinusoidal function). The differences between SSA (based on SVD) and Fourier is that SSA does not make assumption on the shape (or waveform) of the independent components. But your input signal is a sum of cosine (exactly like a Fourier sum), thus the result are the same. If you use another orthogonal functions to generate data, the results will be different. I hope that the comparison with the Fourier decomposition is useful.

Anyway the SVD find an alternative solution. This solution has independent components.

A further question could be this: if SVD autoamtically compute the modes, why does the SVD "choose" 2 orthogonals mode (sine and cosine) with same frequency and half amplitude instead of one with a doubled amplitude?

I believed that it is due to the constraint of minimization of the norm of the matrix of the eigenvalues (or eigemode).

Francesco Barone Thank you for the suggestion. Your understanding to the question is correct. I will check about it.

Francesco Barone You are right. I have refined my code by manually combining together the two duplicated harmonics with the same frequency. From the attached picture you will see now that even the toroidal/poloidal propagation of the n = 3 and n = 2 modes can be observed clearly. This problem maybe due to that SVD has decomposed the n = 3 rotational mode into two orthogonal standing wave harmonics.

Thank you very much!

Nan