A Non-Linear Generalization of Singular Value Decomposition and its Applications to Mathematical Modelling and Chaotic Cryptanalysis

# A Non-Linear Generalization of Singular Value Decomposition and its Applications to Mathematical Modelling and Chaotic Cryptanalysis *Tanmay Bankar* $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ *Anurag Tendulkar* *2020A7PS0976G* $~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$ *2020A7PS1010G* ### Motivation SVD is a powerful tool for data analysis in linear algebra, which has a lot of applications in various fields such as signal processing, statistical analysis, biomedical engineering etc. However, it cannot distinguish a chaotic time series from its surrogates; the stochastic counterparts of data which have the same power spectrum. Chaotic Signals are generated by lower dimensional deterministic non-linear dynamical systems. The resemble noise even though an entirely deterministic non-linear system generates them. Noise on the other hand is very stochastic and this calls in for a way to distinguish between the two. ### Problem Definition This paper proposes an extension of SVD for both the qualitative detection and quantitative determination of non-linear time series. It demonstrates an application of non-linear SVD to identify parameters when the signal is generated by nonlinear transformation. ### Methodology The method is to augment the embedding matrix with additional nonlinear columns derived from the initial embedding vectors and extract the nonlinear relationship using SVD. This is done by using the standard Takens Embedding: a method of reconstruction of the state space with time delayed data segments called embedding vectors. This summary won’t delve deep into the mathematics as it would be overwhelming at this stage, but instead discuss the results that have been obtained by this methodology. ### Summary of Results The numerical results for noisy data show that non-linear SVD distinguishes the original data from its surrogates even under the presence of noise, provided that the noise level is low. In mathematical modelling, this can be used to fit a non-linear differential equation. The reliability of this method is reflected in the example given, where using delay embedding of data vectors, we were able to obtain the parameter values of the Van der Pol equation. It is also important to note that this data was noise free. In cryptanalysis, by performing SVD on a data set generated by the Duffing equation, the parameter values obtained were in sync with the actual parameters in the generating equation. ### Discussion and Conclusion This paper shows the utility of non-linear SVD for recovering the non-linear relationship from the time series generated by discrete and continuous dynamical systems. The results of Van der Pol Oscillator and Duffing Oscillator provide us conclusive evidence on how this method works quite well in the presence of noise, given the noise level is low. It also includes two applications of the method; one to Mathematical Modelling and other to Chaotic Cryptanalysis.