Sajini Anand | National Institute of Advanced Studies (original) (raw)
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Papers by Sajini Anand
In this paper we propose a method of using nonlinear generalization of Singular Value Decompositi... more In this paper we propose a method of using nonlinear
generalization of Singular Value Decomposition (SVD) to arrive at an upper bound for the dimension of a manifold which is embedded in some RN. We have assumed that the data about its co-ordinates
is available. We would also assume that there exists at least one
small neighborhood with sufficient number of data points. Given
these conditions, we show a method to compute the dimension of a manifold. We begin by looking at the simple case when the manifold is in the form of a lower dimensional affine subspace. In this case, we show that the well known technique of SVD can be used to (i) calculate the dimension of the manifold and (ii) to get
the equations which define the subspace. For the more general case,
we have applied a nonlinear generalization of the SVD (i) to search
for an upper bound for the dimension of the manifold and (ii) to find the equations for the local charts of the manifold. We have included a brief discussion about how this method would be highly useful in the context of the Takens’ embedding which is used in the analysis of a time series data from a dynamical system. We show
a specific problem that has recently been found out when applying this method. One very effective solution is to develop a model which is based on local charts and for this purpose a good estimate of the underlying dimension of an embedded data is required.
Synchronization of two identical chaotic systems which starts with different initial conditions, ... more Synchronization of two identical chaotic systems which starts with different initial conditions, by sending a part of state space to other in a continuous fashion is a well established procedure. This paper discusses synchronization by intermittent driving signals from a part of a system to the other system.
Acta applicandae mathematicae, Jan 1, 2010
Unfortunately a few errors were left uncorrected during the proof process of this paper. They are... more Unfortunately a few errors were left uncorrected during the proof process of this paper. They are listed here:
Acta applicandae mathematicae, Jan 1, 2010
Singular Value Decomposition (SVD) is a powerful tool in linear algebra and has been extensively ... more Singular Value Decomposition (SVD) is a powerful tool in linear algebra and has been extensively applied to Signal Processing, Statistical Analysis and Mathematical Modeling. We propose an extension of SVD for both the qualitative detection and quantitative determination of nonlinearity in a time series. 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. The paper demonstrates an application of nonlinear SVD to identify parameters when the signal is generated by a nonlinear transformation. Examples of maps (Logistic map and Henon map) and flows (Van der Pol oscillator and Duffing oscillator) are used to illustrate the method of nonlinear SVD to identify parameters. The paper presents the recovery of parameters in the following scenarios: (i) data generated by maps and flows, (ii) comparison of the method for both noisy and noise-free data, (iii) surrogate data analysis for both the noisy and noise-free cases. The paper includes two applications of the method: (i) Mathematical Modeling and (ii) Chaotic Cryptanalysis.
Citeseer
This paper discusses cryptography based on the property of chaotic synchronization. Specifically,... more This paper discusses cryptography based on the property of chaotic synchronization. Specifically, it is about Round III of such a cryptographic method. Round I showed the feasibility of using chaotic synchronization for cryptography. Round II consisted of a method to counter attack. This paper is Round III and shows how to counter the counter attacks. First, we show numerical evidence that synchronization is possible between two Lorenz systems if one system sends information about x0 at a slower rate. The second system evolves on its own, except that when it receives a signal from the first system, it replaces its own value of y0 by the received x0. We have found that the two systems eventually synchronize, but often after a long time. Therefore, we have devised a technique to speed-up this synchronization. Once this is done, it is possible for the authorized receiver (with the possession of the initial super-key) to keep synchronizing with slowly sampled inputs, whereas the known methods of Round II do not help an eavesdropper.
In this paper we propose a method of using nonlinear generalization of Singular Value Decompositi... more In this paper we propose a method of using nonlinear
generalization of Singular Value Decomposition (SVD) to arrive at an upper bound for the dimension of a manifold which is embedded in some RN. We have assumed that the data about its co-ordinates
is available. We would also assume that there exists at least one
small neighborhood with sufficient number of data points. Given
these conditions, we show a method to compute the dimension of a manifold. We begin by looking at the simple case when the manifold is in the form of a lower dimensional affine subspace. In this case, we show that the well known technique of SVD can be used to (i) calculate the dimension of the manifold and (ii) to get
the equations which define the subspace. For the more general case,
we have applied a nonlinear generalization of the SVD (i) to search
for an upper bound for the dimension of the manifold and (ii) to find the equations for the local charts of the manifold. We have included a brief discussion about how this method would be highly useful in the context of the Takens’ embedding which is used in the analysis of a time series data from a dynamical system. We show
a specific problem that has recently been found out when applying this method. One very effective solution is to develop a model which is based on local charts and for this purpose a good estimate of the underlying dimension of an embedded data is required.
Synchronization of two identical chaotic systems which starts with different initial conditions, ... more Synchronization of two identical chaotic systems which starts with different initial conditions, by sending a part of state space to other in a continuous fashion is a well established procedure. This paper discusses synchronization by intermittent driving signals from a part of a system to the other system.
Acta applicandae mathematicae, Jan 1, 2010
Unfortunately a few errors were left uncorrected during the proof process of this paper. They are... more Unfortunately a few errors were left uncorrected during the proof process of this paper. They are listed here:
Acta applicandae mathematicae, Jan 1, 2010
Singular Value Decomposition (SVD) is a powerful tool in linear algebra and has been extensively ... more Singular Value Decomposition (SVD) is a powerful tool in linear algebra and has been extensively applied to Signal Processing, Statistical Analysis and Mathematical Modeling. We propose an extension of SVD for both the qualitative detection and quantitative determination of nonlinearity in a time series. 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. The paper demonstrates an application of nonlinear SVD to identify parameters when the signal is generated by a nonlinear transformation. Examples of maps (Logistic map and Henon map) and flows (Van der Pol oscillator and Duffing oscillator) are used to illustrate the method of nonlinear SVD to identify parameters. The paper presents the recovery of parameters in the following scenarios: (i) data generated by maps and flows, (ii) comparison of the method for both noisy and noise-free data, (iii) surrogate data analysis for both the noisy and noise-free cases. The paper includes two applications of the method: (i) Mathematical Modeling and (ii) Chaotic Cryptanalysis.
Citeseer
This paper discusses cryptography based on the property of chaotic synchronization. Specifically,... more This paper discusses cryptography based on the property of chaotic synchronization. Specifically, it is about Round III of such a cryptographic method. Round I showed the feasibility of using chaotic synchronization for cryptography. Round II consisted of a method to counter attack. This paper is Round III and shows how to counter the counter attacks. First, we show numerical evidence that synchronization is possible between two Lorenz systems if one system sends information about x0 at a slower rate. The second system evolves on its own, except that when it receives a signal from the first system, it replaces its own value of y0 by the received x0. We have found that the two systems eventually synchronize, but often after a long time. Therefore, we have devised a technique to speed-up this synchronization. Once this is done, it is possible for the authorized receiver (with the possession of the initial super-key) to keep synchronizing with slowly sampled inputs, whereas the known methods of Round II do not help an eavesdropper.