This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.

Regular and Chaotic Dynamics

In this paper, we propose a novel image encryption algorithm based on a hybrid model of deoxyribonucleic acid (DNA) masking, a Secure Hash Algorithm SHA-2 and the Lorenz system. Our study uses DNA sequences and operations and the chaotic Lorenz system to strengthen the cryptosystem. The significant advantages of this approach are improving the information entropy which is the most important feature of randomness, resisting against various typical attacks and getting good experimental results. The theoretical analysis and experimental results show that the algorithm improves the encoding efficiency, enhances the security of the ciphertext and has a large key space and a high key sensitivity, and it is able to resist against the statistical and exhaustive attacks.

A novel chaos-based image encryption using DNA sequence operation and Secure Hash Algorithm SHA-2

Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control

We present a survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them. After some historical notes on the first attempts for the numerical evaluation of LCEs, we discuss in detail the multiplicative ergodic theorem of Oseledec (102), which pro- vides the theoretical basis for the computation of the LCEs. Then, we analyze the algorithm for the computation of the maximal LCE, whose value has been exten- sively used as an indicator of chaos, and the algorithm of the so-called standard method, developed by Benettin et al. (14), for the computation of many LCEs. We also consider different discrete and continuous methods for computing the LCEs based on the QR or the singular value decomposition techniques. Although we are mainly interested in finite-dimensional conservative systems, i.e., autonomous Hamiltonian systems and symplectic maps, we also briefly refer to the evaluation of LCEs of dissipative systems and time series. The relation of two chaos detection techniques, namely the fast Lyapunov indicator (FLI) and the generalized alignment index (GALI), to the computation of the LCEs is also discussed.

https://www.springer.com/cda/content/document/cda_downloaddocument/9783642044571-c1.pdf?SGWID=0-0-45-855958-p173923511

The Lyapunov Characteristic Exponents and Their Computation

We present a survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them. After some historical notes on the first attempts for the numerical evaluation of LCEs, we discuss in detail the multiplicative ergodic theorem of Oseledec \cite{O_68}, which provides the theoretical basis for the computation of the LCEs. Then, we analyze the algorithm for the computation of the maximal LCE, whose value has been extensively used as an indicator of chaos, and the algorithm of the so--called `standard method', developed by Benettin et al. \cite{BGGS_80b}, for the computation of many LCEs. We also consider different discrete and continuous methods for computing the LCEs based on the QR or the singular value decomposition techniques. Although, we are mainly interested in finite--dimensional conservative systems, i. e. autonomous Hamiltonian systems and symplectic maps, we also briefly refer to the evaluation of LCEs of dissipative systems and time series. The relation of two chaos detection techniques, namely the fast Lyapunov indicator (FLI) and the generalized alignment index (GALI), to the computation of the LCEs is also discussed.

The Lyapunov Characteristic Exponents and their computation

A comparative study of computation of Lyapunov spectra with different algorithms

The article published below draws attention to the work of the Kerala school of astronomers, particularly Nilakanta (/500 AD) in modelling planetary motion. In the exchanges between authors and referee, it became clear that this school did not stop with :opying their predecessors but attempted to wrestle with the problems of the old (geocentric) system. Whether their work constituted a clean break towards a true heliocentric system. as proposed by Srinivas and colleagues. appears to hinge upon some subtle points of interpretation of the original texts. For example, did the Kerala astronomers maintain the distinction between the mean and the centre of the epicycle of an interior planet. even though both move together in the sky? They could be at different distances, as a referee suggests. In any case. one cannot but note the vitality of this tradition of mathematics and astronomy which even studied infinite series some years later. while the rest of the cOl/ntry was going through an academic dark age.

Modification of the earlier indian planetary theory by the kerala astronomers (c. 1500 ad) and the implied heliocentric picture of planetary motion

Recently a geometric description of chaos in Hamiltonian systems has been formulated using the tools of Riemannian geometry. Here, Hamiltonian chaos is explained in terms of the curvature properties of the configuration space manifold. In particular, it has been claimed that the average of an appropriately defined sectional curvature ${(K}^{(2)})$ over a constant energy manifold is a measure of the global extent of chaoticity for systems with a small number of degrees of freedom. We investigate the relations between this quantity ${K}^{(2)}$ and the maximal Lyapunov exponent $\ensuremath{\lambda}$ for some Hamiltonian systems of physical interest with two degrees of freedom. We find that there is a close relation between ${K}^{(2)}$ and ${\ensuremath{\lambda}}^{2}.$ Both the quantities scale as ${E}^{1/2}$ for quartic potentials, where E is the energy. They are expected to scale as ${E}^{(n\ensuremath{-}2)/n}$ for a general potential of degree n. However, we find that though ${K}^{(2)}$ is a global indicator of chaos, it is not a sufficiently accurate measure of order-chaos transitions, in all cases.

Global geometric indicator of chaos and Lyapunov exponents in Hamiltonian systems.

Mean longitudes of planets.- True longitudes of planets.- Gnomonic shadow.- Lunar eclipse.- Solar eclipse.- Vyatipata.- Reduction to observation.- Elevation of lunar horns.- Appendices.

Tantrasaṅgraha of Nīlakaṇṭha Somayājī

Here the concept of "space" is discussed from two different streams of thought: (1) the view held by Advaita Vedānta, as expounded by Śaṇkara, and (2) the view that emerges from the ongoing debates in modern philosophy of physics. The emphasis is on addressing the following question: is space created or not? To set the necessary backdrop for a better appreciation of the debate that evolved within the Indian tradition, we first examine how the Vaiśeṣika and Sāṃkhya schools of thought unfold the concept of ākāśa in relation to their metaphysics. We then carefully analyze a section of Śaṇkara's Brahmasūtrabhāṣya (BSB II.3.1–7), wherein the creation of ākāśa is discussed at great length. The second part of this essay attempts to highlight the ongoing struggle among physicists to arrive at an understanding of the nature and origin of space, with reference to the general theory of relativity and quantum field theory. It ends with a critical reflection on parallels and differences that come forth from the study of Advaita Vedānta and modern physics.

K. Ramasubramanian

Papers

A comparative study of computation of Lyapunov spectra with different algorithms

Modification of the earlier indian planetary theory by the kerala astronomers (c. 1500 ad) and the implied heliocentric picture of planetary motion

Global geometric indicator of chaos and Lyapunov exponents in Hamiltonian systems.

Tantrasaṅgraha of Nīlakaṇṭha Somayājī

Is Space Created? Reflections on Śaṇkara's Philosophy and Philosophy of Physics