## Chaos, Nonlinearity, Complexity: The Dynamical Paradigm of NatureI think the next century will be the century of complexity. We have already discovered the basic laws that govern matter and understand all the normal situations. We don’t know how the laws ?t together, and what happens under extreme conditions. But I expect we will ?nd a complete uni?ed theory sometime this century. There is no limit to the complexity that we can build using those basic laws. Stephen Hawking, January 2000. We don’t know what we are talking about. Many of us believed that string theory was a very dramatic break with our previous notions of quantum theory. But now we learn that string theory, well, is not that much of a break. The state of physics today is like it was when we were mysti?ed by radioactivity. They were missing something absolutely fundamental. We are missing perhaps something as profound as they were back then. Nobel Laureate David Gross, December 2005. This volume is essentially a compilation of papers presented at the Int- national Workshop on Mathematics and Physics of Complex and Nonlinear Systems that was held at Indian Institute of Technology Kanpur, March 14 – 26, 2004 on the theme ChaNoXity: The Nonlinear Dynamics of Nature. |

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### Contents

Chaos Periodicity and Complexity on Dynamical Systems | 1 |

Foundations of Nonextensive Statistical Mechanics | 53 |

Critical Attractors and the Physical Realm of qstatistics | 72 |

NonBoltzmannian Entropies for Complex Classical Systems Quantum Coherent States and Black Holes | 114 |

Power Law and Tsallis Entropy Network Traffic and Applications | 162 |

The Role of Chaos and Resonances in Brownian Motion | 179 |

### Other editions - View all

Chaos, Nonlinearity, Complexity: The Dynamical Paradigm of Nature Ashok Sengupta Limited preview - 2006 |

### Common terms and phrases

behavior bifurcation black hole Boltzmann canonical canonical ensemble chaos chaotic coeﬃcient complex system conﬁgurations consider constraint convergence corresponding critical attractors deﬁned deﬁnition diﬀerential dynamical system eﬀect emergence energy ensemble equilibrium ergodic event horizon evolution ﬁeld ﬁnd ﬁnite ﬁrst ﬁxed point ﬂuctuations function Gibbs Hamiltonian increasing inﬁnite initial conditions interaction inverse irreversible isolated horizon iterates lattice leadership Lett limit limsup linear logistic map Lyapunov Lyapunov exponent Math measure preserving transformation metric entropy microcanonical microstates neighbourhood system nonlinear obtained orbits organizational organizations parameter particle periodic structure phase space Phys physical power-law processes properties Proposition q-exponential q-statistics quantum Renyi distribution Renyi entropy resonance result self-organization signiﬁcant spacetime speciﬁc stable statistical mechanics subsets subsystem temperature Theorem thermal thermodynamic thermostatistics tion topological entropy topology traﬃc trajectories transition Tsallis entropy variables