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The study of complexity in world politics began painstakingly in the 1980s on the initiative of authors such as James Rosenau, and looks at different types of complexity such as non-linear interactions, the interaction between actors at different levels (turbulence), the emergence of structures. The chapter intends to analyze the contribution the study of competition processes can provide by means of the theory of dynamical systems. For this purpose, nonlinear equations derived from Richardson's and equations formulated in the framework of population theories concerning crime or terrorism are considered. Finally, the need to move on to the theory of self-organization and emergent structures is indicated.

Dynamical systems.

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"We study dynamics of the horocycle flow on strata of translation surfaces, introduce new invariants for ergodic measures, and analyze the interaction of the horocycle flow and real Rel surgeries. We use this analysis to complete and extend results of Calta and Wortman classifying horocycle-invariant measures in the eigenform loci. In addition we classify the horocycle orbit-closures and prove that every orbit is equidistributed in its orbit-closure. We also prove equidistribution results describing limits of sequences of measures. Our results have applications to the problem of counting closed trajectories on translation surfaces of genus 2"--

Topological dynamics. --- Ergodic theory. --- Random dynamical systems.

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Il presente volume costituisce un trattato di meccanica lagrangiana e hamiltoniana, e completa la rassegna sui sistemi dinamici iniziata nel primo, di cui è la naturale continuazione. Il testo è rivolto a studenti di un corso di laurea triennale in matematica o in fisica, ed è al contempo di potenziale interesse per studenti di un corso di laurea magistrale o di dottorato, nonché per ricercatori intenzionati a lavorare nel campo. Oltre agli argomenti di base, sono infatti affrontati anche argomenti avanzati, per i quali sono comunque forniti gli strumenti matematici utilizzati in modo da rendere la trattazione autocontenuta e accessibile ai meno esperti. I temi discussi sono: formalismo lagrangiano, principi variazionali, metodo di Routh e teorema di Noether, teoria delle piccole oscillazioni, moto dei corpi rigidi pesanti, formalismo hamiltoniano, trasformazioni canoniche, metodo di Hamilton-Jacobi, teoria delle perturbazioni, sistemi quasi-integrabili, studio delle serie perturbative e teorema KAM. Il testo è corredato di un ampio numero di esempi illustrativi, di applicazioni e, alla fine di ogni capitolo, di un'ampia scelta di esercizi, per la maggior parte dei quali è fornita la soluzione. .

Mathematical physics. --- Dynamical systems. --- Mathematical analysis. --- Mathematical Physics. --- Dynamical Systems. --- Analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Physical mathematics --- Mecànica analítica --- Funcions de Lagrange --- Sistemes hamiltonians

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This book offers a complete and detailed introduction to the theory of discrete dynamical systems, with special attention to stability of fixed points and periodic orbits. It provides a solid mathematical background and the essential basic knowledge for further developments such as, for instance, deterministic chaos theory, for which many other references are available (but sometimes, without an exhaustive presentation of preliminary notions). Readers will find a discussion of topics sometimes neglected in the research literature, such as a comparison between different predictions achievable by the discrete time model and the continuous time model of the same application. Another novel aspect of this book is an accurate analysis of the way a fixed point may lose stability, introducing and comparing several notions of instability: simple instability, repulsivity, and complete instability. To help the reader and to show the flexibility and potentiality of the discrete approach to dynamics, many examples, numerical simulations, and figures have been included. The book is used as a reference material for courses at a doctoral or upper undergraduate level in mathematics and theoretical engineering.

Dynamics. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Mathematics. --- Math --- Science

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Dynamics --- Nonlinear theories --- Dynamics. --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics