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The central theme of this book is the extent to which the structure of the free dynamical boundaries of a system controls the evolution of the system as a whole. Applying three orthogonal types of thinking - mathematical, constructivist and morphological, it illustrates these concepts using applications to selected problems from the social and life sciences, as well as economics.  In a broader context, it introduces and reviews some modern mathematical approaches to the science of complex systems. Standard modeling approaches (based on non-linear differential equations, dynamic systems, graph theory, cellular automata, stochastic processes, or information theory) are suitable for studying local problems. However they cannot simultaneously take into account all the different facets and phenomena of a complex system, and new approaches are required to solve the challenging problem of correlations between phenomena at different levels and hierarchies, their self-organization and memory-evolutive aspects, the growth of additional structures and are ultimately required to explain why and how such complex systems can display both robustness and flexibility.   This graduate-level text also addresses a broader interdisciplinary audience, keeping the mathematical level essentially uniform throughout the book, and involving only basic elements from calculus, algebra, geometry and systems theory.  .

Ecology. --- Atomic Physics --- Physics --- Physical Sciences & Mathematics --- Chaotic behavior in systems. --- Chaos in systems --- Chaos theory --- Chaotic motion in systems --- Differentiable dynamical systems --- Dynamics --- Nonlinear theories --- System theory --- Engineering. --- Biological models. --- Applications of Nonlinear Dynamics and Chaos Theory. --- Complex Systems. --- Complexity. --- Systems Biology. --- Theoretical Ecology/Statistics. --- Earth System Sciences. --- Balance of nature --- Biology --- Bionomics --- Ecological processes --- Ecological science --- Ecological sciences --- Environment --- Environmental biology --- Oecology --- Environmental sciences --- Population biology --- Models, Biological --- Construction --- Industrial arts --- Technology --- Ecology --- Statistical physics. --- System theory. --- Computational complexity. --- Systems biology. --- Ecology . --- Physical geography. --- Geography --- Computational biology --- Bioinformatics --- Biological systems --- Molecular biology --- Complexity, Computational --- Electronic data processing --- Machine theory --- Systems, Theory of --- Systems science --- Science --- Mathematical statistics --- Philosophy --- Statistical methods

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Nonlinear science is the science of, among other exotic phenomena, unexpected and unpredictable behavior, catastrophes, complex interactions, and significant perturbations. Ocean and atmosphere dynamics, weather, many bodies in interaction, ultra-high intensity excitations, life, formation of natural patterns, and coupled interactions between components or different scales are only a few examples of systems where nonlinear science is necessary. All outstanding, self-sustained, and stable structures in space and time exist and protrude out of a regular linear background of states mainly because they identify themselves from the rest by being highly localized in range, time, configuration, states, and phase spaces. Guessing how high up you drive toward the top of the mountain by compiling your speed, road slope, and trip duration is a linear model, but predicting the occurrence around a turn of a boulder fallen on the road is a nonlinear phenomenon. In an effort to grasp and understand nonlinear phenomena, scientists have developed several mathematical approaches including inverse scattering theory, Backlund and groups of transformations, bilinear method, and several other detailed technical procedures. In this Special Issue, we introduce a few very recent approaches together with their physical meaning and applications. We present here five important papers on waves, unsteady flows, phases separation, ocean dynamics, nonlinear optic, viral dynamics, and the self-appearance of patterns for spatially extended systems, which are problems that have aroused scientists’ interest for decades, yet still cannot be predicted and have their generating mechanism and stability open to debate. The aim of this Special Issue was to present these most debated and interesting topics from nonlinear science for which, despite the existence of highly developed mathematical tools of investigation, there are still fundamental open questions.

diffusion --- viral infection --- non-Newtonian fluid --- convergence --- Navier–Stokes–Voigt equations --- existence --- Lyapunov functional --- Faedo–Galerkin approximations --- probability distribution --- strong solutions --- stability --- multigrid method --- parabolic equations --- long-time behavior --- Fokker–Planck equation --- viscoelastic models --- Cauchy problem --- unconditionally gradient stable scheme --- uniqueness --- existence and uniqueness theorem --- continuum spectrum pulse equation --- Stokes operator --- Lagrangian scheme --- Cahn–Hilliard equation --- Feller equation

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Compact spaces. --- Spaces, Compact --- Topological spaces

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Differential geometry. Global analysis --- Mathematical physics --- Gases handling. Fluids handling --- Computer science --- differentiaal geometrie --- informatica --- wiskunde --- fysica --- vloeistoffen

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The central theme of this book is the extent to which the structure of the free dynamical boundaries of a system controls the evolution of the system as a whole. Applying three orthogonal types of thinking - mathematical, constructivist and morphological, it illustrates these concepts using applications to selected problems from the social and life sciences, as well as economics. In a broader context, it introduces and reviews some modern mathematical approaches to the science of complex systems. Standard modeling approaches (based on non-linear differential equations, dynamic systems, graph theory, cellular automata, stochastic processes, or information theory) are suitable for studying local problems. However they cannot simultaneously take into account all the different facets and phenomena of a complex system, and new approaches are required to solve the challenging problem of correlations between phenomena at different levels and hierarchies, their self-organization and memory-evolutive aspects, the growth of additional structures and are ultimately required to explain why and how such complex systems can display both robustness and flexibility. This graduate-level text also addresses a broader interdisciplinary audience, keeping the mathematical level essentially uniform throughout the book, and involving only basic elements from calculus, algebra, geometry and systems theory. .

Discrete mathematics --- Numerical analysis --- Classical mechanics. Field theory --- Statistical physics --- Physics --- Geology. Earth sciences --- Biomathematics. Biometry. Biostatistics --- General ecology and biosociology --- Biology --- Biotechnology --- Computer science --- Computer. Automation --- Physical geography --- grafentheorie --- biologie --- computers --- informatica --- statistiek --- biotechnologie --- ecologie --- systeemtheorie --- biometrie --- geologie --- fysica --- fysische geografie --- aarde (astronomie) --- dynamica