Introduction to Modern Dynamics: Chaos, Networks, Space and Time
David D. Nolte
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The best parts of physics are the last topics that our students ever see. These are the exciting new frontiers of nonlinear and complex systems that are at the forefront of university research and are the basis of many high-tech businesses. Topics such as traffic on the World Wide Web, the spread of epidemics through globally-mobile populations, or the synchronization of global economies are governed by universal principles just as profound as Newton's laws. Nonetheless, the conventional university physics curriculum reserves most of these topics for advanced graduate study. Two justifications are given for this situation: first, that the mathematical tools needed to understand these topics are beyond the skill set of undergraduate students, and second, that these are specialty topics with no common theme and little overlap.
Introduction to Modern Dynamics dispels these myths. The structure of this book combines the three main topics of modern dynamics - chaos theory, dynamics on complex networks, and general relativity - into a coherent framework. By taking a geometric view of physics, concentrating on the time evolution of physical systems as trajectories through abstract spaces, these topics share a common and simple mathematical language through which any student can gain a unified physical intuition. Given the growing importance of complex dynamical systems in many areas of science and technology, this text provides students with an up-to-date foundation for their future careers.
the conservation of phase-space volume along a trajectory, known as Liouville’s theorem. One of the most valuable features of Hamiltonian dynamics is the ability to find transformations, known as canonical transformations, of dynamical variables into new variables that simplify the description of the dynamical system. Action–angle variables on multidimensional tori are among the most important of these new variables, and are the starting point for understanding nonlinear Hamiltonian dynamics that
and to the finite rate of the sweeping control parameter. The Lyapunov exponent, which is the relaxation rate, approaches zero at the threshold, and there is not enough time for the system to relax as the control parameter is changed through successive values. Once the system is far enough from the threshold, the oscillation amplitude follows the square-root prediction. An interesting thing happens to the system dynamics when the sign of the r 2 term in Eq. (3.53) is flipped to (c + r 2 ). The
orbits could display chaotic behavior. continued 118 Introduction to Modern Dynamics Example 3.5 continued 4 4 α = 0.6 Unstable manifold 3 Stable manifold Saddle 2 α = 0.9 Unstable manifold 3 Stable manifold 2 Saddle 1 y y 1 Unstable fixed point 0 –1 0 Limit cycle –1 –2 –3 –2 –2 –1 0 1 2 –3 –2 3 –1 0 x 1 2 3 x 4 α = 0.75 Unstable manifold 3 2 Saddle Stable manifold y 1 Unstable f ixed point 0 –1 Homoclinic orbit –2 –3 –2 –1 0 1 2 3 x Figure
that attract all nearby trajectories and to which all nearby trajectories relax exponentially. When the dimension of the state space expands to 3D, there can still be stable nodes and limit cycles, but in addition there now are attracting sets imbedded in the state space. These sets tend to be diffuse collections of points with varying density, including some regions where no points occur at all. These sets are embedded in the dynamical space, but do not fill it, with a measured volume 131 132
networks lies at the boundary between two worlds. On the one hand, there is the fully holistic viewpoint in which complex systems of many variables and high dimension cannot be separated into subunits without destroying the essential behavior that is to be understood. On the other hand, there is the reductionist viewpoint in which each component of the system is isolated and studied individually. Reality often lies between, in which individual systems and their properties are understood in