Strogatz, Steven H. (Steven Henry). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering / Steven. Strogatz, Steven H. (Steven Henry). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering / Steven H. Strogatz. p. cm. S T U DIES IN NON LINEAR IT Y. NON LINEAR. DYNAMICS. AND CHAOS. With Applications to. Physics, Biology, Chemistry, and Engineering. STEVEN.

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An introductory text in nonlinear dynamics and chaos, emphasizing applications in several areas of science, which include vibrations, Physics Today (PDF). Strogatz, Steven H. (Steven Henry) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering / Steven H. Strogatz. p. cm. Nonlinear Dynamics and Chaos Steven H. Strogatz - Ebook download as PDF File .pdf) or read book online.

Otherwise the system would be nonlinear. This converts the problem to a linear one, which can then be solved easily. But by restricting to small x, we're throwing out some of the physics, like motions where the pendulum whirls over the top. Is it really necessary to make such drastic approximations? It turns out that the pendulum equation can be solved analytically, in terms of elliptic functions. But there ought to be an easier way. After all, the motion of the pendulum is simple: at low energy, it swings back and forth, and at high energy it whirls over the top.

A few years later, May found examples of chaos in iterated mappings arising in population biology, and wrote an influential review article that stressed the pedagogical importance of studying simple nonlinear systems, to counterbalance the often misleading linear intuition fostered by traditional education.

Next came the most surprising discovery of all, due to the physicist Feigenbaum. He discovered that there are certain universal laws governing the transition from regular to chaotic behavior; roughly speaking, completely different systems can go chaotic in the same way.

His work established a link between chaos and 1. Finally, experimentalists such as Gollub, Libchaber, Swinney, Linsay, Moon, and Westervelt tested the new ideas about chaos in experiments on fluids, chemical reactions, electronic circuits, mechanical oscillators, and semiconductors. Although chaos stole the spotlight, there were two other major developments in dynamics in the s. Mandelbrot codified and popularized fractals, produced magnificent computer graphics of them, and showed how they could be applied in a variety of subjects.

And in the emerging area of mathematical biology, Winfree applied the geometric methods of dynamics to biological oscillations, especially circadian roughly hour rhythms and heart rhythms. By the s many people were working on dynamics, with contributions too numerous to list.

Table 1. First we need to introduce some terminology and make some distinctions. Differential equations describe the evolution of systems in continuous time, whereas iterated maps arise in problems where time is discrete. Differential equations are used much more widely in science and engineering, and we shall therefore concentrate on them.

Later in the book we will see that iterated maps can also be very useful, both for providing simple examples of chaos, and also as tools for analyzing periodic or chaotic solutions of differential equations.

Now confining our attention to differential equations, the main distinction is between ordinary and partial differential equations. For instance, the equation for a damped harmonic oscillator 1. That is, there is only one independent variable, the time t. In contrast, the heat equation is a partial differential equation-it has both time t and space x as independent variables. Our concern in this book is with purely temporal behavior, and so we deal with ordinary differential equations almost exclusively.

A very general framework for ordinary differential equations is provided by the system - Here the overdots denote differentiation with respect to t. The variables x, ,. The functions A , Hence the equivalent system 2 is This system is said to be linear, because all the x, on the right-hand side appear to the first power only. Otherwise the system would be nonlinear. Typical nonlinear terms are products, powers, and functions of the x , , such as x,x2 , x, ', or cos X 2.

For example, the swinging of a pendulum is governed by the equation where x is the angle of the pendulum from vertical, g is the acceleration due to gravity, and L is the length of the pendulum.

This converts the problem to a linear one, which can then be solved easily. But by restricting to small x , we're throwing out some of the physics, like motions where the pendulum whirls over the top. Is it really necessary to make such drastic approximations? It turns out that the pendulum equation can be solved analytically, in terms of elliptic functions.

But there ought to be an easier way. After all, the motion of the pendulum is simple: at low energy, it swings back and forth, and at high energy it whirls over the top. There should be some way of extracting this information from the system directly. This is the sort of problem we'll learn how to solve, using geometric methods.

Here's the rough idea. Suppose we happen to know a solution to the pendulum system, for a particular initial condition. This solution would be a pair of functions x, t and x, t , representing the position and velocity of the pendulum.

Figure 1. The phase space is completely filled with trajectories, since each point can serve as an initial condition. Our goal is to run this construction in reverse: given the system, we want to 1. In many cases, geometric reasoning will allow us to draw the trajectories without actually solving the system!

Differential equations are used much more widely in science and engineering, and we shall therefore concentrate on them. Later in the book we will see that iterated maps can also be very useful, both for providing sim- ple examples of chaos, and also as tools for analyzing periodic or chaotic solutions of differential equations. Now confining our attention to differential equations, the main distinction is be- tween ordinary and partial differential equations.

For instance, the equation for a damped harmonic oscillator 1. That is, there is only one independent variable, the time t. In contrast, the heat equation is a partial differential equation-it has both time t and space x as independent variables.

Our concern in this book is with purely temporal behavior, and so we deal with ordinary differential equations almost exclusively. A very general framework for ordinary differential equations is provided by the system - Here the overdots denote differentiation with respect to t.

The variables x, ,. The functions A , Hence the equivalent system 2 is This system is said to be linear, because all the x, on the right-hand side appear to the first power only. Otherwise the system would be nonlinear. Typical nonlin- ear terms are products, powers, and functions of the x , , such as x,x2 , x, ', or cos X 2. For example, the swinging of a pendulum is governed by the equation where x is the angle of the pendulum from vertical, g is the acceleration due to gravity, and L is the length of the pendulum.

This converts the problem to a linear one, which can then be solved easily. But by restricting to small x , we're throwing out some of the physics, like motions where the pendulum whirls over the top.

Is it really necessary to make such drastic approximations? It turns out that the pendulum equation can be solved analytically, in terms of elliptic functions. But there ought to be an easier way. After all, the motion of the pendulum is simple: at low energy, it swings back and forth, and at high energy it whirls over the top. There should be some way of extracting this information from the system directly.

Why are nonlinear systems so much harder to analyze than linear ones? The essential difference is that linear systems can be broken down into parts. This idea allows a fantastic simplification of complex problems, and underlies such meth- ods as normal modes, Laplace transforms, superposition arguments, and Fourier analysis. In this sense, a linear system is precisely equal to the sum of its parts. But many things in nature don't act this way.

Whenever parts of a system inter- fere, or cooperate, or compete, there are nonlinear interactions going on. Most of everyday life is nonlinear, and the principle of superposition fails spectacularly. If you listen to your two favorite songs at the same time, you won't get double the plea- sure!

Within the realm of physics, nonlinearity is vital to the operation of a laser, the formation of turbulence in a fluid, and the superconductivity of Josephson junctions.

Our goal is to show the log- ical structure of the entire subject. The framework presented in Figure 1.

The framework has two axes. One axis tells us the number of variables needed to characterize the state of the system. Equivalently, this number is the dimension of the phase space. The other axis tells us whether the system is linear or nonlin- ear. For example, consider the exponential growth of a population of organisms.