7: Nonlinear Systems
- Page ID
- 89126
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"The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful." - Jules Henri Poincaré (1854-1912)
- 7.1: Introduction
- SOME OF THE MOST INTERESTING PHENOMENA in the world are modeled by nonlinear systems. These systems can be modeled by differential equations when time is considered as a continuous variable or difference equations when time is treated in discrete steps.
- 7.2: The Logistic Equation
- In this section we will explore a simple nonlinear population model. Typically, we want to model the growth of a given population, y(t), and the differential equation governing the growth behavior of this population is developed in a manner similar to that used previously for mixing problems.
- 7.3: Autonomous First Order Equations
- In this section we will study the stability of nonlinear first order autonomous equations. We will then extend this study in the next section to looking at families of first order equations which are connected through a parameter.
- 7.5: The Stability of Fixed Points in Nonlinear Systems
- We next investigate the stability of the equilibrium solutions of the nonlinear pendulum which we first encountered in section 2.3.2 . Along the way we will develop some basic methods for studying the stability of equilibria in nonlinear systems in general.
- 7.6: Nonlinear Population Models
- WE HAVE ALREADY ENCOUNTERED SEVERAL MODELS Of population dynamics in this and previous chapters. Of course, one could dream up several other examples. While such models might seem far from applications in physics, it turns out that these models lead to systems od differential equations which also appear in physical systems such as the coupling of waves in lasers, in plasma physics, and in chemical reactions.
- 7.7: Limit Cycles
- So far we have just been concerned with equilibrium solutions and their behavior. However, asymptotically stable fixed points are not the only attractors. There are other types of solutions, known as limit cycles, towards which a solution may tend. In this section we will look at some examples of these periodic solutions.
- 7.8: Nonautonomous Nonlinear Systems
- IN THIS SECTION WE DISCUSS NONAUTONOMOUS SYSTEMS. Recall that an autonomous system is one in which there is no explicit time dependence.
- 7.9: The Period of the Nonlinear Pendulum
- RECALL THAT THE PERIOD OF THE SIMPLE PENDULUM
- 7.10: Exact Solutions Using Elliptic Functions
- THE SOLUTION IN EQUATION 7.9.9 OF THE NONLINEAR PENDULUM EQUATION led to the introduction of elliptic integrals.
Thumbnail: A sample trajectory through phase space is plotted near a Lorenz attractor with σ = 10, ρ = 28, β = 8/3. The color of the solution fades from black to blue as time progresses, and the black dot shows a particle moving along the solution in time. Initial conditions: x(0) = 0, y(0) = 2, z(0) = 20. 0 < t < 35. The 3-dimensional trajectory {x(t), y(t), z(t)} is shown from different angles to demonstrate its structure. (CC BY-SA 3.0; Dan Quinn via Wikipedia)