Lecture 1: A Preliminary to Nonlinear Dynamics and Chaos Autonomous Systems A set of coupled autonomous 1st-order ODEs. Here "autonomous" means that the right hand side of the equations does not explicitly depend on time. This kind of coupled ODEs can be expressed in a matrix form where and are -dimensional vectors defined by and. This kind of coupled differential equations is very popular not only in physics but also in other fields such as chemistry and biology. In physics, Hamiltonian dynamics (Newtonian dynamics) is exactly like this.. where is the degrees of freedom. Note that there are differential equations. (Remark: Even when no Hamiltonian exists, for example, due to dissipation, we still can write down Newton equations as a set of coupled first-order ODEs.) Remark: If some of ODEs are not coupled to the remaining ODEs, there are two independent sets of the ODEs. Then, they can be investigated separately. (See coupled harmonic oscillators in coming lecture.) Our goal is to find a trajectory -dimensional space for a given initial condition (or in the -dimensional space.) However, in 1963, two researchers, Edward Lorenz and Yakov Sinai, independently shown that such an objective is fruitless. In most cases, it is impossible
to predict a trajectory for a large. When two trajectories begin with very slightly different initial conditions, and, the two trajectories deviate from each other as time goes. The two trajectories look totally different tepid that they started almost the same places. Now, we call what they discovered "Chaos". Since the discovery, our understanding of classical mechanics has changed dramatically. This is just a conceptual image of chaos (two-dimensional flow cannot be chaotic in real systems. ) Non chaotic Chaotic One of the reasons why we did not find chaos earlier is that it does not exist in and. Even when, if the equations can be decoupled, again the Chaotic behaviors disappear. Unfortunately, most of examples studied in traditional classical mechanics courses are such systems where no chaos is possible. First we will show that the systems with and have simple trajectories and they are fundamentally different from the systems with. The new understanding is mathematically challenging due to the nonlinearity inherent in Newton equation (note that Schrödinger equation is linear). it is very difficult to understand it without the help of computer (numerical solution and computer graphics). Throughout the lectures, we investigate from simple to very complicated trajectories using Maple. Visualizing trajectories Phase space and phase trajectory For, the phase space is a line
x 1 t x 1 For, t x 2 x 1 t, x 2 t x 1 For
x 3 x 1 t, x 2 t, x 3 t x 2 x 1 It is difficult to show the phase trajectories of higher dimension. Remarks 1. Do not be confused with the trajectory of a particle in the position space. The variable can be any dynamical variable including velocity and even time. 2. A phase trajectory never crosses itself. (Uniqueness of the solution.) Projection to a plane and Stroboscopic map It is difficult to plot a curve in a high dimensional phase space. One way to visualize it is to project a trajectory on a plane, for example.
x 3 x 1 x 2 Stroboscopic Map x 3 t 0 C 3 t t 0 C 4 t t 0 C 2 t x 1 t t 0 C t 0 x 2 Poincarè map It is difficult to plot a curve in a high dimensional phase space.
Intersects of a trajectory with a plane (Poincarè section). Poincarè section Poincaré map Bifurcation diagram In general, the behavior of the system depends on parameters such as frictional coefficient and strength of the force. We are particularly interested in the cases where the behavior changes qualitatively as the system parameters change. An famous example is dripping faucet.
low flow high t 1 t 2 t 4 t 3 t 1 t 1 t 2 t 2 t 1 t 1
(This is just a conceptual image and not an actual data.) N=1, If does not depend on, we have an autonomous equation analytically. We can solve this problem However, we are interested in more general aspects of the equation. The non-crossing rule for one-dimensional phase severely restricts types of the trajectories. In general, either monotonically increases or decreases until it reaches a fixed point or infinity. That means cannot oscillate. For, increases as increases. For, is negative. Hence, decreases as increases. For,. That means remains at the root of
From this flow diagram, it is clear that the 1st order autonomous ODE has only two types of solutions: one monotonically approaches to a fixed point (attractor) and the other monotonically diverges. Chaotic motion is not possible in one-dimensional autonomous dynamics. Example: Free falling Consider a particle of mass m subject to a frictional force is falling under a uniform gravity g. Assume that its velocity is at. The equation of motion for the particle is and its integral solution is (3.1.1) Solving for, we obtain (3.1.2)
Since (terminal velocity) is only the attractor in this equation, all solutions monotonically approach to the terminal velocity. Using Maple ODE solver: (3.1.3) where _C1 is a constant of integration. To specify a boundary condition, (3.1.4) in agreement with Eq. (3.1.2). You can plot solutions as a flow diagram using DEplot Regardless of the initial value of, all the velocities approach the terminal velocity.
N=2, Possible trajectories near fixed points: unstable fixed point stable fixed point saddle point stable spiral unstable spiral center node stable limit cycle unstable limit Poincaré-Bendixson theorem If a phase trajectory is confined in a region in which there is no fixed point, the trajectory must be a close orbit. This theorem is valid only for N=2. (This theorem is also due to the non-crossing rule.) Newton equation for a particle moving in one dimensional space: can be written as a coupled first order ODEs:
. Hence, one-dimensional classical dynamics correspond to a dynamics in two-dimensional phase space of (x,v). Example: pendulum Equation of motion where. where and. *** No dissipation ***
*** With dissipation ***
Why one-dimensional Newton equations with autonomous force does not show chaotic behavior? 1. If there is dissipation, energy must decrease. At the end, the system lost all energy and come to holt. 2. When there is no dissipation, energy must conserve:. This means that and are not independent. Therefore, the conservation of energy makes the system effectively one dimension. In general, conservation laws apply strict restriction to the motion which reduce the dimensionality. N=3, Three dimension or above, the geometric restriction is greatly reduced. A aperiodic trajectory can be confined within a finite regions. Chaotic behavior is also possible. Example: Lorenz attractor
Lorentz Equation Parameter Values Initial Conditions Numerically solve the equation Plot the result
Lorenz attractor (chaotic attractor)
Hence, autonomous Newtonian dynamics can be chaotic only when the system has two degrees of freedom. N=4 The systems with do not have the geometric constraints the systems with. The dimension of the autonomous Newtonian system is even. Hence, the smallest (time-independent) Hamiltonian systems that can show chaotic behavior is. If there is no dissipation, the conservation of the energy reduces the dimension by one. If there is another conserving quantity, such as angular momentum, then the dimension is further reduced. Then, there is no chaotic behavior. A coming lecture covers this topics using double pendulum and the Henon-Heiles model. Non-autonomous systems
A set of coupled non-autonomous 1st-order ODEs. This set of equations can be transformed to an autonomous form by introducing another dynamical variable. Hence, mathematically speaking, a dimensional non-autonomous problem is equivalent to a dimensional autonomous system. However, for Newtonian dynamics, the presence of timedependent forces introduce new kinds of dynamics. Since time-dependent forces are necessarily exerted by an external agent, the system energy does not have to conserve. It is a completely different game from energy conserving dynamics. Example: driven Duffing oscillator The Duffing oscillator is a non-linear oscillator whose equation of motion is given by Now, an external agent applied a time-dependent force The resulting equations of motion are on the oscillator. where,, and. Hence, this is a problem and the system shows chaotic behaviors. We will discuss this system extensively in a coming lecture.