Introduction to Nonlinear Dynamics for Physicists

The following sections are included:

• Preface

• Contents

What is dynamics? What is nonlinear dynamics? Let us start with the simple example of the swing on a child's playground. The child is raising and lowering herself by pumping her legs back and forth. Up and down, up and down. The swing oscillates up and down, and with strong pumping could even turn upside-down and start rotating…

The following sections are included:

• A ball in a trough

• Our equations of motion can be easily integrated

• The integral curves on which the direction of motion is defined are referred to as phase trajectories

• Analytical description of motion on a separatrix loop

Where dissipation appears, the energy conservation law disappears. But different kinds of behavior change differently when dissipation is turned on. For example close, to a center, where a nondissipative (also called Hamiltonian) system is stable, the behavior will change substantially, while near a saddle point, where the Hamiltonian system is unstable, there will be just small changes. This is clear intuitively. In both cases (with or without friction) a ball will roll down from a hill (a saddlepoint). But, if we have friction, oscillations (around a center) will decay, and we will not see periodic motion…

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So far we have only considered systems which did not depend on time explicitly: these systems are called autonomous. Now let us see what will happen when a nonlinear system is subject to an external force. An example of such a system is shown in Figure 8.1 (an oscillating ferromagnetic ball in a time dependent magnetic field)…

In the previous lecture we became acquainted with a few very interesting phenomena taking place in a forced nonlinear oscillator: nonlinear resonance (periodic oscillations with the frequency equal to that of the external force), pulsation (quasi-periodic oscillations with two incommensurate frequencies whose phase space portrait consists of a dense winding on a two-dimensional torus), multistability (more than one stable stationary solutions corresponding to the resonance)…

Today we are going to discuss how two coupled generators behave. This problem is very important. This problem appears in many branches of physics (coupled modes in a laser resonator, in convection flow in a fluid, in electrical circuits and so on)…

In this lecture we will consider a qualitative mothod which may help us escape sometimes very boring mathematics. It often leads us directly to an answer about the stability of equilibrium points, limit cycles and is very handy as we will see…

We have a pendulum with a spring in its support. This motion is confined to a plane in the gravitation field. What should we expect to see? If the amplitudes are very small, we may guess that the spring vibrations and the angular oscillations will be almost independent. However, if the amplitudes are large enough, the behavior is more interesting. Assume that at the start the spring is almost vertical with a very small angular velocity and is very stretched. We will see something like a parametric resonance. If the frequency of spring vibrations is twice that of the angular oscillations, the latter will absorb energy from the spring and its amplitude will increase. If we have the reverse initial conditions (the spring is at its equilibrium length and makes a large angle with the vertical line), it is clear that the amplitude of the spring vibration will also increase when the frequencies of each have the same ratio…

We already know what will happen if we couple two or three oscillators to each other. Now we are about to find out how an ensemble of coupled oscillators behaves. This is a very difficult question because the behavior depends in general on very many factors: the kind of coupling, the kind of nonlinearity, geometry etc. As usual, we begin with the simplest model one can think of: a one dimensional chain of linear oscillators with nonlinear coupling between them (Figure 13.1)…

Some of you probably noticed that in the previous lecture we considered a rather idealized situation. We neglected dissipation, but that is present in any physical system. Does this mean that everything we said is useless? By no means! We pointed out that solitons are often stable even with respect to small perturbations in the equations we use to describe them. Therefore, if at some time we have a soliton going in some direction, then when a small amount of dissipation is introduced, we will see the same “soliton” but with slowly decreasing amplitude and speed. However, we still have no clear idea about what will happen if, in addition to the dissipation, some energy is pumped into the system, perhaps from the outside world through the boundaries of the system. Of course, there is such a variety of ways to do this that we could spend hours discussing them. Instead let us consider the simplest and, probably, the most important situation when the solution to the driven, dissipative problem is zero at + ∞ (or at −∞) and has some finite value at −∞ (or at +∞). We will again try to find a solution in the form u(x, t) = u(x − vt). If we succeed, we will find a description of ‘shock’ waves in dissipative media…

In the previous lecture we considered the structure of a region where a physical variable suffers a sudden jump. When we consider a particular solution, we always should ask ourselves if this solution can really occur in a physical system. In other words, we must find out if there are realistic paths for the system to evolve towards that solution. Let us consider how shock waves can appear from smooth initial conditions. As an example we will consider gravitational waves on the surface of a body of water (Figure 15.1). We will assume that the wavelength is much larger than the depth (λ > > h₀); that is, the water is shallow. We consider only two spatial dimensions…

We already used the spectral approach to understand the actions of nonlinearity and dispersion, but only a qualitative analysis was presented. Can we use the spectral method for a quantitative description of nonlinear continuous systems? Yes, we can. But the nonlinear spectral method is very different from what we do in linear systems. Specifically, in a nonlinear system different spectral components do not, as a rule, evolve independently. Harmonics with different wave numbers and different frequencies interact with each other. Therefore, we would expect that in this case the equations of evolution for spectral amplitudes will in general be differential or even integral equations and not algebraic as in linear systems. It is a very hard problem to find these equations, but as usual one can often do this in the case of a weak nonlinearity. In this case we can again use the averaging method with a slight change…

So far we observed only regular motions in systems with large numbers of degrees of freedom. This regularity is closely related to the fact that in those systems all modes are synchronized. In general this is not true, and in some cases the energy is transferred among the modes in a very twisted and unpredictable way. The field fluctuates randomly, and we observe what we call turbulence. How can we possibly describe this phenomenon?…

Let us start with a description of the Rayleigh-Benard thermally driven convection experiment…

Before discussing the origin of chaos in dynamical systems, let us list a few relevant facts about the behavior of phase trajectories in one and two dimensional phase space. In one dimensional phase space we have only trivial equilibrium states, that is stable and unstable fixed points. All motions are very simple. The system evolves towards one of the stable points, and having reached this point it stays there. As we go from one to two dimensional phase space we encounter a new type of trajectory: a limit cycle. The behavior of the system near the equilibrium points becomes more complicated and diverse. We know how to use the characteristic exponents to separate the equilibrium states into different groups. But there is one point we have not discussed yet. We did not study how predictability depends on the configuration of the phase plane…

As we saw in the previous lecture, a dynamical system may exhibit chaotic behavior. For a physicist the problem is to describe this behavior somehow and to find how its characteristics depend on the parameters of the system. In general this is a very hard problem and well beyond our present understanding of the dynamcis of nonlinear systems. However, in the system which we considered in the previous lecture we had a small parameter multiplying the time derivative, and we were able to separate the motion into fast and slow parts. After that the system looked simpler. Now we will try to simplify it even further by constructing a Poincaré section…

In the previous lecture we saw that chaotic motions arise when there are no stable periodic motions in the attracting region of phase space. It is clear that the behavior of any system depends on some control parameters. That is, for some values of the control parameter the system behaves in a very predictable way while for other values the behavior is chaotic. This means that at some value of the parameter there are some stable periodic trajectories. When we change the parameter, some of these trajectories become unstable, and at some critical value of the parameter the last trajectory becomes unstable, and the behavior of the system becomes chaotic. Therefore, it is very important to understand how a periodic trajectory losses its stability…

In the previous lecture we discussed the three most important bifurcations of periodic trajectories and said that each of them can lead to chaos. We then studied the transition to chaos through the sequence of period doubling bifurcations using the example of the logistic map. In the logistic map there is only one control parameter and only one route to the onset of chaos. In many physical systems, however, we have more than one control parameter and these systems can approach the chaotic region in many ways. Today and in the next lecture we will consider a system which can become chaotic after a period doubling sequence or after intermittency. This system is the nonlinear oscillator shown in Figure 22.1 coupled parametrically with a control device…

When we discussed bifurcations of periodic motions, we said that each of them correspond to a specific scenario of arising of chaos in the system. The first scenario we considered was the sequence of period doubling bifurcations. In spectral language we start with the spectrum shown in Figure 23.1a; To is the period of original motion. After the first bifurcation the base frequency becomes half of the original, and we observe the spectrum shown in Figure 23.1b. Similarly after the second bifurcation, we will see a spectrum with the base frequency one quarter of the original, Figure 23.1c. At the critical value of the control parameter the spectral lines produced during the bifurcation sequence merge and form a continuous spectrum corresponding to chaotic motion in the system Figure 23.Id…

The following sections are included:

• Fractal Dimension

• Lyapunov Exponents

In this last lecture we will consider a more complicated system that exhibits chaotic behavior. When we discussed the bifurcations leading to the appearance of strange attractors we studied only low dimensional systems which could be described by systems of ordinary differential equations. However, there are many systems which can be described only by partial differential equations. and which can also behave chaotically. Indeed, it may fairly be said that the description of any physical process by ordinary differential equations represents a “lumped” parameter or long wavelength approximation to the dynamics of the underlying fields. In this lecture we will consider an example of such continuous systems called the Ginzburg–Landau model, and for simplicity we will restrict ourselves to one spatial dimension. This model is used to describe motions in a variety of excitable media such as chains of Van der Pol generators, a chain of Taylor-Couette vortices, convection and many other systems. But let us consider for a change a biological example of such a medium to illustrate that nonlinear physics is indeed an interdisciplinary science, and also to illustrate how nonlinear phenomena affect our life…

The following sections are included:

• Lecture One

• Lecture Two

• Lecture Three

• Lecture Four

• Lecture Five

• Lecture Six

• Lecture Seven

• Lecture Eight

• Lecture Nine

• Lecture Ten

• Lecture Eleven

• Lecture Twelve

• Lecture Thirteen

• Lecture Fourteen

• Lecture Fifteen

• Lecture Sixteen

• Lecture Seventeen

• Lecture Eighteen

• Lecture Nineteen

• Lecture Twenty

• Lecture Twenty-one

• Lecture Twenty-two

• Lecture Twenty-three

• Lecture Twenty-four

• Lecture Twenty-five