The path pierces these planes at the points A 1 , A 2 , A 3. Now let us analyze results seen in Fig. The displays x versus time t when transient effects have died out.
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In Figs. In Fig. In these cases we have aperiodic motions which is a characteristic of the deterministic chaos .
Finally, we remark that only for dissipative systems there are set of points attractors or a point on which the motion converges. In chaotic motion, nearby trajectories in phase space are continually diverging from one another following the attractor. This effect is shown in Fig. Due to these attractors, named strange or chaotic attractors, the motions in the phase space are necessarily bounded. The attractors create intricate patterns, folding and stretching the trajectories must occur because no trajectory intersects in the phase space, which is ruled out by deterministic dynamical motion .
The figures reveal a complex folded, layered structure of the attractors. Amplifying figure we would note that the lines are really composed of a set of sub lines.
Amplifying a sub line we would see another set of sub lines and so on … verifying that the strange attractors usually are fractals [ 3 , 31 , 33 ]. Another example of one-dimensional nonlinear motion is the one described by the damped and driven pendulum around its pivot point shown in Fig. Dividing Eq. If we want to deal with Eq. So, let us divide Eq. The new dimensionless variables and parameters are presented in Table 1 :. Using the variables and parameters defined in Table1, we verify that Eq. Integrating numerically Eq.
As an example, in Fig. These solutions correspond to three different periodic attractors. Furthermore, to show the attractor dependence on initial conditions, we present in Fig. For each initial condition we obtain the numerical solution and identify the corresponding atractor, associated with one of the three lines shown in Fig.
Deterministic Chaos Theory: Basic Concepts
Figure 9 a is denominated basin of attraction of teh solutions of Eq. The successive amplifications of the basin of attraction, shown in Fig. In some cases it is very difficult to study the evolution of a nonlinear system integrating their differential equations. Sometimes it is also difficult to construct an exact nonlinear mathematical model to study physical system.
In these cases it is possible to get a good description of the chaotic process using an iterative algebraic model named mapping. To understand the origin of this model let us assume that the motion of a system is described by nonlinear first-order differential equations of the form .
Math 352: Introduction to Complex Analysis
There are innumerous chaotic systems studied with the mapping approach. Famous examples are the map models for ecological and economic interactions: symbiosis, predator prey and competition [ 34 , 35 ]. Malthus, for instance, claimed that the human population p grows obeying the law . Verhulst  argued that the population grow has inhibitory term a p 2 so that Eq. One century later, indicating the population by x the differential equation 17 was substituted by the logistic equation [ 34 , 35 ].
Note that the Eq. An n cycle is an orbit that returns to its original position after n iterations. A more general view of the evolution can be obtained plotting a bifurcation diagram [ 1 , 34 , 35 ] see Fig. Analyzing this figure we verify that for 2. The bifurcation and period doubling continues up to an infinite number of cycles near 3.
Examples of numerical solutions of Eq. To show how the numerical solutions depend on the control parameters, we present in Fig. An interval with a period 5 attractor can be observed in Fig.
In the parameter space of Fig. The amplification in Fig. Such windows are also called shrimps  and have been observed in several dynamical systems [ 38 , 39 ].
Periodic windows are in black. White points represent parameters with chaotic attractor. In gray is a periodic-5 window. The nonlinear terms of the differential equations amplify exponentially small differences in the initial conditions. In this way the deterministic evolution laws can create chaotic behavior, even in the absence of noise or external fluctuations.
In the chaotic regime it is not possible to predict exactly the evolution of the system state during a time arbitrarily long.
This is the unpredictability characteristic of the chaos. The temporal evolution is governed by a continuous spectrum of frequencies responsible for an aperiodic behavior see, for instance, 4. The motions present stationary patterns, that is, patterns that are repeated only non-periodically [ 2 , 3 ]. Lyapunov created a method [ 1 - 3 , 34 ] known as Lyapunov characteristic exponent to quantify the sensitive dependence on initial conditions for chaotic behavior.
It gives valuable information about the stability of dynamic systems. With this method it is possible to determine the minimum requirements of differential equations that are necessary to create chaos see footnote 2. To each variable of the system is a Lyapunov exponent. We want to investigate the possible values of x n after n iterations from the two initial values. From Eq. The difference d 1 between the two initial states is written as. Now, in order to avoid confusion that sometimes is found in the chaotic literature, we remember that.
After a large number n of iterations the difference between the nearby states, using Eq. So, using the derivative chain rule we get. This occurs because when is done an infinite numbers of iterations.
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This kind of calculation for the damped and driven pendulum is seen, for instance, in reference . Using maps these calculations become easier. This is shown in what follows for logistic map and triangular map.