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 . 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.
Determinism and Chaos
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.
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. According to Eq.
For the logistic map we have the logistic equation In reference [ 36 , 40 ] are seen cobweb plots web diagrams or Verhulst diagrams that are graphs that can be used to visualize successive iterations of the function f x. In particular , the segments of the diagram connect the points x , f x , f x , f f x , f f x , f f f x. The diagram is so-named because its straight lines segments anchored to the functions x and f x resemble a spider web.
The cobweb plot is a visual tool used to investigate the qualitative behavior of one-dimensional iterated functions such as the logistic map. With this plot it is possible to infer the long term status of an initial condition under repeated application of a map. This map, represented in Fig.
Introduction to chaos : physics and mathematics of chaotic phenomena - University Of Pikeville
By symmetry the next point of minimum must be 2. Taking into account Eq. As usually seen in basic physic courses [ 4 , 41 ], turbulence is originated from studies of fluid motion in classical mechanics. This equation is a miracle of brevity, relating a fluid's velocity, pressure, density and viscosity . Since Eq. In these conditions Eq.
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In spite of this yet there is some sort of order found within the disorder or turbulence which could be described as self-similar or fractal . An open problem is to find a mathematical formalism able to describe this disordered state [ 25 - 27 ].
Turbulence in fluid dynamics is being understood in infinite dimensional phase space under the flow defined by the Navier-Stokes equation. We have seen that in the finite dimensional phase space physical systems can be described with very good precision by LODE and NLODE that can solved exactly or numerically. They can in principle reveal all detailed structures of the dynamical systems. Turbulence in fluid mechanics is generated by a NLPDE anchored in an infinite dimensional phase space. Is turbulence a chaotic process? Up to nowadays it is well-known that the theory of chaos in finite-dimensional dynamical systems has been well-developed.
Such theory has produced important mathematical theorems and led to important applications in physics, chemistry, biology, engineering, etc . Note that, in the contrary, theory of chaos in PDE has not been well-developed. In spite of extensive investigations it was not possible to prove, in the general case, the existence of chaos in infinite-dimensional systems [ 10 , 17 , 18 , 20 ]. Among the NLPDE there is a class of equations called soliton equations that are integrable Hamiltonian PDE and natural counterparts of finite-dimensional integrable Hamiltonian systems .
Many works have also been developed investigating the existence of chaos in perturbed soliton equations [ 20 , 27 ]. Marion and S. Thompson and H. Alligood, T. Sauer and J. Grebogi, E. Ott and J. Yorke, Phys. Symon, Mechanics Addison-Wesley, London, Lichtenberg and M.
Conway, N. Sloane and E.
Introduction to chaos : physics and mathematics of chaotic phenomena
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