Introductory Chapter: Fixed Points Theory and Chaos

1. Introduction

Among the systems that exhibit dynamical behavior, nonlinear and chaotic systems are the most intriguing, as they exhibit an enormous variety of performances and offer a great opportunity for technological applications. The formal study of chaotic systems begins with the results reported by Lorenz [1]. In this sense, the study and characterization of dynamical systems, especially chaotic systems, is one of the breakthroughs of the last century, although it is a relatively new field of research that is becoming increasingly important in various scientific disciplines [2, 3, 4, 5, 6]. In the case of nonlinear maps, it has been found that chaos can also arise between the dynamic behavior that these maps produce [7, 8, 9]. The study of fixed points could prepare the scientific community to investigate how to stabilize the behavior of multiple dynamical systems that generally exhibit nonlinear behavior, which is of great importance in current issues [10, 11]. The stabilization of fixed points in chaotic systems is one of the most interesting topics in the study of systems with chaotic behavior. Among the systems that have been stabilized are Lorenz, Rössler, and Chua [12, 13, 14]. As mentioned earlier, there are many works in which the chaotic behavior can be controlled by stabilizing the system’s fixed points. However, it is also possible to control the stabilization of the fixed points to obtain stable or multistable behavior of chaotic systems [15, 16, 17]. Moreover, this behavior has been studied in both integer and fractional-order systems [18]. Recently, Echenausía-Monroy et al. [19] presented an interesting method to characterize qualitative changes in the dynamical behavior of a family of piecewise linear systems by controlling the transition from monostable to multistable oscillations around different fixed points by studying the stable and unstable manifolds and their relation to the eigendirections.

2. A brief definition of fixed points

In the field of applied mathematics, fixed-point theory refers to an interdisciplinary topic that can be applied in various disciplines like economics, variational inequalities, approximation theory, game theory, and optimization theory, among other areas of interest. Fixed-point theory is divided into three major areas, as can be seen in Figure 1.

Figure 1.

Fixed points components.

Topological fixed-point theory was developed by L.E.J. Brouwer in 1912 [20]. According to Brouwer “Every continuous function from convex compact subset K of a Euclidian space to K itself has a fixed point.” It has several real-world illustrations. Consider a map of the country. If this map were placed anywhere in that country, there would always be a point on the map representing exactly that point.

One of the pioneering works about fixed points is Henri Poincaré [21], which was proposed as the first work about fixed points in 1886. Although the basic concept of metric fixed-point theory was known to others previously, the Polish mathematician Stefan Banach is credited with making it usable and well-known. The Banach Fixed Point Theorem (also known as the contraction mapping theorem or contraction mapping principle) is a useful tool in the study of metric spaces. It ensures the presence and uniqueness of fixed points of particular self-maps of metric spaces and gives a constructive approach to finding such fixed points [22]. The theorem is named after Stefan Banach (1892–1945) and was first stated by him in 1922. Banach stated that “Let $\left(X\text{,}d\right)$ be a metric space.” A mapping $T:X\to X$ is called Banach contraction mapping if there exists a constant $k\in \left[0\text{,}1\right)$ (s.t)

$$d\left(Tx\text{,}Ty\right)\le k.d\left(x\text{,}y\right)forallx,y\in X$$E1

Some fixed points theorems and different spaces were from the study and generalization of fixed points, as well as the Banach contraction theorem (Figure 2).

Figure 2.

Contraction mapping [23].

The discrete fixed-point theory came from Alfred Traski in 1955. Traski proved that “If F is a monotone function on a nonempty complete lattice, then the set of fixed points of F forms a nonempty complete lattice” [24].

But what is a fixed point? In this sense a short and comprehensive definition and interesting example are given next:

Let $X$ be a nonempty set and $T:X\to X$ be a mapping. Then $x\in X$ is known as fixed point of $T$ if $Tx=x$

Graphically, these are the places at which the graph of $f$ whose equation is $y=f\left(x\right)$, crosses the diagonal, whose equation $y=x.$

Let $y=f\left(x\right)=x^3+4x^2-3x-16,$ then it has three fixed points $x=2,x=-2\wedge x=-4$ as shown in Figure 3.

Figure 3.

Graphically representation of fixed point [25].

A fixed point is a location that stays the same when a map, set of differential equations, etc. are applied to it. Informally, the area of mathematics known as fixed point theory aims to locate all self-maps or self-correspondences in which at least one element is left invariant.

• Fixed Point for single-valued mapping

The fixed point for the mapping $S:R\to R$ defined as $S\left(x\right)=\frac{x}{2}$ is distinct. Obviously, the only fixed point is 0.

• Fixed Point for multi-valued mapping

  1. There are two fixed points in the mapping $S:R\to R$ defined as $S\left(x\right)=\sqrt{x}.$ The only fixed point, in this case, is 0 and 1.

  2. There are infinitely many fixed points in the mapping $S:R^2\to R^2$ defined as $T\left(x\text{,}y\right)=x.$ In fact, all points of $x-axis$ are fixed points.

A Mapping may have a unique fixed point, more than one, or infinitely many fixed points.

Remark: There may exist mapping which not has a fixed point.

Example: Let $X$ be a nonempty set. There is no fixed point in the mapping $S:X\to X$ defined as $S\left(x\right)=x+a$ where ‘a’ is any constant.