*2.1.1 Dynamics of laser map*

Most biological systems exhibit enormous diversity and structurally multicomponent resulting in ecological imbalance and disorder/disharmony in environment. Inspired by articles of Lotka, Volterra, and Allee, numerous articles appeared with diversity in assumptions depending of species and their living envi-

A dynamical system be chaotic then it must be (i) sensitive to initial conditions, (ii) topologically mixing and (iii) its periodic orbits must be dense. In chaotic systems, there exists a strange attractor, a chaotic set, which has fractal structure. Complex systems are also sensitive to their initial conditions and two complex systems that are initially very close together in terms of their various elements and dimensions can end up in distinctly different places. Wide discussions on complex system may found in some pioneer literatures,

similarity that chaotic sets have fractal structure, [60, 68–73].

Chaos measured by Lyapunov exponents, (also called Lyapunov characteristic components or LCEs); LCE > 0 indicates existence of chaos and LCE < 0 indicates regularity, [52–62]. A complex system can better understood by measuring (i) chaos, (ii) Topological entropies and (iii) correlation dimension. Topological entropy, a non-negative number, provides a perfect way to measure complexity of a system. More topological entropy in any system signifies more complexity in it. Actually, it measures the evolution of distinguishable orbits over time, thereby providing an idea of how complex the orbit structure of a system is, [48–50, 61–69]. A system may be chaotic with zero topological entropy. In addition, a significant increase in topological entropy does not justify that it is chaotic. The book by Nagashima and Baba, [62], gives a very clear definition of topological entropy. The correlation dimension provides the dimensionality of the chaotic attractor. Correlation dimensions are non-integers and this is one reasons besides self-

It emerges from a good number of recent researches that chaos appearing in dynamical system be controlled and suggested number techniques to control chaos, [74–88]. These techniques have some limitations depending on the models and

Objective of this article is to investigate the emergence of chaos and complexity

in nonlinear dynamical systems through examples of nonlinear models. Numerical simulations carried out for bifurcation analysis, plotting of LCEs and topological entropies for different systems. Numerical calculations extended to obtain correlation dimensions for certain chaotic attractors emerging in different systems. The study further extended to explain different types of chaos controlling technique. Studies confined to one, two and three-dimensional

Real systems are mostly nonlinear and many of them are with multicomponent structure. Their individual elements possess individual properties. Such systems are

During evolution, a complex system exhibits chaos in some parameter space but also some other phenomena called complexity. This complexity is due to the interaction among multiple agents within the system displayed in the form of coexistence of multiple attractors, bistability, intermittency, cascading effects, exhibit of hysteresis properties etc. Thus, complexity can viewed as its systematic nonlinear properties and it is due to the interaction among multiple agents within the system. Foundation work and elaborate descriptions on complexity can viewed from some pioneer articles on complexity in nonlinear dynamics presented in [45–51]. Study of complexity means to know the results that emerging from a collection of

ronmental conditions in predator-prey models, [20–44].

*A Collection of Papers on Chaos Theory and Its Applications*

termed as the complex system.

interacting parts.

[14, 18, 45, 46, 48, 50, 51].

nature of nonlinearity.

systems only.

**186**

A highly simplified type discrete nonlinear model for laser system, arising from Laser Physics, described in articles, [12, 50, 89–91]. The model describes evolution of certain Fabry-Perot cavity containing a saturable absorber and driven by an external laser represented by

$$\mathbf{x}\_{\mathbf{n}+1} = \mathbf{Q} - \frac{\mathbf{A} \cdot \mathbf{x}\_{\mathbf{n}}}{\mathbf{1} + \mathbf{x}\_{\mathbf{n}}^2}, \forall \mathbf{e} \in \mathcal{R}, \mathbf{n} \in \mathbf{N} \tag{1}$$

Here Q is the normalized input field and A is a parameter depends on the specifics of the parameters and A > 0. The fixed points of the map are the real root of equation

$$\mathbf{x}^3 - \mathbf{Q}\mathbf{x}^2 + (\mathbf{1} + \mathbf{A})\mathbf{x} - \mathbf{Q} = \mathbf{0} \tag{2}$$

This equation has either three real roots or one real and a pair of complex conjugate roots depending on parameter space ð Þ *A*, *Q* . Stability occur in the form of stability and bistability, [89].

#### **Fixed Points and Bifurcations**:

For *Q* fixed, *Q* ¼ 2*:*76, and *A* <4*:*3793, only one stable steady state solution exits and stable two cycle starts when *A* exceeds this value. Thus, approximately, *A* ¼ 4*:*3793, is the bifurcation point. At value *A* ¼ 4*:*3, the stable steady state solution is x\* = 0.720533.

Keeping *Q* ¼ 2*:*76 and varying parameter *A*, bifurcation diagrams are drawn, **Figure 1**, for four different ranges of values of *A*. Similarly, keeping *A* fixed,

**Figure 1.** *Bifurcation diagrams of map (1) for four cases: when Q = 2.76 and parameter A varies.*

*A* ¼ 5*:*4 and varying *Q* in four different ranges, bifurcation diagrams are drawn, **Figure 2**. One observe clearly the appearance of periodic windows within chaotic region of bifurcations as an indication of intermittency and other complex phenomena. Periodic windows become gradually shorter and appearance become more frequent while moving forward in parameter space.

Both time series plots shown in **Figure 3** are for chaotic evolution of system (1) and correspond to parameters (a) ð Þ *A*, *Q* = (5.3, 2.76), due to which an unstable fixed point obtained as *x*� ¼ 0*:*58531, and parameters (b) ) ð Þ *A*, *Q* = (5.4, 2.9), due to which an unstable fixed point obtained as *x*� ¼ 0*:*572218. For both cases, initial point taken is *x*<sup>0</sup> ¼ 0*:*5 which lies nearby these points and so, also, unstable.
