Chaotic dynamics of various continuous and discrete-time mathematical models are used frequently in many practical applications. Many of these applications demand the chaotic behavior of the model to be robust. Therefore, it has been always a challenge to find mathematical models which exhibit robust chaotic dynamics. In the existing literature there exist a very few studies of robust chaos generators based on simple 1-D mathematical models. In this paper, we have proposed an infinite family consisting of simple one-dimensional piecewise smooth maps which can be effectively used to generate robust chaotic signals over a wide range of the parameter values.
- Robust Chaos
- Piecewise-smooth Map
- Lyapunov exponent
- Logistic Map
Application of chaos in various fields of engineering and technology has increased rapidly over the past few decades. Some of the most important such applications include secured communication using chaos, generation of random number in cryptography based on chaos etc. [3, 9, 11,12,13,14,15,16,17, 19]. These applications of chaos have entirely changed the trend of research in many engineering fields. Due to these emerging important applications of chaos, the research related to chaotic systems and chaos generators have became one of the important fields of research these days. Design of chaos generators is one the important topic of interest in this field. Several studies have already been reported on this topic in last few years [1, 4, 6, 7, 10, 20, 21].
Many chaos generators have been designed based on different continuous-time mathematical models. Chua circuit , chaotic circuit based on Lorentz and Rossler systems  are some of such examples existing in the literature. The advanced technology in these days requires the integrability of any such chaos generator on chip. However, most of the chaos generators based on continuous-time mathematical models are difficult to be integrated on chip. Because of this drawback, studies of chaos generators based on discrete-time mathematical models are necessary. As a result of that, various discrete-time chaos generators have been studied [7, 10, 21], which have attracted much attention for having simple design and robust chaotic behaviour. These chaos generators are mainly implemented by imitating the existing 1-D chaotic maps (e.g. Logistic Map, Tent Map etc.) and have been described to be effective techniques of generating chaotic signals [20, 21]. However, many practical applications demands these chaotic signals to be robust [5, 8].
Let us discuss the phenomenon robust chaos in brief. If there exist a neighbourhood in the parameter space such that a system has a unique chaotic attractor throughout that neighbourhood, i.e., if the system has no periodic windows or coexisting attractors inside that neighbourhood, then such type of dynamics is called robust chaos. In general, dynamics of a chaotic map is not always robust chaotic. As an example, we know that the dynamics of the Logistic map shows a period-doubling root to chaos. However, if we continuously tune the parameter value of the Logistic map inside any neighbourhood entirely lying within its chaotic range, the stable chaotic dynamics is destroyed very often due to the appearance of periodic windows. Therefore the Logistic map does not exhibit robust chaotic dynamics. In fact, mathematical models which exhibit robust chaotic dynamics are rare. Therefore, it is in general difficult to guarantee the robustness of the chaotic behaviour of any chaos generator. Hence, it is necessary to study simple discrete-time mathematical models which exhibit robust chaos. In the literature there are very few studies on mathematical models exhibiting robust chaos and their circuit implementation [2, 5, 18].
In this paper, we have proposed a family of one-dimensional piecewise smooth maps which exhibit robust chaotic behaviour in a wide range of system parameter values. This family of maps have been constructed via a minor modification of the Logistic map, which is one of the very simple discrete-time mathematical model used to generate chaotic signals. Although, due to the lack of robust chaotic behaviour the circuits based on the Logistic map is unable to produce robust chaotic signal. However, we have shown that a minor modification of the Logistic model changes the entire scenario nicely. This modification allows to produce a family of 1-D maps displaying not only chaotic but also robust chaotic behaviour over a sufficiently wide range of the parameter values.
The Logistic map is defined on the closed and bounded interval [0,1] of the real line by
Now consider the one-dimensional piecewise smooth discontinuous map
Our claim is that for each fixed value of the border
Next we move on to show that the map (2) satisfies the property ‘sensitive dependence on initial condition’. We shall present a brief idea in support of our claim. Suppose
As the map (2) is not differentiable at
Moreover it is desired that more number of iterates of (2) should fall in the left hand side compartment as the left hand side map
Another point we want to specify here is that the border must lie to the left of the point
Now in the map (2),
Here we discuss one thing that what if
Now we verify our claim numerically. We show the graph of Lyapunov exponents and the bifurcation diagrams of the map (2) for the the corresponding values of
Next we vary the parameter
Now we give an numerical evidence in support of our mathematical proposition that
Summarizing the above discussions we conclude that the equation in (2) actually represents an infinite family of robust chaotic maps. For each fixed value of