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Intelligent Automation & Soft Computing
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DOI: 10.32604/iasc.2023.031575

Article

A Chaotic Pulse Train Generator Based on Henon Map

Babu H. Soumya1,*, N. Vijayakumar2 and K. Gopakumar3

1College of Engineering, Trivandrum, 695016, India
2Government Engineering College, Barton Hill, Trivandrum, 695035, India
3APJ Abdul Kalam Technological University, Trivandrum, 695016, India
*Corresponding Author: Babu H. Soumya. Email: soumiyam@gmail.com
Received: 21 April 2022; Accepted: 17 August 2022

Abstract: The Henon map forms one of the most-studied two-dimensional discrete-time dynamical systems that exhibits chaotic behavior. The Henon map takes a point (Xn,Yn) in the plane and maps it to a new point (Xn+1,Yn+1). In this paper, a chaotic pulse generator based on the chaotic Henon map is proposed. It consists of a Henon map function subcircuit to realize the Henon map and another subcircuit to perform the iterative operation. The Henon map subcircuit comprises operational amplifiers, multipliers, delay elements and resistors, whereas, the iterative subcircuit is implemented with a simple design that comprises of an edge forming circuit followed by a monostable multivibrator and a voltage controlled switch without the use of any clock control. The proposed design can be used to realize the Henon map and also to generate a chaotic pulse train, with a controllable time interval and pulse position. The proposed circuit is implemented and simulated using Multisim 13.0 and MATLAB R2019b. The chaotic nature of the generated pulse train and also the time interval between the consecutive pulses is verified by the calculation of its Lyapunov exponents.

Keywords: Chaos; iterated maps; henon map; chaotic pulse train; lyapunov exponents

1  Introduction

Chaotic systems are deterministic nonlinear dynamical systems that consist of a set of possible states, together with a rule that determines the present state in terms of past states. It is required that the rule be deterministic, to uniquely determine the present state from the past states. When the governing rule is applied at discrete times, it is a discrete-time dynamical system or the so-called iterated maps and when the rule is a set of differential equations, it is a continuous-time dynamical system [1]. These systems are characterized by high sensitivity to initial conditions, presence of dense orbits and must be topologically transitive, which makes the system output completely different for even a very small change in the input applied to the system [2]. A chaotic map or the so-called iterated map is an evolution function exhibiting chaotic behavior, and can be parameterized by a discrete-time or a continuous-time parameter. It mostly takes the form of iterative functions and can be one-dimensional, two-dimensional or multi-dimensional. But much attention is given to one-dimensional and two-dimensional quadratic mappings, as they exhibit all the interesting properties a map can have.

Such pseudo-random behavior of chaotic systems makes these systems highly regarded for various applications like pseudo-random sequence generation, encryption and secure communication. Moreover, these systems are extensively used for the generation of chaotic signals and chaotic pulse train, that can be used as a control signal and modulating signal, and has several applications in cryptography [3], data processing [4], secure communication [5], steganography [6], bio-medical processing and its extended applications [7,8].

Several works have been done in the past for the generation of chaotic pulse train. In 2008, Wang et al. introduced a method to generate chaotic pulse sequence by chaotic phase trajectory [9]. In 1994, Balmforth et al. found that under certain particular conditions, the third-order nonlinear ODE can take the form of a chaotic pulse sequence [10]. Later in 2005, methods were proposed for generating chaotic pulse for reaction-diffusion systems, mainly focused on algorithm implementation and not appropriate for circuit implementation [11]. Dmitriev et al. in 2005 proposed a method to generate the chaotic pulse trains using a dynamical system that uses a periodic signal [12]. In most of these dynamic systems with circuits implemented by resistors, capacitors, monostable multivibrators, analog switches etc. one–dimensional return maps are used. But in these methods, it is difficult or not possible to control the generated pulse parameters like pulse position, pulse interval etc. The major drawback of such a realization is the requirement of an external control signal to drive the circuits and the difficulty in controlling various parameters such as pulse width and frequency of the generated chaotic pulse sequence. In the last decade, another one-dimensional piecewise linear map referred to as the Tent map has been used extensively in these applications [13,14] and such circuit implementations [15,16]. In 2000, Tanaka et al. proposed the voltage mode based integrated circuits of Logistic map and Tent map chaotic generators [17]. In 2009, Campos-Canton et al. proposed a voltage mode based electronic circuit to implement the Tent map [18]. Though these circuits are easy to implement, chaotic pulse sequence could not be produced. Later in 2015, Zhang et al. proposed a chaotic pulse generator based on the Tent map which overcomes the drawbacks of the previous methods and can be used for the generation of chaotic pulse sequence with control over the pulse parameters [19].

However, this article proposes the implementation of a chaotic pulse train based on a two-dimensional chaotic map referred to as the Henon map. In past literatures, only a few one-dimensional chaotic maps like logistic map, Tent map etc. have been realized and analyzed for such applications [18,20,21]. But, very few works have been done to realize the two-dimensional iterated maps, and analyze its chaotic and complex dynamics [22,23]. Even though all the circuits mentioned in these literatures can be used to generate chaotic series, and some of them are even easy to implement, since these methods use periodic clock signals to implement the iterative process, the chaotic pulse train or sequence may not be produced. This article deals with the implementation of a circuit to realize the Henon map and thus generate two sets of chaotic time series and also to generate the chaotic pulse sequence with a control on the pulse parameters like pulse interval and pulse position. Also, the entire circuit is realized more concisely compared to the existing literatures and has a reasonable design without any clock control.

The Henon map forms one of the most studied two-dimensional discrete-time dynamical systems that exhibit chaotic behavior. It was introduced as a simplified model of the Poincare section of Lorenz model [24]. Due to its simple form and interesting chaotic behaviors, the map has always been attractive as a mathematical model to study deterministic chaos. The map takes a point (Xn,Yn) in the plane and maps it to a new point (Xn+1,Yn+1), where (Xn, Yn) are the initial conditions or the initial state of the map, whereas, Xn+1 and Yn+1  forms the next state of the map after iteration. The map also depends on two bifurcating parameters ‘a’ and ‘b’. For the classical Henon map, the bifurcating parameters are set to the values, a=1.41 and b=0.3, which keeps the map in the chaotic regime [25]. The Henon map also exhibits a strange horseshoe shaped attractor with a fractal structure, and a constant value for the Jacobian determinant | J |=− b. This existence of chaotic behavior and a transversal homoclinic orbit for the map for some specific parameter values was analytically proved by Marotto [26].

2  Circuit Implementation

This article focuses on the circuit implementation of a chaotic pulse generator. A 2-D chaotic Henon map based chaotic pulse train generator is proposed, which does not require an external control signal. The design of the proposed circuit consists of mainly two parts (A) Henon map subcircuit realization (B) a circuit to realize the iterative operation. The Henon map subcircuit realization is done more precisely compared to the ones in the existing literatures whereas, the iterative subcircuit is implemented with a simple design without any clock control. Instead of using an external control signal, an edge signal forming circuit is designed to generate a positive and negative edge signal that, in turn, triggers a monostable multivibrator (MM). Upon triggering, a pulse signal is generated at the output of the MM, which is used to close an analog switch that passes the output of the Henon map to its input.

Fig. 1 shows the overall block diagram of the proposed circuit. It consists of a simple circuit designed to set initial condition for the Henon map, followed by a circuit to realize the Henon map. The Henon map realization is followed by an iterative subcircuit which feeds back the iterated output of the Henon map so that, it serves as the next input to the Henon map.

Figure 1: Block diagram of the proposed circuit

2.1 Henon Map Function Subcircuit

The Henon map function takes a point (Xn ,Yn) in the plane and maps it to a new point (Xn+1, Xn+1) as given by the Eqs. (1) and (2) where (Xn,Yn) represens the initial conditions of the map and Xn+1  and Yn+1  represent the next state of the map, obtained by iterating the map once. Eqs. (1) and (2) describe a classical Henon map with ‘a’ and ‘b’ as the bifurcating parameters. For the chaotic classical Henon map, the value of ‘a’ is set to 1.41 and ‘b’ to 0.3, for the Henon map to be in the chaotic regime. The Henon map can be chaotic, intermittent or can even converge to a periodic orbit for other values of ‘a’ and ‘b’ [27,28].

Xn+1=1 – a Xn2+Yn(1)

Yn+1=b⋅Xn(2)

Fig. 2 shows the circuit diagram of the proposed subcircuit of the Henon map. The Henon map is realized using an operational amplifier U3A, four resistors R2 to R5, two multipliers A1 and A2, a delay element A3 and power supplies. Two multipliers A1 and A2 are used to set the values of the parameters ‘a’ and ‘b’. Since for the classical Henon map, a = 1.41 and b=0.3, one of the inputs to the multipliers A1 and A2 are set to −1.41 and 0.3 respectively, so as to design the Henon map as described by Eqs. (1) and (2). The operational amplifier U3A, configured in non-inverting mode of operation along with resistors R2 to R5, is configured to act as a summing amplifier. In order to make the op-amp output equal to the sum of the inputs at the non-inverting terminal of it, the closed-loop gain of the op-amp is made equal to three, which is equal to the number of summing inputs [29].

Figure 2: Circuit diagram of the Henon map subcircuit realization

The closed-loop gain of the summing amplifier is given by Av=1+(R6R5). Since, Av=3, the values of resistors are chosen to be R6=10KΩ and R5=5KΩ. Also, if the input resistances R2, R3 and R4 are all made equal, the circulating currents cancel out since they cannot flow into the high impedance non-inverting input of the op-amp and the output voltage of the op-amp becomes equal to the sum of its inputs at the non-inverting terminal. Therefore, R2=R3=R4=10KΩ. Thus, the map is realized with less number of components when compared to those in the existing literatures. Two sets of chaotic sequences can be generated from the Henon map for a particular value of ‘a’ and ‘b’, though only one set of sequences (X0, X1, X2, X3,…) is utilized in this article to be used as one of the inputs to the edge forming circuit.

2.2 Iterative Function

In this article, an ingenious design consisting of an edge signal forming circuit, a switch and a monostable multivibrator (MM) is used to develop a circuit that performs an iterative operation as described in the Fig. 3. The edge signal generated is used to trigger the MM, that in turn produces the pulse signal at its output. This pulse output from the MM is used as control signal to control the analog switch, that pass on the signal Xn+1 onto Xn.

Figure 3: Flow diagram of the proposed iterative operation

3  Working Principle

The overall circuit diagram of the proposed circuit to generate the chaotic pulse train is shown in the Fig. 4. It consists of a simple circuit to set the initial condition of the Henon map followed by the Henon map realization subcircuit, a comparator circuit using LM311D, a monostable multivibrator CD4538BCM and a voltage-controlled SPST switch S2.

Figure 4: Overall circuit diagram of the proposed circuit

The initial condition X0 for the Henon map is set using the resistor R1, Key K1, capacitor C1 and the potentiometer P1. The value of X0 is selected in the range [0, 1] by suitably selecting the values of the different components, with the switch K1 closed and then fed as the non-inverting input to an operational amplifier U4A that acts as a voltage follower. The output of the voltage follower is fed as the initial condition of the Henon map. Even though two sets of chaotic sequences are generated from Henon map, only one set of chaotic sequences given by [X0, X1, X2……] is taken into consideration to the next stage of the system. Initially, the initial condition X0, and the iterated output X1  are fed to the next stage of the system that forms the iterative circuit.

In the implementation of the iterative circuit, the design of the edge forming circuit forms the crucial part. It is implemented by making use of an operational amplifiers U5A, a comparator U6 using LM311D, a switch S2, a capacitor C2 and a potentiometer P3. The initial state X0, and the corresponding output state of the Henon map, Xn+1 are fed as inverting and non-inverting inputs of the op amp U5A respectively, whereas the output is connected to a capacitor C2 through a potentiometer P3. The voltage across the capacitor C2 is fed as the inverting input of the comparator circuit using LM311 whereas Xn+1 forms the non-inverting input of the comparator.

When Xn+1 is greater than Xn, the op amp U5A outputs a high positive power supply voltage and U6, the comparator circuit produces a high output voltage level. Now, the capacitor C2 gets charged through the potentiometer P3 and when the voltage across the capacitor rises to Xn+1, which is in turn fed as the inverting input to the comparator U6, it produces a low output voltage. This results in the appearance of a falling edge at the output of the comparator. Similarly, when Xn+1 is lesser than Xn, the op amp U5A produces a low power supply voltage and U6 produces a high voltage level. Now, the capacitor C2 starts to discharge through the generated negative power supply voltage. When the voltage across the capacitor C2 discharges to Xn+1, the output of the comparator U6 rises to high level, resulting in the appearance of a rising edge at its output. This output of the comparator LM311 is in turn fed as the trigger input to the monostable multivibrator circuit, which results in the generation of the chaotic pulse train. The generated pulse from the MM is used to close the voltage-controlled SPST switch S2, which in turn passes Xn+1 to Xn.

Since both positive and negative triggering are used to trigger the monoshot CD4538BCM IC, a dual precision monostable multivibrator is chosen for the circuit, which has two triggering inputs to enable both rising and falling edge triggering. The time period of the output pulse of the MM is controllable with the use of external resistor RM and capacitor CM, whose values are designed accordingly. By repeating this process, an iterative process can be thus accomplished.

The time interval between the nth and (n+1)th chaotic pulse generated is denoted by Tn. The expression for evaluating Tn can be obtained by using the basic charging equations of a capacitor ‘C’ given by Eq. (3), which represents the voltage VC  across a capacitor ‘C’, with initial voltage Vi, charged from a voltage source Vf through a resistor R [30].

VC=Vf+(Vi − Vf ) e−T/RC(3)

In this case, when Xn < Xn+1, VC= Xn+1, Vi=Xn and Vf=V and when Xn > Xn+1, VC =Xn+1, Vi= Xn and Vf=−VDD. Thus, based on the charging principle of the capacitor C2 with initial voltage Xn, getting charged to V1 through a potentiometer P3, the time interval Tn can be expressed as

Tn=P3C2ln⁡[V1−XnV1−Xn+1] if Xn<Xn+1Tn=P3C2ln⁡[V1+XnV1+Xn+1] if Xn>Xn+1(4)

where Xn and Xn+1 represent the nth and (n + 1)th sequence generated from the Henon map and V1 = 5 V, represents the power voltage. From Eq. (4), it is evident that the time interval between the consecutive pulses is directly proportional to the product of P3 and C2.

4  Simulation and Analysis

Table 1 summarizes the various components and devices used in the circuit along with its values. First step is the setting of an appropriate value of initial condition for the Henon map in the range [0, 1] by suitably adjusting the values of the potentiometer P3, resistor R1 and power supply. In this article, the electronic circuit to generate the chaotic sequences and chaotic pulse train is simulated and analyzed using Multisim 13.0, whereas, the Lyapunov exponents of the generated sequence and the pulse interval between the consecutive pulses are measured using MATLAB R2019b. The simulations were performed for an initial condition of (Xn, Yn) = (0.981, 0) by setting P1=2.5KΩ,R1=10KΩ and V1 = 5 V. The initial condition specified, was chosen randomly so as to be in the range [0, 1].

Figs. 5a and 5b shows the chaotic waveforms generated from the Henon map, realized with bifurcating parameters a = 1.41 and b=0.3, with an initial condition of (0.981, 0). It represents the generated chaotic output waveforms of Xn and Yn, indicated in red and green respectively. The figure clearly shows that the generated waveforms vary chaotically, which can be verified further by computing its Lyapunov exponents.

Figure 5: The chaotic waveforms Xn+1 and Yn+1 generated from the realized Henon map circuit

Fig. 6 shows the Henon attractor obtained from the simulated results Xn+1 and Yn+1 of the Henon map circuit, plotted in Figs. 5a and 5b. It shows the peculiar well-known horseshoe shaped Henon attractor.

Figure 6: The Henon map attractor obtained from the simulated results of Xn+1 and Yn+1

The influence of the potentiometer P3 on the time period of the chaotic series generated is indicated in Fig. 7. It show the waveforms of the Henon map series Xn and the chaotic pulse train Pn, generated by the test circuit for an initial condition X0 = 981 mV where, the value of the potentiometer is changed from 16kΩto32KΩ as shown respectively in Figs. 7a and 7b. Based on the previous discussion, as indicated by Eq. (4), the figure clearly indicates that the time period between the generated chaotic pulses is directly proportional to the value of the potentiometer P3, i.e.; TnαP3, since the amount of pulse generated with P3=32kΩ is twice that is generated with P3=32kΩ. Also, the values of the generated Henon map series is in good agreement with the theoretical values and that whenever the Xn value mutates, the chaotic pulse appears.

Figure 7: The waveforms of Henon map series Xn (red) and chaotic pulse train Pn (green) generated from the circuit for initial condition X0 = 981 mV with (a) P3 = 16 KΩ (b) P3 = 32 KΩ

The chaotic nature of the pulse train generated from the MM can be verified by calculating its Lyapunov exponents [31]. The Lyapunov exponents provide a quantitative as well as qualitative characterization of dynamical systems and are also related to the nearby orbit’s quick divergence or convergence in phase space. The Lyapunov exponents of a discrete-time dynamical system Xn+1 = f (Xn) is expressed as L(X0) = limn→∞1n∑i=0n−1⁡ln⁡|f′(Xi)|. If L  >  0, then {Xn} denotes a chaotic sequence. The Lyapunov exponents of both, the pulse train generated from the MM and the time interval between nth and n + 1th pulse, represented by Tn can be calculated using the method proposed by Wolf et al. [32]. Figs. 8a and 8b show the plot of Lyapunov exponents of the pulse train generated and Tn against the bifurcating parameter ‘a’ calculated from the time series generated from the realized Henon map for an initial condition of (X0,Y0) = (0.982, 0.294) respectively. The figure clearly indicates that the Lyapunov exponents of both Xn and Tn are positive, thus indicating the chaotic nature of the generated pulse train [33].

Figure 8: (a, b) Plot of Lyapunov Exponents calculated for Xn and Tn with the bifurcating parameter from the time series for an initial condition X0 = 0.982 V and Y0 = 0.294 V

5  Comparison with Other Benchmark Algorithms

In the electronic circuit designed, simulated and analyzed in the article, less number of components are utilized to implement the overall circuit thus, when compared to other benchmark circuits, the overall circuit is more concisely realized. The same circuit can be used to realize the Henon map and also to generate two sets of chaotic series and a chaotic pulse train. Moreover, the position and time interval of the generated chaotic pulse are controllable by varying the variable components used in the design. Another important feature of the design is the implementation of a circuit to perform iterative operation. To implement an iterative operation, a reasonable design comprising an edge forming circuit, analog switch and monostable multivibrator is employed which eliminates the need of a periodic clock control in contrast to existing works.

6  Conclusion

In this article, a two-dimensional Henon map based chaotic sequence and pulse generator is proposed. The designed and simulated circuit can be used to realize the Henon map function and also to generate two sets of chaotic sequences and a chaotic pulse train. The proposed circuit is simple since less number of components are used to implement the system. The circuit is designed using various electronic components such as operational amplifier, comparator, monostable multivibrator, analog switch and other passive components. The key feature of this realization is that the generated chaotic pulse train from the monoshot multivibrator can be controlled in terms of pulse interval and also pulse position, by changing the values of various electronic components used in the circuit. The chaotic nature of both, the generated pulse train and the time interval between the consecutive pulses, is verified by its calculated positive Lyapunov exponents and plotted against the bifurcating parameter.

Funding Statement: The authors received no specific funding for this study.

Availability of Data and Materials: The data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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