An Exponential Chaotic Oscillator Design and Its Dynamic Analysis

Xiaoyuan Wang,,Chenxi Jin,Xiaotao Min,Dongsheng Yu,,and Herbert Ho Ching Iu,Senior

Abstract—After years of development,chaotic circuits have possessed many different mathematic forms and multiple realization methods. However, in most of the existing chaotic systems, the nonlinear units are composed of the product terms. In this paper, in order to obtain a chaotic oscillator with higher nonlinearity and complexity to meet the needs of utilization, we discuss a novel chaotic system whose nonlinear term is realized by an exponential term. The new exponential chaotic oscillator is constructed by adding an exponential term to the classical Lü system. To further investigate the dynamic characteristics of the oscillator,classical theoretical analyses have been performed,such as phase diagrams,equilibrium points,stabilities of the system,Poincaré mappings, Lyapunov exponent spectrums, bifurcation diagrams. Then through the National Institute of Standards and Technology (NIST)statistical test, it is proved that the chaotic sequence generated by the exponential chaotic oscillator is more random than that produced by the Lü system. In order to further verify the practicability of this chaotic oscillator, by applying the improved modular design method, the system equivalent circuit has been realized and proved by the Multisim simulation.The theoretical analysis and the Multisim simulation results are in good agreement.

I.Introduction

IN nature,chaos is a ubiquitous phenomenon.In the 1960s,the famous meteorologist Lorenz found that the solution of a completely determined set of third-order ordinary differential equations,when selecting a specific range of parameters,becomes very irregular and uncertain[1].Subsequently,many scientists and scholars have conducted extensive and in-depth research on the Lorenz system,and successively discovered many systems with chaotic properties.In the late 1990s,Chenet al.built the Chen system[2].Later,Lüet al.built the Lü system on the basis of the Lorenz system and Chen system[3],and showed the evolution between these two systems.In order to enrich the type of chaotic systems and meet the needs of their application,it is necessary to improve the existing chaotic systems to get the systems with higher nonlinearity and complexity.In a chaotic system, the nonlinear term can affect the complexity of the system[4].The optimizations of chaotic systems are usually made by the following two modifications to the nonlinear term of the existing system: one method is to make simple adjustments to the nonlinear terms without affecting its order [5],[6],another way is to modify the nonlinear term to a higher order nonlinear term,such as changing the product term to an exponential or logarithmic function [7]–[10].

Since chaotic systems are aperiodic,unpredictable and extremely sensitive to the initial conditions,they can produce high-performance pseudo-random sequences and can be widely used in the fields such as secure communication[11]–[14],image encryption[15]–[18],and random number generators[19],[20].The randomness of the sequence generated by the chaotic system is mainly related to the complexity and nonlinearity of the system equations.The higher the complexity and nonlinearity of the system are,the more random the resulting sequence is.Therefore,it is valuable to design a chaotic system with stronger nonlinearity(such as an exponential chaotic system).

In this paper,the Lü system is improved to obtain a more complex chaotic system based on an exponential function.In Section II,the novel exponential chaotic system is proposed and verified by drawing the phase diagrams and calculation of the Lyapunov exponents.In Section III, dynamic characteristics analyses of the new system are carried out,including equilibrium points, stabilities of the system, Poincaré mappings,Lyapunov exponent spectrums,and bifurcation diagrams. Especially in Section III-E, through the NIST test, the randomness of the sequence generated by the exponential chaotic system is quantitatively analyzed.In Section IV,the mathematical model of this chaotic system is transformed into a circuit model and the Multisim circuit simulation is carried out.Finally,some conclusions are drawn in Section V.

II.Establishment of Exponential Chaotic System

The product nonlinear termxyof the third equation of the Lü system is replaced by an exponential nonlinear term exyto construct a new exponential chaotic system.Its mathematical equations can be described as

III.Dynamic Characteristics Analysis of the System

Finally, by fixing the parametersa=20.75,b=6.05 and changing the parametercin the interval[8,16],the Lyapunov exponent spectrum and the bifurcation diagram of the state variablexcan be obtained, which is shown in Fig.6.

Fig.6.(a)Lyapunov exponent spectrum changes with c;(b)Bifurcation diagram of state variable x.

As can be seen from Fig.6, with parametercincreasing,the system gradually changes from periodic state to chaotic state,and finally returns to periodic state.Whenc∈[8.3,13.45]∩[13.95,14.55],the system changes from quasiperiodic state to chaotic state.Finally,aftercis greater than 16.75,the system returns to the periodic state.

In order to further study the influence of parameters,taking the parameteraas an example to illustrate how the system goes from period-1 to chaotic state by period-doubling bifurcations phase diagrams.Whenais equal to 17.6, the system is in period-1 as shown in Fig.7(a). As the parameteragradually increases,the system goes through period-2, periodn,and finally be in chaotic state.

Fig.7.Period-doubling bifurcations phase diagrams.(a) period-1with a=17.6;(b)period-2 with a=18;(c) period-n with a=18.09;(d)chaotic state with a= 20.75.

E. Performance Analysis of Exponential Chaotic Sequences

In order to further analyze the performance of the exponential chaotic system in practical applications, the Statistical Test Suite provided by the National Institute of Standards and Technology of the United States was used for analysis.The version of the Statistical Test Suite used in this analysis is 2.1.2.Due to the data generated by the chaotic system needs to be discretized into a binary sequence in this test.Here,the variablexis selected for generating the binary sequence.So after calculating the sequence of the variablexby the MATLAB simulation,we generate the binary sequence through the following equation:

wherexmaxandxminare the maximum and minimum values of the variablex, round represents the rounding operation,and mod means modular arithmetic.Finally,1000 sets of binary sequences with a length of 1 000 000 were obtained through the MATLAB.The test results are shown in Table I.

TABLE I Randomness Test Of Exponential Chaotic System Sequence and Lü System Sequence

The P-value is the probability that a perfect random number generator would have produced a sequence less random than the sequence that was tested.If the P-value is greater than 0.01 and Proportion is greater than 0.98,it is acceptable that the sequence is random. As can be seen from Table I,the exponential chaotic system passed all fifteen tests, but the Lü system passed only fourteen of them.Also the exponential chaotic system has 9 tests with P-values greater than those of the Lü system in all 15 tests.It shows that the sequence generated by the exponential chaotic system is more random than that produced by the Lü system.

IV.Circuit Realization of the Chaotic System

A.Circuit Design of the Exponential Chaotic System

In order to further verify the effectiveness of the designed system in actual circuits,the corresponding analog circuit is built.Because the maximum value of the system state variablezis greater than 40 in Fig.2,which exceeds the linear dynamic range of the operational amplifier LF347 powered by±13.5 V.So it is necessary to perform proportional compression transformation on each state variable,so we setx→mx,y→my,z→mz,wheremis the state variable proportional compression factor,and takem= 10, the following equation can be obtained:

Then,in order to make the trajectories of the chaotic circuit denser,transformation on time scale should be done.Lett→τ0τ,whereτ0is the time scale transformation factor,and settingτ0equal to 100,we can convert the system equation as follows:

According to(10),the equivalent circuit is designed as shown in Fig.8,where Fig.8(a)is an inverter circuit.The circuits shown in Fig.8(b)and 8(c)both have the functions of addition and integration.In Fig.8(d),operation amplifier U4 and resistors R7,R8 constitute the proportional operation circuit;the diode D1,operational amplifier U5,and resistor R9 form the exponential operation circuit; the operation amplifier U6,capacitor C3,and resistors R10,R11 form the addition circuit and the integration circuit.The outputs of all circuits are as follows:

B. Multisim Simulation Experiment

According to the chaotic circuit diagrams in Fig.8, the Multisim simulations are done by setting suitable parameters and choosing correct components as shown in Fig.9.The simulation results are shown in Fig.10.It can be seen that the circuit simulation results are consistent with the numerical simulation results of the MATLAB.

V.Conclusion

Fig.8.Equivalent chaotic circuit.

Fig.9.Multisim circuit design diagram of exponential chaotic system.

Fig.10.Multisim simulation result graphs:(a) x-y;(b) x-z;(c) y-z.

In this paper,a novel exponential chaotic system is successfully constructed and the dynamic properties of the system are studied by classical dynamic analysis methods and NIST test.The results indicate that by changing the product term to its exponential form in classical Lü system, the randomness and complexity of the presented system has been improved effectively compared with the original system,which indicates that the exponential chaotic system we proposed is more suitable for real applications,such as secure communications,information security,information hiding and chaos cryptography.Moreover,the equivalent circuit of the proposed system has been built to verify the theoretical analysis,and the simulation results are consistent with the theoretical analyses.