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Enzeng Dong, Xiaodong Jiao, Shengzhi Du, Zengqiang Chen, Guoyuan Qi, "Modeling, Synchronization, and FPGA Implementation of Hamiltonian Conservative Hyperchaos", Complexity, vol. 2020, Article ID 4627597, 13 pages, 2020. https://doi.org/10.1155/2020/4627597
Modeling, Synchronization, and FPGA Implementation of Hamiltonian Conservative Hyperchaos
Conservative chaotic systems have potentials in engineering application because of their superiority over the dissipative systems in terms of ergodicity and integer dimension. In this paper, five-dimension Euler equations are constructed by integrating two of sub-Euler equations, which are contributory to the exploration of higher-dimensional systems. These Euler equations compose the conservative parts from their antisymmetric structure, which have been proved to be both Hamiltonian and Casimir energy conservative. Furthermore, a family of Hamiltonian conservative hyperchaotic systems are proposed by breaking the conservation of Casimir energy. The numerical analysis shows that the system displays some interesting behaviors, such as the coexistence of quasi-periodic, chaotic, and hyperchaotic behaviors. Adaptive synchronization method is used to realize the hyperchaos synchronization. Finally, the system passed the NIST tests successfully. Field programmable gate array (FPGA) platform is used to implement the proposed Hamiltonian conservative hyperchaos.
Since the discovery of Lorenz attractor in 1963 , the interesting dynamic behavior of chaotic attractors has attracted intensive attention, with various mathematical analyses. Conventional analyses on the chaotic system include the determination of Lyapunov exponents (LEs), bifurcation diagram , phase portrait, ultimate boundary estimation, and topological horseshoe analysis [3, 4], which illustrate the chaotic state intuitively. LEs are commonly used as an indicator of chaotic systems. The bifurcation diagram focuses on the evolution of the dynamics of a chaotic system when parameters or initial values are changed. The phase portrait describes the phase space trajectory of a chaotic system. The ultimate boundary estimation and topological horseshoe analysis [3, 4] reveal the abundant characteristics of a chaotic system.
Generally, chaotic systems can be categorized into dissipative or conservative chaotic systems. A dissipative chaotic system (DCS) owns strange attractor that has sensitive dependence on initial conditions. A conservative chaotic system (CCS) does not even have any attractor with integer dimension. The ergodicity of CCS is usually greater than DCS. According to the sign of the coefficients in the dissipation term, the DCS can be further divided into two types: Rayleigh and non-Rayleigh DCSs [5, 6], as there must be at least one positive dissipation term in a non-Rayleigh DCS . There are two types of CCSs too. A Hamiltonian CCS (HCCS) meets conditions of zero-sum LEs, and the Hamiltonian energy and the phase space volume are both conservative, while a non-Hamiltonian CCS (non-HCCS) only meets zero-sum LE condition, such as Sprott A system in  and Cang-case B system in . HCCSs are further categorized to traditional HCCS (traditional-HCCS) [9, 10] and generalized HCCS (generalized-HCCS) . DCS has received intensive attention but much little for CCS. In the early stage of chaos research, there are few reports on the mechanism of chaos. Recently, analyses on energy cycling in chaotic systems have made some progresses. For instance, Vinicio Pelino, Filippo Maimone, and Pasini described the energy cycle of Lorenz attractor . The energy cycle of the Qi four-wing chaotic system  was explained by using the system entropy and the Casimir function as the kernel of Lie–Poisson bracket. In this process, the concerned system was converted to Kolmogorov form containing the conservative term, dissipative term, and an external force [5, 12]. As a measuring index of the orbital mode, the rate of change of Casimir energy, Casimir power, is analyzed in detail. The conservative term is interpreted by Euler equations. As a basic equation in inviscid hydrodynamics, Euler equation gives a Hamiltonian description in 3D form. However, higher-dimensional Euler equations (such as 4D and 5D) are essential to be studied.
In research of chaotic systems, an important and useful implementation is encryption algorithms [13–16] because of the complex properties, such as extremely sensitive dependency on initial conditions, topologically mixing and density of periodic orbits, broadband, pseudo-randomness, and white-noise-like phenomenon . Since the first four-dimensional hyperchaotic system  was proposed, researchers found that a hyperchaotic system has potential in engineering applications, especially in control [18, 19], encryption [20, 21], communication , synchronization , optical system , and biological network . Research on hyperchaotic systems has become a hot topic. One of the most important phenomena in these dynamical systems [26–28] is the multistability occurring in physics, chemistry, biology , economics , and nature. There are two kinds of systems with multistability including hidden attractors  and infinite attractors [32, 33]. Both display complex dynamical behavior such as the coexistence attractor [11, 34]. All CCSs do not have attractors, but some have line equilibrium which is referred as hidden attractors. Therefore, it is interesting to test whether CCSs can show multistability behavior as well. For application, chaos can be controlled and synchronized, usually including designing a controller to stabilize the chaotic system and tracking the chaotic system. Chaos synchronization is of vital significance to the practical application of chaos. Important works in chaos synchronization for the chaotic system and its applications in control and tracking have been done recently and in the past [35, 36]. Design and implementation of the pseudo-random digital signal generator is one of the most important chaos-based application. With the advantages of digital integrated circuits, superior computing power, and reconfigurable designs , FPGA is considered as one of the most suitable platforms to implement a chaotic system.
While intensive research studies were made on 3D and 4D chaotic systems, this paper focuses on the vital importance to expand the research to higher-dimensional systems, for instance, 5D systems, which display rich dynamics and lay an important foundation for higher-dimensional chaotic systems’ study.
The rest of this paper is listed as follows: in Section 2, five 5D Euler equations are constructed, and a family of HCCSs are proposed; in Section 3, the nonconservation property of the proposed HCCS and dynamics are analyzed in detail by numerical simulation; in Section 4, adaptive synchronization is achieved for system ; in Section 5, FPGA digital development platform is used to generate the pseudo-random number generator; and Section 6 concludes the paper.
2. Modeling of HCCS
2.1. Model of 5D Euler Equations
Euler equations govern the rotation of a rigid body and control the motion of an inviscid fluid, which are also applicable to incompressible fluid. 5D Euler equations are imperative to satisfy the Lie–Poisson structure. Thus, from the view point of the 3D free rotational rigid body, 5D Euler equations governing the 5D rigid body or fluid systems can be constructed satisfying the Lie–Poisson bracket for the generalized Hamiltonian systems or Kolmogorov systems. In , six types Euler equations are proposed but the modeling work just considering the four-dimensional case. On the contrary, only the system dimension of a conservative chaotic system is greater than or equal to 5, and the dynamics of the system may be hyperchaotic state according to the requirements of LEs.
To construct the 5D Euler equations, the following 3D rigid subbodies are considered: subbody defined in the space spanned from the axes ijk (taking the values of 123, 124, 125, 145, 245, and 345). Suppose there is a 5D rigid body with five axes, with the principle moment of inertia , , and angular momentum , is the angular velocity, i = 1, 2, 3, 4, 5. Then, the six 3D rigid subbodies can be extended to 5D by leaving another two dimensions uncoupled, and one gets the generalized 5D sub-Euler equations; these six 5D sub-Euler equations can be used to construct the five types of 5D Euler equations as summarized in Table 1.
As an example of Table 1, by coupling subbody and subbody with the third common axis, the 5D Euler equation of body is demonstrated in a Hamiltonian vector field form as equation (1). Bodies , , , and can be developed in the similar way. The general form of the five 5D Euler equations is , where .withand its cross-product form is
The Hamiltonian energy of these 5D systems is
As a significant physical quantity, similiar to the enstrophy or potential vorticity in the context of hydrodynamics, the Casimir function, , is a valid way in globally describing a dynamical system and analyzing stability conditions. The Casimir function is defined as the kernel of the Lie–Poisson bracket :in which is the structural matrix of a generalized Hamiltonian system, which meets , i.e., , .
For a Hamiltonian system with a constant of the motion, one gets . The Casimir energy is conservative when there is no dissipative torque and external torque [5, 39]. The Casimir function (energy) of a generalized 5D Euler equations can be defined as
The rate of change of the Casimir energy is called the Casimir power , which is defined asin which is an antisymmetric matrix, which represents the energy conservative part of a system.
Remark 1. The constructed 5D Euler equations (i = 1, 2, 3, 4, 5) are both Hamiltonian and Casimir energy conservative.
The proposed five 5D Euler equations satisfy the vector field shown in equation (1), which provides the symplectic structure. Because the proposed 5D Euler equations are conservative for both the Hamiltonian and Casimir energy, they cannot produce chaos. However, they can be used as the basic framework to construct the HCCS.
2.2. The Proposed 5D HCCS
To generate chaotic behavior, conservation of the proposed generalized 5D Euler equations has to be broken. Replace one zero element by a in the upper triangle and −a in the lower triangle of , respectively. Because still keeps the skew-symmetric form, the conservations of Hamiltonian are preserved, but Casimir energy is broken. Correspondingly, the five systems generate a family of Hamiltonian conservative chaotic systems:
As an example, for , one getswith is a constant. The resulting system is a HCCS.
Force and energy analysis can be used to explain the mechanism underlying chaos and its dynamic states . For system , the Casimir energy is not conservative, which means there must exist an external torque that breaks the conservation. According to Kolmogorov–Arnold–Moser perturbation theorem , when a Hamiltonian system whose H(x) function is perturbed by the inclusion of an interaction term , the coupled Hamiltonian system probably generates conservative chaos with Hamilton function because of the energy exchange between the two Hamiltonian functions (H(x) and ).
Consider as a perturbation term, which represents a kind of external torque. According to equation (7), the Casimir power of system is
The term of system is conservative in both Hamiltonian and Casimir energy. Therefore, this term is an inertial torque, which can be a kind of fictitious torque generated by a free rotational rigid body without external torque [5, 38]. But, the term is a nonconservative torque and leads to the loss of Casimir energy conservation. Rewrite system in vector form to illustrate the conservative and nonconservative torque aswhere the first vector term on the right is the initial torque, and the second vector term is the external term causing the energy exchanges between dissipative and supplied energy.
3. Dynamic Analysis on the Proposed 5D HCCS
For simplicity, we only analyze system :
System has six adjustable parameters: (i = 1, 2, 3, 4, and 5) and a having influences on the system dynamic motion. The initial values determine Hamiltonian equation (4) when is fixed. From equation (12), parameters and determine the change rate of Casimir energy, which will impact the degree of chaos with fixed .
3.1. Equilibrium Point Analysis on System
With , the equilibrium points of system are (p, 0, 0, 0, 0), (0, 0, r, 0, 0), (0, 0, 0, 0, q), and (p, 0, 0, 0, q), in which . Thus, the equilibrium points of system are in the form of three lines and one plane. Substituting equilibrium points (0, 0, 0, 0, 0), (p, 0, 0, 0, 0), (0, 0, r, 0, 0), (0, 0, 0, 0, q), and (p, 0, 0, 0, q) into the Jacobian matrix and calculating the characteristic polynomial, one getsrespectively. Let these characteristic polynomials equal to zero; then, we obtain the eigenvalues of these polynomials, as shown in Table 2.
System has nonhyperbolic equilibrium points, which are rare in chaotic systems . Another finding from Table 2 is that different kinds of equilibrium points all have zero eigenvalues, so the equilibrium points of are all unstable (exponential) .
3.2. Numerical Investigations
Let and initial values . When a = 0, system becomes , it is both Hamiltonian energy (H(t) = 4.595) and Casimir energy (C(t) = 0.385) conservative with a periodic trajectory (the red curve shown in Figure 1), and the Casimir power is zero (the red line in Figure 2). When a = 0.1, is still Hamiltonian conservative (H(t) = 4.595) but not Casimir energy, and the Casimir power is not zero as the blue curve shown in Figure 2. The orbit becomes chaotic as shown by the blue trajectory in Figure 1.
Usually, the bifurcation diagram evolution process is “source—periodic orbit—double periodic orbits—multiperiodic orbits—chaos.” However, system does not. It enters the chaotic orbit almost immediately when the parameter a 0, as shown in the bifurcation diagram in Figure 3(a), ; correspondingly, Figure 3(b) shows the Lyapunov exponent spectrum as a increases, in which there are two LEs greater than 0.
Remark 2. When the Casimir energy is conservative, the system may be either periodic or quasi-periodic states.
Considering with a 0, we have ; hence, the system remains conservative both in the Hamiltonian and the Casimir energy with H(t) = 2.72 and C(t) = 0.385, where , , and the system is quasi-periodic state with five zero Les; the corresponding phase portrait and Poincare section are shown in Figure 4.
3.3. Coexistence of System
The dynamics of classical chaotic systems contain periodic motion, quasi-periodic motion, chaos, and hyperchaos, which can be verified by LEs [41, 42]. A hyperchaotic system is characterized by the presence of two or more positive LEs [41, 43]. For a dynamical system with conservative flows, the sum of all LEs must be zero . There must be two positive LEs for a 5D autonomous conservative hyperchaotic system.
Basin of attraction is an important tool for analyzing the dissipative coexisting attractors [44, 45], which change with initial values. Various initial values can be used to get full information about the typical regimes and their localization; as the same idea of the basin of attraction, the dynamical evolution map is used for this purpose.
The dynamical evolution map is generated for system to study the coexistence phenomenon. Fixed system parameters , the sampling period = 0.1 s, and changed the third and fourth initial values, i.e., (0.1, 0.5, , 0.1), where and , the step of the initial values changing is 0.05. The dynamical evolution map is shown in Figure 5, which shows , and of each point with , respectively. Various coexisting orbits are highlighted with different colors in Figure 4. The values of region A are close to zero. The rest area is denoted as region B, where the values of LEs are greater than zero.
Comparing the dynamical evolution map, one finds that the LEs are symmetric about their 0 LE, and the dynamics of system switch among hyperchaos, chaos, and quasi-periodic motion. When the initial values of and are located in region A, all of the five LEs are approximating to zero, which indicates the system undergoes a quasi-periodic motion. When the initial values of and are located in region B, the largest LE is greater larger than zero, and the second one is greater than or equal to zero, which means the system is chaotic or hyperchaotic. As a demonstration, Figures 6 and 7 show the coexistence of quasi-periodic state with , , and hyperchaotic state (with the five LEs 4.4358, 0.1288, 0, −0.1288, and −4.4358) for and .
When system evolves into the hyperchaotic state, the system is still Hamiltonian conservative but not in Casimir energy. The phase portraits with Hamiltonian and Casimir energy represented by color are shown in Figures 8(a) and 8(b) which confirm this point. Therefore, the break of the conservation of Casimir energy is an effective indicator of the chaos-generating mechanism.
4. Adaptive Synchronization of System
The property of chaotic synchronization gives the ability to exercise control over the dynamics of the chaotic system. As a result, such systems are of vital importance for secure communication. In this part, the adaptive synchronization method is used to realize the synchronization. The adaptive synchronization method has a mature theoretical and application basis, which is widely used in the electronic field [36, 46, 47]. This method has the following advantages: small synchronization error, the control intensity can be adjusted adaptively according to the error, and if it is applied to chaotic communication, the security is strong.
Relabeling system as the master system, one getswhere are state variables and are parameters. Consider the slave system aswhere are state variables and are controllers to be configured. The synchronization errors are determined by , and the adaptive controllers are set as , in which .
Lyapunov function is constructed as the following form:and one getsin which , are the upper bound of the corresponding state variables, , and is the matrix corresponding to quadric form . Hence, to guarantee the asymptotic stability of synchronization errors, must be a positive definite matrix, that is to say, is a negative definite quadratic form, which means exponentially as times goes on.
Set the control parameters for master system (22) and slave system (23) as , a = 7.5, b = 30, the gains , and the initial values are , under which the master system keeps hyperchaotic states, , respectively. By calculation, matrix has five positive characteristic roots, and is a negative definite quadratic form. To test the validity of synchronization, simulations are implemented, and the first 5 seconds are time sequences of master and slave systems without synchronization. From the 5th second, synchronization is achieved, see Figures 9(a)–9(c); for the first 5 seconds, and conduct different time evolution, and after synchronization is achieved, and carry on synchronous evolution, and other two variables proceed the similar evolution. Figure 9(d) depicts the time history of , which converge to 0 quickly, and the state variables move to be synchronized.
5. NIST Test and FPGA Implementation
5.1. NIST Test of System
The National Institute of Standards and Technology (NIST) provides 15 statistical tests for random or pseudo-random generators, namely, the SP800-22 standard. Widely used in the test of pseudo-random sequences, the SP800-22 standard is considered as a criterion for evaluating the statistical performance of pseudo-random sequences [48, 49]. The tests comprehensively analyze the performance of the pseudo-random sequence. A sequence passes the test only when the following conditions are satisfied: all values are greater than the significance level, = 0.01; the relevant proportions are within the acceptance interval ; and the distribution of values obeys the uniformity.
From Table 3, all values obtained in the statistical tests are greater than the significance level, , so condition one is met. The relevant proportions lie within the acceptance interval , and hence, the second condition holds. In addition, the distribution of values must pass the uniformity test. For simplicity, we consider the nonoverlapping template (Test No. 8 in Table 3) as an example. The distribution of values is examined to validate the uniformity which can be visually illustrated using the histogram. Figure 10 shows the distribution of values for the nonoverlapping template which is uniform. Other 14 tests obtained a similar uniformity. Therefore, the third condition is also satisfied. The conclusion is that the proposed Hamiltonian conservative hyperchaotic system is suitable as a pseudo-random generator.
5.2. FPGA Implementation for System
Although some important works have been implemented on FPGA [37, 50], hardware implementation of conservative chaotic systems is more difficult than common dissipative chaotic systems since the former is highly sensitive to initial conditions and computational errors. In this section, we perform the implementation of system , as an example, based on the FPGA platform. In order to accommodate digital computers, the first-order difference algorithm is used to discrete the system . For system with , one obtains
The discrete model was built by DSP-builder in MATLAB (R2015b). Then, system (24) is solved by the fourth-order the Runge–Kutta method in Matlab/Simulink/DSP-builder, with the fixed step of , initial values set as Section 3.3, and sampling time of . The system models in equation (24) are converted to the VHDL code (Verilog hardware description language), synthesized, and compiled to the RTL (register transfer level) code.
To compare the results of computer simulation with digital circuit implementation, the phase portraits of system are shown in Figures 11 and 12. It shows that the phase portraits of system implemented by FPGA are roughly consistent with the computer simulation (shown in Figure 5), which means the conservative hyperchaotic system is implemented in the digital circuit with an acceptable error level.
This paper constructed a family of 5D Hamiltonian conservative hyperchaotic systems. Some interesting properties are revealed by theoretical and numerical analyses. The paper started from extending the 3D Euler equations of rigid bodies to generalized 5D sub-Euler equations. Five 5D Euler equations were then obtained by combining any two of the generalized 5D sub-Euler equations that with one common axis. These 5D Euler equations were proved to be both Hamiltonian and Casimir energy conservative, which can be regarded as the conservative term in a Kolmogorov system or a 5D generalized Hamiltonian system because of their antisymmetric structure matrix. So, these generalized 5D Euler equations are useful to study the dynamics of inviscid fluid and quantum mechanics and rigid body dynamics systems.
The mechanism of the proposed 5D Hamiltonian conservative chaotic systems was found by analyzing the Casimir energy and the Casimir power, which is an effective indicator of chaos generating. Using the dynamical evolution map, the coexistence of quasi-periodic and hyperchaotic was revealed, which showed the abundant dynamic characteristics of the proposed 5D Hamiltonian conservative chaotic systems. By the adaptive synchronization method, the hyperchaotic system was synchronized effectively.
To validate the application value of the proposed systems, NIST tests on system were performed to verify the feasibility as a random number sequence generator. The system is implemented based on FPGA digital development platform, which was a contributing work for this field due to the difficulties of hardware implementation of conservative hyperchaos. These proposed systems are expected to be further helpfully applied to secure communication, neural network, and economics.
The data used to support the conclusions of this article can be computed by the relative equations and parameters given in the article using ODE45 solver in Matlab.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Enzeng Dong and Xiaodong Jiao contributed equally to this work.
This work was partially supported by the Natural Science Foundation of China (no. 61873186) and the Natural Science Foundation of Tianjin (no. 18JCYBJC87700).
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