In this paper, the Adomian decomposition method (ADM) is applied to solve the fractional-order system with line equilibrium. The dynamics of the system is analyzed by means of the Lyapunov exponent spectrum, bifurcations, chaotic attractor, and largest Lyapunov exponent diagram. At the same time, through the Lyapunov exponent spectrum and bifurcation graph of the system under the change of the initial value, the influence of fractional order
Three hundred years ago, fractional-order calculation was proposed as a classical mathematical problem. Fractional calculation has no practical background and has not been applied in engineering. Therefore, researchers and scientists are not interested in fractional research. Recently, it have been found that many engineering and physical systems, electronic systems, etc. exhibit their own fractional-order characteristics, and the fractional order can more accurately reflect natural phenomena, such as material memory and damping characteristics [
In the numerical calculation of fractional chaotic systems, namely, discretization of fractional chaotic systems, many scholars have made some achievements based on frequency-domain method (FDM) [
On the other hand, hot research on chaotic systems mainly focuses on chaos control [
In this paper, we propose a new fractional-order chaotic system with line equilibrium. Discretization of the proposed system is performed using Adomian decomposition scheme. The rest of the paper is organized as follows. In Section
Reference [
A 3D integer-order chaotic system is defined by
The characteristic equation of matrix is
The eigenvalue at the line equilibrium point
If
If
If
When
Phase diagrams in different projections of chaotic systems (
Fix
Bifurcation diagram and Lyapunov exponents spectrum of system (
Fix
The chaos diagram of the largest Lyapunov exponents with
System (
The initial condition is
Thus, the solution of systems (
When
Phase diagrams in different projections of fractional-order chaotic systems from initial values (1, 2, 3), red, and (1, 2, −1). (a) Phase portraits in the (
Based on the ADM solution of the fractional‐order chaotic system, we focus on the analysis of the influence of parameters on the system. In this paper, the system bifurcation diagram and Lyapunov exponent are used to analyze the dynamic system. The bifurcation diagram is obtained by using the maximum value of state variables. In this section, the influence of three parameters on the system is studied.
Fix
Bifurcation diagram and Lyapunov exponents spectrum of the fractional-order chaotic system (
Fix
Bifurcation diagram and Lyapunov exponents spectrum of system (
For further research, influence of coexistence of attractors in fractional derivative
Phase diagrams in different projections of chaotic systems from initial values (1, 2, 20) and (1, 2, 3). (a) Phase diagrams in integer-order system (
The fractional chaotic system (
Dynamical properties of the fractional-order system with
Phase diagrams of the systems with different
Secondly, bifurcation diagrams of the fractional-order system (
QR decomposition method [
Here,
Fix
Lyapunov exponents spectrum of fractional order with
Fix
By calculating the energy distribution in the Fourier transform domain and combining it with the Shannon entropy, spectral entropy is obtained [
Sample entropy (SampEn) [
In general, for the specific calculation method of sample entropy of time series {
Fix
Fixed
Fix
Complexity is consistent with Lyapunov exponent, but Figure
Complexity of the system with parameter varying. (a)
Fix
C0 complexity in the
The complexity of the system is calculated by the time sequence of a certain state variable. Compared with Lyapunov exponent calculation, the calculation of complexity is faster and saves resources, but there are also some errors in complexity, especially when SE complexity is in a periodic state.
In this paper, the accurate approximate solution for the fractional-order system with line equilibrium is obtained based on the Adomian decomposition method. Dynamical behaviors of the systems are analyzed using the chaotic attractor, bifurcation diagram, Lyapunov exponent spectrum, and largest Lyapunov exponent diagram. Chaotic range and periodic windows are determined. Both the system parameter and the fractional order can be taken as bifurcation parameters, which shows that the fractional-order system has more complex and abundant dynamics than its integral-order counterpart. Besides, integer-order systems do not have the phenomenon of attractors coexistence, while fractional-order systems have it, so fractional-order systems have better effects when applied to communication security systems.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This work was supported financially by the Key Research and Development Plan of Shaanxi Province (No. 2018GY-091), the Special Fund for High Level Talents of Xijing University (XJ19B03), the Major Scientific and Technological Innovation Projects of Shandong Province (Grant No. 2019JZZY010111), the Natural Science Foundation of Shandong Province (Grant No. ZR2017PA008), the Key Research and Development Plan of Shandong Province (Grant No. 2019GGX104092), and the Science and Technology Plan Projects of Universities of Shandong Province (Grant No. J18KA381).