Exploring the dynamics feature of robust chaotic system is an attractive yet recent topic of interest. In this paper, we introduce a three-dimensional fractional-order chaotic system. The important finding by analysis is that the position of signal
Over the past few decades, the dynamics and property of chaotic system have been extensively studied from different points of views as an active topic [
Although fractional calculus has a history of more than 300 years, it was not really applied to physics, life sciences, and psychology until the last decade [
In this paper, we attempt to explore some new dynamics properties of robust chaotic system by constructing a three-dimensional fractional chaotic system and to further consider the synchronization problem. Analysis of the derived system shows that the parameter
The introduced fractional system is written in the form
In system (
The corresponding characteristic equation for any equilibrium point
Specially, when selecting the parameter set
With the aid of stability theory of commensurate fractional system [
(a) 3D phase portrait; (b) Poincaré map with
Our analysis found that the introduced system has the robust chaos of constant Lyapunov exponents with the variation of parameters
The linear transformation of
Therefore, the control parameter
We substitute the nonzero equilibrium point
Since (
(a), (b), and (c) Bifurcation diagram and (d) Lyapunov exponent spectrum versus parameter
When introducing the linear transformation of
Therefore, the position of signal
(a), (b), and (c) Bifurcation diagram and (d) Lyapunov exponent spectrum versus parameter
Influence of
In this section, some concepts and techniques are recalled for the stability analysis of fractional system.
When function
Let
Let
Considering the following fractional-order system with the Caputo definition
Furthermore, if there exists another scalar class-
The partial projective synchronization of the proposed system is studied here by taking the advantage of the property of amplitude modulation.
Let system (
The state variables of systems (
The synchronization errors are set as
For the drive system (
Let us propose the Lyapunov function
It requires
Therefore, we finally obtain inequalities (
The appropriate selection of parameters
And for the sake of simplicity, we only consider the maximal power consumption of synchronous controller, as below
Therefore, the optimal synchronization region will be the coupling-parameter region (
In the numerical analysis, we choose the parameter values as
The distribution of the maximal power consumption of synchronous controller.
The synchronization result with
Partial projective synchronization with
Partial projective synchronization with
Select the drive system (
And when considering the property of position modulation, the errors of partial phase synchronization are expressed as
For the drive system (
Here we skip the proof process for brevity since it resembles the one of Theorem
Likewise, to effectively evaluate the optimal synchronization region, we consider the distribution of coupling parameters by defining the maximal power consumption of synchronous controller, as follows:
In the numerical analysis, we choose the parameter values as
The distribution of the maximal power consumption of synchronous controller.
As the examples to explain our analysis, we select two sets of coupling parameters (
Partial phase synchronization with
Partial phase synchronization with
Partial phase synchronization with
Robust chaotic system is a recent research interest yet a promising candidate for signal processing, image encryption, chaotic radar, and chaotic communication. Therefore, it is worthwhile to explore the dynamics feature of robust chaotic system. In this paper, we introduced a robust fractional-order chaotic system and revealed the significant dynamics of position modulation and amplitude modulation.
The linear control scheme is designed to realize the partial projective synchronization and partial phase synchronization, by considering the property of amplitude and position modulation. The distribution of optimal synchronization region in the control-parameter space is evaluated by searching the minimum power consumption of the linear controller. Numerical experiments are executed to confirm the theoretical analysis. Since the relation of synchronization variables only depends on the system parameters, it is not easy to attack and accurately reconstruct the drive system. Therefore, it could be significant in secure communication for masking signals.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China under Grant no. 51577046, the State Key Program of National Natural Science Foundation of China under Grant no. 51637004, the National Key Research and Development Plan “Important Scientific Instruments and Equipment Development” under Grant no. 2016YFF0102200, Equipment Research Project in Advance under Grant no. 41402040301, the Research Foundation of Education Bureau of Hunan Province of China under Grant no. 16B113, and Hunan Provincial Natural Science Foundation of China under Grant no. 2016JJ4036.