^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

In this paper, a method for determining the initial value of the hidden attractors in the Chua system is studied. The initial value of the hidden attractors can be calculated quickly and accurately by the proposed method, and the hidden attractors can be found by numerical simulation. Then, the initial values of the hidden attractors are set accurately by digital signal processor (DSP), so as to the circuit realization of the chaotic system with hidden attractors is performed. The results show that the numerical simulation results of Matlab are consistent with the experimental results of DSP.

In the last three decades, chaos has been widely used in neural networks [

Because the domain of attraction of the hidden attractor does not intersect with any small neighbourhood of the equilibrium point, there is no general method to predict the existence of the hidden attractor, so it is of great theoretical and practical significance to study the hidden attractor in the field of machinery and so on [

In 2016, Bao et al. designed the chaotic circuit of the chaotic system and found the hidden attractor of the system by PSIM simulation [

In this paper, we study the method to determine the initial value of the hidden attractors in the Chua system. Its initial value of the hidden attractors can be set accurately by DSP, and the circuit realization of the chaotic system with hidden attractors is performed. The results show that the numerical simulation results of Matlab are consistent with the experimental results of DSP.

The rest of this work is organized as follows. Section

According to the initial value determining algorithm for the chaotic system with hidden attractors in [

Now, system (

Let

Using nonsingular linear transformation

The transfer function of equation (

The transfer functions of system (_{H}_{0})^{–1}. From the equivalence of the transfer functions of systems (

System (

Let

For the infinitesimal number

From equation (

In this way, the initial value of system (_{0} can be calculated as

According to the algorithm in [_{0} = [5.9, 0.3720, −8.4291] and _{00} = [−5.9, −0.3720, 8.4291]. Based on the calculated initial value, the phase diagram is shown in Figure

Phase diagrams of system (

Attractor basin of system (

From Figure

The realization of the chaotic system by hardware circuit is the most common method to verify the design of new chaotic system, including analog circuit and digital circuit. Analog circuit mainly adopts discrete components [

In this part, the chaotic system with hidden attractors is implemented on DSP platform. The block diagram of the working principle is shown in Figure

Working principle for implementing the Chua system with hidden attractors on DSP.

The flowchart of the programming is shown in Figure _{0}, _{0}, _{0}]. To keep the iteration not being affected by data processing, it is necessary to push the results of each iteration into the stack. Data processing includes two steps. Firstly, an appropriate positive number is added to all data to make sure all data is greater than zero. Secondly, all data is rescaled and truncated to make the output adapting the 16 bit digital-to-analog converter.

Flowchart for DSP implementation of the Chua system with hidden attractors.

According to the required iterative equation in Figure

According to the values of _{i}, and _{i} and calculating the values of _{1}, _{2}, _{3}, and _{4}, we can attain the value of _{i+1}. Three equations of system (

We set _{0} = [5.9 0.3720–8.4291] and _{00} = [5.9 0.3720 8.4291], when

Hardware part of DSP implementation.

Hidden attractors by DSP implementation for

Hidden attractors by DSP implementation; initial values: (5.9 0.3720–8.4291) (magenta) and (–5.9 –0.3720 8.4291) (green): (a)

From Figures

In this paper, we calculate the initial values of the Chua system with hidden attractors, find its hidden attractors, and obtain its phase diagram and attractor basin. Since the analog circuit cannot accurately set its initial state and cannot achieve the calculated initial value of the hidden attractor, this paper uses DSP to realize the chaos system with hidden attractors. The results show that the numerical simulation is consistent with the experimental results of DSP, which provides a practical method for the circuit implementation of the chaotic system with hidden attractors. In the next work, we will study the hidden attractor applied to secure communication.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

The work presented in this paper was a collaboration of all authors. Xianming WU contributed the idea and wrote the paper. Weijie Tan and Huihai Wang did the simulation analysis and reviewed the paper.

This work was supported by the National Natural Science Foundation of China (no. 61741104), Science and Technology Foundation of Guizhou Province of China (no. [2018]1115), Science and Technology Plan Project of Guizhou Province of China (no. [2018]5769), and Doctoral Scientific Research Foundation of Guizhou Normal University (2017).