CN104821797A - Simple Chua's chaotic circuit realized by bridge diode pair - Google Patents

Abstract

The invention discloses a simple Chua's chaotic circuit realized by a bridge diode pair. The simple Chua's chaotic circuit comprises the components of a resistor G, a capacitor C1, a capacitor C2, an inductor L and a diode pair, wherein the anode end and the cathode end of the resistor G are respectively connected with the anode end and the cathode end of the capacitor C1 for forming an active RC oscillating circuit. The inductor L is parallelly connected with the capacitor C2 for forming an LC oscillating circuit. The diode pair is bridged between the LC oscillating circuit and the active RC oscillating circuit. The simple Chua's chaotic circuit realized by the bridge diode pair can generate a single-scroll chaotic attractor in which double scrolls and branches coexist through adjusting circuit parameters, thereby forming the simple Chua's chaotic circuit and performing a relatively large promotion function on development of a chaotic system.

Description

Simple Chua's chaotic circuit realized by bridge diode pair

Technical Field

The invention relates to a method and a device for realizing a simple Chua's chaotic circuit by a bridge diode pair.

Background

The theory of chaos is a subject that has developed vigorously over the last fifty years. The chaos phenomenon is ubiquitous and it penetrates almost every corner of the human society. A large number of researches show that the chaos has wide application prospects in the fields of bioengineering, mechanical engineering, electronic engineering, data encryption, secret communication, dynamic analysis and protection of power grids and the like. Early chaotic system generation models, such as the Lorenz atmospheric turbulence equation, Logistic population model, zeiss chaotic circuit, and the like. In general, the simplicity of the physical implementation of the chaotic circuit and the complexity of the topological structure of the attractor generated by the chaotic circuit are two important directions for developing the research of the chaotic circuit. The Chua's circuit is a very simple non-linear chaotic circuit named after the last name of Malus zettata, a chinese scientist at berkeley division, california, usa. Chua's circuit topology: the chaotic circuit mainly comprises an inductor, two capacitors, a linear resistor and a nonlinear resistor, has a simple structure, and can generate complex chaotic characteristics, so that the chaotic circuit becomes a main object of research in the chaotic field.

The current Chua's circuit mainly comprises a classical Chua's circuit, an improved Chua's circuit and a standard Chua's circuit. The Chua's diode is a nonlinear negative resistor with a piecewise linear function form and plays a key role in the generation of a chaotic attractor by the Chua's circuit. Therefore, the invention analyzes the action mechanism of the piecewise linear function, further provides a simple Chua's chaotic circuit realized by adopting a diode pair to realize piecewise linear action and designs a bridge diode, thereby playing a promoting role in simplifying the Chua's chaotic circuit and realizing chaotic application research.

Disclosure of Invention

The invention aims to solve the technical problem of how to provide a method for realizing a simple Chua's chaotic circuit by a bridge diode pair, wherein the circuit adopts the coupling of the bridge diode pair between a passive LC oscillating circuit and an active RC oscillating circuit, thereby realizing a novel Chua's chaotic circuit.

In order to solve the technical problem, the invention provides a simple Chua's chaotic circuit realized by a bridge diode pair, which has the following structure:

a simple Chua's chaotic circuit realized by a bridge diode pair comprises a negative resistance-G and a capacitor C₁Capacitor C₂An inductor L, a diode pair; wherein the positive and negative terminals of the negative resistor-G are respectively connected with the capacitor C₁The positive and negative terminals of the active RC filter are connected (respectively marked as a terminal and b terminal) to form an active RC filter; one end of inductor L and C₂The positive terminal of (c) is connected (denoted as terminal c); the other end of the inductor L and C₂Is connected to the negative terminal (denoted as terminal d); inductors L and C₂The LC filters are connected in parallel to form an LC filter; the diode pair is bridged between the terminals a and c. The diode pair includes: diode D₁Diode D₂(ii) a Wherein the diode D₁And diode D₂The positive electrode end is connected; diode D₁Anode terminal and diode D₂The negative electrode end is connected. The negative resistance-G realizing circuit comprises an adder and a resistor R₁Resistance R₂A resistor R, wherein the positive input end and the negative input end of the adder are respectively connected with the resistor R₁And a resistance R₂Are connected (denoted as e and f, respectively); the output end of the adder is respectively connected with the resistor R₁And a resistance R₂The other end of (a) is connected (denoted as end g); one end of the resistor R is connected with the end e, and the other end of the resistor R is connected with the end b. The b terminal is grounded.

The Chua's chaotic circuit designed by the invention contains three state variables, namely a capacitor C₁Voltage v across₁Capacitor C₂Voltage v across₂Flowing an inductor L current i_L。

The invention has the following beneficial effects:

the bridge diode provided by the invention can generate a single-scroll chaotic attractor with coexisting double scrolls and bifurcations by adjusting the circuit parameters of the simple Chua's chaotic circuit, so that the simple Chua's chaotic circuit becomes a simple Chua's chaotic circuit and has a great propulsion effect on the development of a chaotic system.

Drawings

In order that the present disclosure may be more readily and clearly understood, reference is now made to the following detailed description of the present disclosure taken in conjunction with the accompanying drawings, in which:

FIG. 1 is a simple Chua's chaotic circuit implemented with a bridge diode pair;

when the circuit element parameter R in fig. 2(a) is 1.55k Ω, the simple Chua's chaotic circuit is in v₁-v₂Phase rail diagram on the plane;

when the circuit element parameter R of fig. 2(b) is 1.90k Ω, the simple Chua's chaotic circuit is at v₁-v₂Phase rail diagram on the plane;

when the circuit element parameter R of fig. 2(c) is 2.15k Ω, the simple Chua's chaotic circuit is at v₁-v₂Phase rail diagram on the plane;

when the circuit element parameter R of fig. 2(d) is 2.35k Ω, the simple Chua's chaotic circuit is at v₁-v₂Phase rail diagram on the plane;

in fig. 3(a), when R is 1.55k Ω, v is experimentally measured₁-v₂Phase rail diagram on the plane;

in fig. 3(b), when R is 1.90k Ω, v is experimentally measured₁-v₂Phase rail diagram on the plane;

in fig. 3(c), when R is 2.15k Ω, v is experimentally measured₁-v₂Phase rail diagram on the plane;

in fig. 3(d), when R is 2.35k Ω, v is experimentally measured₁-v₂Phase rail diagram on the plane;

state variable v when circuit element parameter R of FIG. 4 changes₁A bifurcation diagram of (1);

FIG. 5 shows a Lyapunov exponential spectrum with variations in the circuit element parameter R;

Detailed Description

Mathematical modeling: diode D as described in the circuit of fig. 1_kThe constitutive relation of (A) can be described as

<math> <mrow> <msub> <mi>i</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>I</mi> <mi>S</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow> <mn>2</mn> <msub> <mi>&rho;v</mi> <mi>k</mi> </msub> </mrow> </msup> <mo>-</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </math>

Wherein k is 1,2, ρ is 1/(nV)_T),v_kAnd i_kRespectively, through a diode bridge D_kVoltage and current of_SN and V_TRespectively representing diode reverse saturation current, emission coefficient and thermal voltage.

According to the circuit shown in fig. 1, the constitutive relation of kirchhoff's voltage and current law and circuit elements is used to obtain:

$C_1\frac{{\mathrm{dv}}_1}{\mathrm{dt}}=i+{\mathrm{Gv}}_1$

$C_2\frac{{\mathrm{dv}}_2}{\mathrm{dt}}=-i_L-i$

(2)

$L\frac{{\mathrm{di}}_L}{\mathrm{dt}}=v_2$

wherein,ρ＝1/(2nV_T)，v＝v₂-v₁(I_Sn and V_TRepresenting diode reverse saturation current, emission coefficient and thermal voltage, respectively). In FIG. 1, two diodes are 1N4148 with parameter I_S＝6.89nA,n＝1.83,V_T＝26mV。

With reference to equation (2), a set of equations for the dynamics of the simple Chua's chaotic circuit is obtained, and is described as follows:

$\frac{{\mathrm{dv}}_1}{\mathrm{dt}}=(i+{\mathrm{Gv}}_1)C_1^{-1}$

$\frac{{\mathrm{dv}}_2}{\mathrm{dt}}={(-i_L-i)C}_2^{-1}$

(3)

$\frac{{\mathrm{di}}_L}{\mathrm{dt}}=v_2{L}^{-1}$

numerical simulation: using the MATLAB simulation software platform, a numerical simulation analysis can be performed on the circuit described by equation (3). And solving the system equation by adopting a Runge-Kutta (ODE45) algorithm to obtain a phase-track diagram of the circuit state variable. Typical circuit parameters: r1.55 k omega, L45 mH, C₁＝80nF、C₂When the initial state value of the circuit state variable is (0.0001V, 0A) at 150nF, the circuit can generate a chaotic attractor with a complex topology, and the corresponding MATLAB numerical simulation phase trajectory diagrams in different phase planes are shown in fig. 2, wherein fig. 2(a) shows the corresponding circuit element parameter R at V1.55 k Ω₁-v₂Projection on plane, fig. 2(b) is a graph showing the value of v when the circuit element parameter R is 1.90k Ω₁-v₂Projection on plane, fig. 2(c) is a graph at v when the circuit element parameter R is 2.15k Ω₁-v₂Projection on plane, fig. 2(d) is a graph showing a circuit element parameter R of 2.35k Ω at v₁-v₂Projection on a plane.

In order to further analyze the dynamic behavior of the circuit, the circuit parameters are selected, and the circuit parameter R is selected to be a variable parameter, namely the parameter value of the resistor R is adjustable. According to the formula (3), a circuit bifurcation diagram and a Lyapunov exponential spectrum can be simulated by using MATLAB, so that the dynamic characteristics of the circuit when the circuit parameter R changes are analyzed. When R is changed within the range of 1.40k omega-2.60 k omega, the state variable v of the simple Chua's chaotic oscillator₁The bifurcation diagram of (a) is shown in fig. 4; accordingly, the lyapunov exponential spectrum calculated using the Wolf algorithm is shown in fig. 5. For clarity, in FIG. 5, LE is shown in its entirety₁And LE₂The first 2 Lyapunov indices.

As can be seen from fig. 5, as the parameter R increases gradually, the system leads from chaos to cycle 1 from the reverse multiple cycle bifurcation road. It is worth noting that in a non-linear circuit, the attractor is formed from a trajectory starting from the unstable saddle focus of the corresponding index 2. In the simple Chua's chaotic circuit, when R is larger than or equal to 1.82k omega and smaller than or equal to 2.60k omega, different variable initial values are selected to form attractors which are independent in two non-zero equilibrium point attraction domains, so that the system has two different bifurcation modes, wherein the attractors on the left side of the bifurcation diagram represent that the initial values are (-0.0001V,0V and 0A), and the attractors on the right side represent that the initial values are (0.0001V,0V and 0A), and the two coexisting bifurcation modes are caused by the existence of two different non-zero equilibrium points in the system described by the formula (3). Meanwhile, in the middle of the chaotic region, a narrow periodic window appears on the bifurcation diagram, and the periodic window plays an important role in the dynamic behavior evolution of the chaotic system. FIG. 3 shows that when the parameters R of the circuit elements take different values, the simple Chua's chaotic circuit is at v₁-v₂A phase trajectory diagram on a plane, wherein fig. 2(a) is a double-vortex chaotic attractor (R ═ 1.55k Ω); fig. 2(b) a branched single-vortex attractor (R ═ 1.90k Ω); fig. 2(c) a cycle 2 limit cycle (R ═ 2.15k Ω) with branching coexistence; fig. 2(d) shows a cycle 1 limit cycle (R ═ 2.35k Ω) with branching. Here, v is selected₁-v₂The plane serves as a projection plane. Fig. 2(a) -3(d) show a double-vortex chaotic attractor, a bifurcation coexisting single-vortex chaotic attractor, a bifurcation coexisting double-cycle limit cycle and a bifurcation coexisting single-cycle limit cycle, respectively.

The bifurcation diagram shown in fig. 4 was analyzed in comparison with the Lyapunov exponential spectrum shown in fig. 5. In fig. 5, the maximum lyapunov exponent has some zero intervals, which is the reason why the periodic window occurs in the chaotic parameter interval of the system shown in fig. 5. The system dynamics behavior of both mappings is consistent.

Circuit simulation: in order to further verify the feasibility of the simple Chua's chaotic circuit, the invention carries out experimental verification by building the circuit shown in FIG. 1. The experimental circuit adopts a precise adjustable resistor, a monolithic capacitor, a manually wound inductor and a 1N4148 diode, and adopts an OP07CP operational amplifier, and the working voltage is +/-15V. The measured waveforms were captured using a Tektronix DPO3034 digital storage oscilloscope, using a current probe implemented by a combination of Tektronix TCP312 and Tektronix TCPA 300. Different parameters are selected to perform experimental verification on the partial phase trajectory diagram shown in fig. 2, and the corresponding experimental results are shown in fig. 3. FIG. 3 shows the experimental measurements of v for different values of the element parameter R₁-v₂In the phase diagram on the plane, fig. 3(a) shows 1.55k Ω, fig. 3(b) shows 1.90k Ω, fig. 3(c) shows 2.15k Ω, and fig. 3(d) shows 2.35k Ω. And comparing the numerical simulation analysis with the experimental result, and showing that the circuit experimental result is basically consistent with the numerical simulation result of the corresponding system equation.

The result further proves the feasibility of analyzing the chaos phenomenon generated by the simple Chua's chaotic circuit realized by adopting the bridge diode pair.

In the simple Chua's chaotic circuit realized by the bridging diode pair, the bridging diode pair is adopted between the passive LC oscillating circuit and the active RC oscillating circuit. The single-scroll chaotic attractor with double scrolls and bifurcation coexisting can be generated by adjusting circuit parameters, so that the simple Chua's chaotic circuit becomes a simple Chua's chaotic circuit, and finally, the simple Chua's chaotic circuit with simple structure, stable performance and complex chaotic behavior is constructed. The invention has a great propulsion effect on the development of the chaotic system.

The above examples are merely illustrative for clearly illustrating the present invention and are not intended to limit the embodiments of the present invention. Variations or modifications in other variations may occur to those skilled in the art based on the foregoing description. And are neither required nor exhaustive of all embodiments.

Claims (4)

1. The utility model provides a simple and easy Chua's chaotic circuit of bridge diode pair realization which characterized in that: comprises a negative resistance G and a capacitor C₁Capacitor C₂An inductor L and a diode pair; wherein the positive and negative terminals of the negative resistor G are respectively connected with the capacitor C₁The positive and negative terminals of the capacitor are connected to form an active RC oscillation circuit; inductors L and C₂The LC oscillating circuits are connected in parallel to form an LC oscillating circuit; the diode pair is bridged between the LC oscillating circuit and the active RC oscillating circuit.

2. A bridge as claimed in claim 1The simple Chua's chaotic circuit realized by the pole tube pair is characterized in that: the diode pair includes: diode D₁And a diode D₂A simple nonlinear component is formed; wherein the diode D₁And diode D₂The positive electrode end is connected; diode D₁Anode terminal and diode D₂The negative terminals are connected and are denoted as a and b, respectively.

3. The simple Chua's chaotic circuit implemented by the bridge diode pair according to claim 1, characterized in that: the negative resistance G realization circuit comprises an adder and a resistor R₁Resistance R₂A resistor R, wherein the positive input end and the negative input end of the adder are respectively connected with the resistor R₁And a resistance R₂Is connected with the output end of the adder and the output end of the adder is respectively connected with a resistor R₁And a resistance R₂The other end of the resistor R is connected with the negative input end of the adder, and the other end of the resistor R is connected with the end c.

4. A simple zaa chaotic circuit implemented with a bridged diode pair according to claim 1,2 or 3, characterized in that: comprising three state variables, each being a capacitor C₁Voltage v across₁Capacitor C₂Voltage v across₂Flowing an inductor L current i_L。