CN111211885A - Multi-stability chaotic system with impulse function form Lyapunov exponent - Google Patents
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Abstract
The invention belongs to the field of electronic communication research, and particularly relates to a multi-stability chaotic system with an impulse function type Lyapunov exponent, which is formed by a four-dimensional memristive chaotic system formed by introducing a memristor to a three-dimensional chaotic system, wherein the four-dimensional memristive chaotic system comprises an integral channel I, an integral channel II, an integral channel III and an integral channel IV, and v in each integral channelxThe ports are connected together, v in each integration channelyThe ports are connected together, v in each integration channelzThe ports are connected together, and-v in each integral channelxCompared with a general chaotic system, the chaotic system has the advantages that the novel system is simple in structure, easy to realize circuits and more complex in dynamic characteristics, and has important theoretical physical significance and engineering application value for researching coexistence multiple attractors and hardware circuit realization thereof.
Description
Technical Field
The invention belongs to the field of electronic communication research, and particularly relates to a multi-stability chaotic system with an impulse function type Lyapunov exponent.
Background
Chaos refers to the random behavior produced by a deterministic system. From the philosophy, chaos is a unity of determinism and randomness. The chaotic signal has the characteristics of inherent randomness, initial value sensitivity, ergodicity, boundedness and the like, can generate a broadband signal similar to white noise, and has wide application in the fields of information encryption, secret communication, chaotic radar and the like. The chaotic signal source is an important component of various chaotic application systems, and the research and development of a novel chaotic signal source is very important for the practicability of a chaotic theory.
In 1971, the chinese scientist professor zeitseuda begonia predicted the existence of the fourth basic circuit component based on the completeness of the circuit variables and named it as a memristor. The physical nature of memristors was not discovered by the HP lab for the first time until 2008. Since the memristor is a nonlinear device and has a memory function, the memristor is widely applied to many fields including researches on low-power-consumption flash memories, neural synapse architecture design, neural network construction, novel chaotic system construction and the like. The memristor with various nonlinear characteristics is introduced into the classical chaotic circuit, a new chaotic circuit can be easily constructed, and the dynamic behavior of the chaotic circuit is more complex than that of the original chaotic system.
In recent years, the multi-stability of chaotic systems has become a research hotspot. The phenomenon that multiple attractors coexist under the same system parameter setting is referred to. Multi-stability is a common phenomenon in many non-linear systems. Compared with a general chaotic system, the chaotic system has more complex dynamic behavior and has wide application prospect in the fields of secret communication, image video encryption and the like.
In the existing multi-stability chaotic system, Lyapunov exponential spectrums are in both chaotic states and non-chaotic states, and the state transition is smooth. However, in the multistable chaotic system provided by the invention, when the system parameters are set to be (x (0),1,0,0), the Lyapunov exponent spectrum is kept constant except for x (0) ═ 0, a sudden change occurs at x (0) ═ 0, and the maximum Lyapunov exponent is similar to the form of a shock function. It is noted that the non-chaotic region of the system is approximately
Close to x (0) ═ 0, the degree of mutation is very close to the impulse function. The violent mutation phenomenon has very important significance in the research field of chaotic systems. The dynamic behavior of the new system is more complex, and the method has great application prospect in the fields of information encryption, secret communication and the like.
Disclosure of Invention
The invention aims to solve the technical problem of designing a multi-stability chaotic system with an impulse function type Lyapunov exponent and realizing the multi-stability chaotic system by a hardware circuit.
In order to solve the technical problem, the invention provides a multi-stability chaotic system with an impulse function type Lyapunov exponent, and a corresponding hardware circuit is designed, specifically as follows:
a multi-stability chaotic system with an impulse function form Lyapunov exponent is formed by a four-dimensional memristive chaotic system formed by introducing a memristor to a three-dimensional chaotic system, wherein the four-dimensional memristive chaotic system comprises an integral channel I, an integral channel II, an integral channel III and an integral channel IV, and v in each integral channelxThe ports are connected together, v in each integration channelyThe ports are connected together, v in each integration channelzThe ports are connected together, and-v in each integral channelxThe ports are all connected together.
The three-dimensional chaotic system corresponds to the following equations:
where a, b, c and d are constants and x, y and z are state variables.
The four-dimensional memristor chaotic system corresponds to the following equation:
wherein e and h are two constants, W (w) is a memory conductance function of the magnetic control memristor model, and the relation between the charge and the magnetic flux is established, and the expression is as follows:
W(w)＝f+3gw2
where f and g are two positive real numbers and w is a state variable.
The first integration channel has 3 input ends, v respectivelyx、vy、vz；
Input terminal vxA resistor R7 is connected in series with the inverting input end of the operational amplifier U1; a resistor R8 is connected in parallel between the inverting input end and the output end of the U1, and the output end of the U1 outputs-vx(ii) a A resistor R1 is connected in series between the output end of the U1 and the inverting input end of the operational amplifier U2; input terminal vyAnd vzAfter being multiplied by the multiplier M1, the mixed signal is connected in series with a resistor R2 and is connected to the inverting input end of the operational amplifier U2; a capacitor C1 is connected in parallel between the inverting input end and the output end of the U2, and the output end of the U2 outputs v at the momentx(ii) a The non-inverting inputs of the operational amplifiers U1 and U2 are both tied to ground.
The second integration channel has 4 input ends which are respectively 2-vx1vyAnd 1vz；
Input terminal vyA resistor R3 is connected in series with the inverting input end of the operational amplifier U3; input terminal-vxAnd vzAfter being multiplied by the multiplier M2, the mixed signal is connected in series with a resistor R4 and is connected to the inverting input end of the operational amplifier U3; input terminal-vxA memristor is connected in series with the inverting input end of the operational amplifier U3; a capacitor C2 is connected in parallel between the inverting input end and the output end of the U3, and the output end of the U3 outputs v at the momenty(ii) a The non-inverting input of the operational amplifier U3 is terminated at ground.
The third integration channel has 3 input ends which are respectively-vx、vyAnd vz；
Input terminal vzA resistor R5 is connected in series with the inverting input end of the operational amplifier U4; input terminal-vxAnd vyAfter being multiplied by the multiplier M3, the mixed signal is connected in series with a resistor R6 and is connected to the inverting input end of the operational amplifier U4; a capacitor C3 is connected in parallel between the inverting input end and the output end of the U4, and the output end of the U4 outputs v at the momentz(ii) a The non-inverting input of the operational amplifier U4 is terminated at ground.
The four integrating channels only have 1 input end-vx；
Input terminal-vxA resistor Ra is connected in series with the inverting input end of the operational amplifier U5; a capacitor C4 is connected in parallel between the inverting input end and the output end of the U5, and the output end of the U5 outputs v at the momentw(ii) a Output terminal v of U5wThe output v is multiplied by a multiplier Ma and squaredw 2；vw 2And-vxThe series resistor Rb is connected to the inverting input end of the operational amplifier U3 after being multiplied by the multiplier Mb; input terminal-vxThe series resistor Rc is also connected to the inverting input of the operational amplifier U3.
The invention has the beneficial effects that:
compared with a general chaotic system, the chaotic system has a simple structure, is easy to realize, has more complex dynamic characteristics, and has important theoretical physical significance and engineering application value for researching coexistence multiple attractors and hardware circuit realization thereof.
Drawings
FIG. 1 is a circuit diagram of integration channel one;
FIG. 2 is a circuit diagram of integration channel two;
FIG. 3 is a circuit diagram of integration channel three;
FIG. 4 is a bifurcation diagram of the state variable w with the system parameters set to (x (0),1,0,0), and x (0) changed;
FIG. 5 is a Lyapunov exponent spectra with system parameters set to (x (0),1,0,0), x (0) varying;
FIG. 6 is a diagram of a numerical simulation phase trajectory of an x-y multistable chaotic system;
FIG. 7 is a diagram of a numerical simulation phase trajectory of an x-z multistable chaotic system;
FIG. 8 is a diagram of a numerical simulation phase trajectory of a y-z multistable chaotic system;
FIG. 9 is a diagram of a numerical simulation phase trajectory of an x-w multistable chaotic system;
FIG. 10 is a diagram of a numerical simulation phase trajectory of a y-w multistable chaotic system;
FIG. 11 is a numerical simulation phase-trajectory diagram of a multi-stability chaotic system of z-w;
FIG. 12 is vx-vyThe PSpice circuit simulation result diagram of the multi-stability chaotic system;
FIG. 13 is vx-vzThe PSpice circuit simulation result diagram of the multi-stability chaotic system;
FIG. 14 is vy-vzThe PSpice circuit simulation result diagram of the multi-stability chaotic system;
FIG. 15 is vx-vwThe PSpice circuit simulation result diagram of the multi-stability chaotic system;
FIG. 16 is vy-vwThe PSpice circuit simulation result diagram of the multi-stability chaotic system;
FIG. 17 is vz-vwThe simulation result diagram of the PSpice circuit of the multi-stability chaotic system.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
The newly proposed multi-stability chaotic system has the Lyapunov exponent in the impulse function form and also has a discrete bifurcation diagram, and can generate two infinite coexisting attractors which have different structures and are distributed in parallel along a fourth axis.
A multi-stability chaotic system with an impulse function type Lyapunov exponent introduces a memristor on the basis of a three-dimensional chaotic system, and is transformed into a new four-dimensional memristor chaotic system. Through analysis of a phase orbit diagram, a Lyapunov exponent spectrum and a bifurcation diagram, the chaotic behavior of the system is researched, and the analysis shows that the new system can generate two infinite coexisting attractors which are different in structure and distributed in parallel along a fourth axis, and the system has the Lyapunov exponent and the discrete bifurcation diagram in the form of an impulse function.
As shown in fig. 1, the main circuit includes: the integration channel I, the integration channel II, the integration channel III and the integration channel IV. The first integration channel has 3 input ends, respectively 1' vx", 1" vy"and 1" vz", respectively through an inverter and a multiplier, then through an integrator, and finally outputting" vx"; integration channel two has 4 inputsEnds, 2 respectively "-vx", 1" vy"and 1" vzFirst through multiplier and memristor, then through integrator, and finally output "vy"; the third integration channel has 3 input ends, respectively 1 "-vx", 1" vy"and 1" vz", the final output" v "is passed through a multiplier and integratorz"; the fourth integrating channel is a memristor part in the block of FIG. 2, which has only 1 input end "-vx", by the integrator output" vwAfter passing through two multipliers, the signal passes through an integrator in an integrating channel 2, and finally "v" is outputy"; the non-inverting inputs of the operational amplifiers U1, U2, U3, U4 and U5 are all tied to ground.
In integration channel one, input terminal "vx"a resistor R7 is connected in series with the inverting input terminal of the operational amplifier U1; a resistor R8 is connected in parallel between the inverting input end and the output end of the U1, and the output end outputs "-v at the momentx"; a resistor R1 is connected in series between the output end of the U1 and the inverting input end of the operational amplifier U2; input terminal "vy"and" vz"after multiplied by multiplier M1, a resistor R2 is connected in series with the inverting input terminal of operational amplifier U2; a capacitor C1 is connected in parallel between the inverting input end and the output end of the U2, and the output end of the U2 outputs' vx"; the non-inverting inputs of the operational amplifiers U1 and U2 are both tied to ground.
In the second integration channel, input terminal "vy"a resistor R3 is connected in series with the inverting input terminal of the operational amplifier U3; input terminal "-vx"and" vz"after multiplied by multiplier M2, a resistor R4 is connected in series with the inverting input terminal of operational amplifier U2; input terminal "-vx"connect a memristor (i.e. the circuit in the block) in series to the inverting input of the operational amplifier U3; a capacitor C2 is connected in parallel between the inverting input end and the output end of the U3, and the output end of the U3 outputs' vy"; the non-inverting input of the operational amplifier U3 is terminated at ground.
In integration channel three, input terminal "vz"a resistor R5 is connected in series with the inverting input terminal of the operational amplifier U4; input terminal "-vx"and" vy"after multiplied by multiplier M3, a resistor R6 is connected in series with the inverting input terminal of operational amplifier U4; a capacitor C3 is connected in parallel between the inverting input end and the output end of the U4, and the output end of the U3 outputs' vz"; the non-inverting input of the operational amplifier U4 is terminated at ground.
In integration channel four (i.e., memristor circuit in block), input "-vx"a resistor Ra is connected in series to the inverting input terminal of the operational amplifier U5; a capacitor C4 is connected in parallel between the inverting input end and the output end of the U5, and the output end of the U5 outputs' vw"; output terminal "v" of U5w' output v after multiplication by multiplier Ma and square operationw 2”；“vw 2"and" -vx"the series resistance Rb is connected to the inverting input terminal of the operational amplifier U3 after being multiplied by the multiplier Mb; input terminal "-vx"series resistance Rc is also connected to the inverting input of operational amplifier U3; a capacitor C2 is connected in parallel between the inverting input end and the output end of the U3, and the output end of the U3 outputs' vy"; the non-inverting input of the operational amplifier U3 is terminated at ground.
The invention provides a multi-stability chaotic system with an impulse function form Lyapunov exponent. The method has the main idea that a memristor is introduced on the basis of a three-dimensional chaotic system, the memristor is transformed into a new four-dimensional memristor chaotic system, and the chaotic behavior of the system is researched through phase trajectory diagram, Lyapunov exponential spectrum and bifurcation diagram analysis. Analysis shows that the new system can generate two infinite coexisting attractors which are different in structure and distributed in parallel along the fourth axis, and has the Lyapunov exponent in the form of an impulse function and a discrete bifurcation diagram.
The three-dimensional chaotic system corresponds to the following equations:
where a, b, c and d are all constants and x, y and z are state variables.
A memristor is introduced into the three-dimensional chaotic system and is transformed into a four-dimensional memristor chaotic system, and the corresponding equation of the transformed four-dimensional chaotic system is as follows:
where e and h are two normal numbers. W (w) is a memristive function of a magnetic control memristor model, and a relation between electric charge and magnetic flux is established, wherein the expression is shown as the following formula.
W(w)＝f+3gw2
Where f and g are two positive real numbers and w is a state variable.
Basic kinetic analysis of New System
(1) Symmetry property
Symmetry is widely present in chaotic systems with an even number of attractors. The symmetry of the memristive chaotic system (2) is consistent with that of the original three-dimensional chaotic system (1), namely the system (2) is transformed
The lower layer remains unchanged. This means that the attractors in the state space must be symmetric with respect to the z-axis.
(2) Balance point and stability
It can be seen that the equilibrium point of the proposed multistable chaotic system depends only on x, y and z, and is not related to w. The system has a linear balance point:
O＝{(x,y,z,w)x＝y＝z＝0,w＝k}
where k is an arbitrary constant.
The system (2) is linearized at the origin O, and its jacobian matrix can be obtained:
the characteristic equation of the system is shown in the following formula.
λ(λ-a)(λ+c)(λ+d)＝0
The characteristic root of the system can be found:
λ1＝0,λ2＝a,λ3＝-c,λ4＝-d
when the parameters a, c and d are all positive real numbers, the characteristic root λ3And λ4Are all negative, and the characteristic root λ2Always positive. Therefore, the system has one positive real root, one zero root and two negative real roots, namely the system (2) has unstable saddle points.
(3) Dissipative property
The dissipative nature of system (2) can be shown by the following equation.
When the parameters a, c and d satisfy a-c-d < 0, the system (2) is dissipative.
Numerical simulation of a system
According to the multi-stability chaotic circuit with the impulse function type Lyapunov exponent shown in the figure 1, a simulation software platform is utilized to perform numerical simulation analysis on the described system. When the system parameters a-4, b-6, c-20, d-5, e-0.01, f-1, 3 g-0.1, and h-0.1, the initial values are set to (x (0),1,0,0), giving x (0) at [ -10 ]4,104]The bifurcation map and Lyapunov exponent spectra of the state variable w in the interval are shown in fig. 4 and 5.
Initial values set to (1,1,0,0), (± 10,1,0,0), (± 50,1,0,0), (± 100,1,0,0) respectively, the system (2) may generate seven different coexistence attractors, as shown in fig. 6-11. Where green corresponds to an initial value of (1,1,0,0), blue and red correspond to initial values of (10,1,0,0), (-10,1,0,0), pink and cyan correspond to initial values of (50,1,0,0), (-50,1,0,0), and yellow and black correspond to initial values of (100,1,0,0), (-100,1,0, 0). It can be found that the phase trace plot of the system has good agreement with the bifurcation plot and the Lyapunov exponential spectrum above.
Circuit implementation of a system
(1) And performing variable ratio compression transformation on the multi-stability chaotic system. The power supply voltage of the power supply is +/-15V, the saturation voltage of the operational amplifier is +/-13.5V, and the voltage range of the multiplier is +/-10V. The dynamic range of the chaotic attractor variables may exceed the saturation voltage of the element. Therefore, compressing the system (2) state variables by a factor of 10 yields:
(2) the time scale transformation is carried out on the multi-stability chaotic system, and a dimensionless equation of the system can be expressed as follows:
(3) and (3) building a circuit according to the transformed system state equation, so as to obtain:
wherein v isx，vy，vzAnd vwRespectively the voltage over each capacitor. Corresponding resistance and capacitance expressions can be solved. C1＝C2＝C3＝C4＝C，R1＝R/a，R2＝R/10b，R3＝R/c，R4＝R/10，R5＝R/d，R6＝R/10，Ra＝R/h，Rb＝R/(e*3g*100)，Rc＝R/ef。
Let R100 k Ω and C10000 nF. The system parameters of the four-dimensional memristive chaotic system are that a is 4, b is 6, c is 20, d is 5, e is 0.01, f is 1, 3g is 0.1, h is 0.1, and the system initial value is set to (1,1,0, 0). Therefore, the corresponding resistance values are as follows. R1＝25kΩ，R2＝1.67kΩ，R3＝5kΩ，R4＝10kΩ，R5＝20kΩ，R6＝10kΩ，Ra＝1000kΩ，Rb＝1000kΩ，Rc10000k Ω. Capacitor C1And C2The initial voltage of (2) is set to 0.1V, and the initial voltage of the other capacitors is maintained at 0V. It is worth noting that the state variables of the chaotic system are compressed by 10 times, and the initial values of the system should be compressed by 10 times. The circuit simulation result diagrams are shown in fig. 12-17.
The invention discloses a multi-stability chaotic circuit with an impulse function type Lyapunov exponent. The main idea is that a memristor is introduced on the basis of a three-dimensional chaotic circuit and is transformed into a new four-dimensional memristor chaotic circuit. The circuit consists of an inversion, addition, integration and other operation modules formed by an operational amplifier and a resistor or a capacitor, and a nonlinear operation module formed by an analog multiplier and the operational amplifier. The circuit can generate two infinite coexisting attractors which are different in structure and distributed in parallel along a fourth axis under certain system parameters, and the circuit has a Lyapunov exponent in an impulse function form and a discrete bifurcation diagram. Compared with a general chaotic system, the chaotic system has more complex dynamic characteristics. Meanwhile, the method has important theoretical physical significance and engineering application value for researching the coexistence of multiple attractors and the realization of hardware circuits thereof.
The multistable chaotic system with the Lyapunov exponent in the impulse function form is simple in structure, can generate two infinite coexisting attractors which are different in structure and distributed in parallel along a fourth axis, has the Lyapunov exponent in the impulse function form, and further has a discrete bifurcation diagram.
Claims (7)
1. A multi-stability chaotic system with an impulse function form Lyapunov exponent is formed by a four-dimensional memristor chaotic system formed by introducing a memristor to a three-dimensional chaotic system, and is characterized in that: the four-dimensional memristive chaotic system comprises an integral channel I, an integral channel II, an integral channel III and an integral channel IV, wherein v in each integral channelxThe ports are connected together, v in each integration channelyThe ports are connected together, v in each integration channelzThe ports are connected together, and-v in each integral channelxThe ports are all connected together.
2. The multistable chaotic system with the impulse function form Lyapunov exponent as claimed in claim 1 is characterized in that the equation corresponding to the three-dimensional chaotic system is as follows:
where a, b, c and d are constants and x, y and z are state variables.
3. The multistable chaotic system with the impulse function form Lyapunov exponent as claimed in claim 1, wherein the four-dimensional memristive chaotic system corresponds to an equation:
wherein e and h are two constants, W (w) is a memory conductance function of the magnetic control memristor model, and the relation between the charge and the magnetic flux is established, and the expression is as follows:
W(w)＝f+3gw2
where f and g are two positive real numbers and w is a state variable.
4. The multistable chaotic system with the impulse function form Lyapunov exponent as claimed in claim 1, wherein the first integration channel has 3 input ends, v is v, respectivelyx、vy、vz；
Input terminal vxA resistor R7 is connected in series with the inverting input end of the operational amplifier U1; a resistor R8 is connected in parallel between the inverting input end and the output end of the U1, and the output end of the U1 outputs-vx(ii) a A resistor R1 is connected in series between the output end of the U1 and the inverting input end of the operational amplifier U2; input terminal vyAnd vzAfter being multiplied by the multiplier M1, the mixed signal is connected in series with a resistor R2 and is connected to the inverting input end of the operational amplifier U2; a capacitor C1 is connected in parallel between the inverting input end and the output end of the U2, and the output end of the U2 outputs v at the momentx(ii) a The non-inverting inputs of the operational amplifiers U1 and U2 are both tied to ground.
5. The multistable chaotic system with the impulse function form Lyapunov exponent as claimed in claim 1, wherein the integral channel is twoHas 4 input ends, respectively 2-vx1vyAnd 1vz；
Input terminal vyA resistor R3 is connected in series with the inverting input end of the operational amplifier U3; input terminal-vxAnd vzAfter being multiplied by the multiplier M2, the mixed signal is connected in series with a resistor R4 and is connected to the inverting input end of the operational amplifier U3; input terminal-vxA memristor is connected in series with the inverting input end of the operational amplifier U3; a capacitor C2 is connected in parallel between the inverting input end and the output end of the U3, and the output end of the U3 outputs v at the momenty(ii) a The non-inverting input of the operational amplifier U3 is terminated at ground.
6. The multistable chaotic system with the impulse function form Lyapunov exponent as claimed in claim 1, wherein three of the integration channels have 3 input ends, and the three input ends are respectively-vx、vyAnd vz；
Input terminal vzA resistor R5 is connected in series with the inverting input end of the operational amplifier U4; input terminal-vxAnd vyAfter being multiplied by the multiplier M3, the mixed signal is connected in series with a resistor R6 and is connected to the inverting input end of the operational amplifier U4; a capacitor C3 is connected in parallel between the inverting input end and the output end of the U4, and the output end of the U4 outputs v at the momentz(ii) a The non-inverting input of the operational amplifier U4 is terminated at ground.
7. The multistable chaotic system with the impulse function form Lyapunov exponent as claimed in claim 1, characterized in that the four integration channels only have 1 input end-vx；
Input terminal-vxA resistor Ra is connected in series with the inverting input end of the operational amplifier U5; a capacitor C4 is connected in parallel between the inverting input end and the output end of the U5, and the output end of the U5 outputs v at the momentw(ii) a Output terminal v of U5wThe output v is multiplied by a multiplier Ma and squaredw 2；vw 2And-vxThe series resistor Rb is connected to the inverting input end of the operational amplifier U3 after being multiplied by the multiplier Mb; input terminal-vxA series resistor Rc connected to the inverting input of an operational amplifier U3And (4) an end.
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