CN104486064A - Memory resistance chaotic signal producing circuit with self-excitation attractor and hidden attractor - Google Patents
Memory resistance chaotic signal producing circuit with self-excitation attractor and hidden attractor Download PDFInfo
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Abstract
The invention discloses a memory resistance chaotic signal producing circuit with a self-excitation attractor and a hidden attractor. The memory resistance chaotic signal producing circuit comprises a generalized memristor GM formed by a negative resistor G, a resistor R1, a capacitor C1, a capacitor C2, an inductor L and a diode bridge cascade first-order RC filter, wherein the negative resistor G, the generalized memristor GM and the capacitor C1 are connected in parallel, and the positive end and the negative end are marked as the a end and the b end; the inductor L and the capacitor C2 are connected in parallel, and the positive end and the negative end are marked as the c end and the d end; a coupling resistor R1 is connected between the a end and the c end in a bridging mode. Circuit parameters are adjusted so that the circuit can generate the self-excitation attractor and the hidden attractor with complex non-linear phenomena, the hidden attractor phenomenon are not dependent on the original state of the circuit, and observation can be carried out in an experimental circuit. In this way, the circuit has the scientific theoretical foundation and the physical realizability, can be applied to the engineering fields of secret communication, chaotic encryption and the like, and has a positive pushing effect on the development of chaotic application systems.
Description
Technical field
The present invention relates to a kind of new chaotic signal generating circuit based on FIRST ORDER GENERALIZED DISTRIBUTED PARAMETER memristor, by regulating circuit parameter, this circuit can produce conventional self-excitation attractor, can produce again the observable hidden attractor of experiment.
Background technology
Chaos is a kind of complex oscillation between rectilinear oscillation and noise that the certainty annuity of nonlinear equation description produces.Chaotic signal has the features such as intrinsic stochasticity, initial value sensitivity, broadband, ergodic and boundedness, can produce the broadband signal of similar white noise, has a wide range of applications in fields such as information encryption, secure communication and chaotic radars.Chaos signal source is the important component part of all kinds of chaos applications system, practical most important to chaology of research and development new chaotic signal source.
1971, the scientist Cai Shaotang of Chinese origin of University of California Berkeley, according to variable combinatorial completeness principle, predicted the existence recalling resistance theoretically, and proposed memory resistor and recall resistance system.2008, the Willliams team of Hewlett-Packard is the reported first physics realization of nanoscale memristor on " Nature " magazine.Memristor is a kind of special nonlinear circuit element, can be used for producing chaotic signal, is the most effective element building nonlinear circuit and system.Adopt the memristor with various nonlinear characteristic to substitute nonlinear device in Classical Chaos circuit, the Generation of Chaotic Signals made new advances can be constructed easily.Wherein, cai's circuit is simple due to topological structure, and can produce the characteristic of the chaos of complexity, pays close attention to widely recalling to hinder in chaos signal generator research field to obtain.Cai Shi recalls resistance chaos circuit may have line balance point set, can also observe the Special Nonlinear such as Transient Chaos, hyperchaos phenomenon.These nonlinear characteristics are many to be produced by unstable equilibrium point, and corresponding chaos attractor is called as self-excitation attractor.In recent years, a kind of novel attractor obtains to be paid close attention to widely, and its domain of attraction is not crossing with any unstable equilibrium point, is therefore called as hidden attractor.The non linear system producing hidden attractor may have line balance point set, does not have balance point or only has stable equilibrium point, cannot try to achieve its corresponding domain of attraction as self-excitation attractor by calculating unstable equilibrium point.Therefore, be no matter the discovery of hidden attractor phenomenon, or the estimation of its corresponding parameter area all seem particularly difficulty.Based on this characteristic of hidden attractor, find hidden attractor phenomenon hiding in each nonlinear systems, specify its parameter area, or structure can produce the new chaotic signal generating circuit of hidden attractor, has great importance in practical engineering application.
According to having been reported, classical Cai Shi chaos system, under particular initial state condition, can observe the hidden attractor excited by stable equilibrium point.But the initial condition of above-mentioned hidden attractor to classical Cai Shi chaos system has sensitive dependence, is difficult to verify by experiment.The present invention is namely for this problem, employing single order diode bridge broad sense memristor substitutes the Cai Shi diode in classical cai's circuit, construct a kind of new chaotic signal generating circuit, wherein can observe self-excitation attractor and hidden attractor, and above-mentioned phenomenon is all verified by experiment test.
Summary of the invention
Technical problem to be solved by this invention is to provide one can produce self-excitation attractor and hidden attractor, and produce hidden attractor can the new chaotic signal source of experimental observation.
For solving the problems of the technologies described above, what the present invention adopted diode bridge cascaded RC filters to form recall resistance equivalent electric circuit replaces in classical cai's circuit Cai Shi diode, construct a kind of novel Cai Shi and recall resistance Generation of Chaotic Signals, comprising: Cai Shi recalls resistance chaos circuit main circuit, the broad sense memristor G that diode bridge cascade single order RC filter is formed
mwith negative resistance G.
The broad sense memristor G that described diode bridge cascade single order RC filter is formed
mas shown in Figure 1, comprising: diode D
1, diode D
2, diode D
3, diode D
4, resistance R, electric capacity C.Wherein, diode D
1negative pole end and diode D
2negative pole end is connected, and is denoted as 2 ends; Diode D
2positive terminal and diode D
3negative pole end is connected, and is denoted as 3 ends; Diode D
3positive terminal and diode D
4positive terminal is connected, and is denoted as 4 ends; Diode D
4negative pole end and diode D
1positive terminal is connected, and is denoted as 1 end; Resistance R is in parallel with electric capacity C forms single order RC filter, and the positive and negative terminal of electric capacity is designated as 5 ends and 6 ends respectively.2 ends of diode bridge, 4 ends are connected with 5 ends of single order RC filter, 6 ends respectively; 1 end of diode bridge, 3 ends are as broad sense memristor G
minput.
The realizing circuit of described negative resistance G as shown in Figure 2, comprises operational amplifier, resistance R
a1, resistance R
a2, resistance R
b, its input is labeled as 1 end, 2 ends respectively; The electrode input end of operational amplifier and negative input respectively with resistance R
a1with resistance R
a2one end be connected; The output of operational amplifier respectively with resistance R
a1with resistance R
a2the other end be connected; Resistance R
bone end be connected with the electrode input end of operational amplifier, the other end is connected with input 2.
Described Cai Shi recalls resistance chaos circuit main circuit as shown in Figure 3, primarily of negative resistance G, coupling resistance R
1, electric capacity C
1, electric capacity C
2, inductance L, diode bridge cascade single order RC filter form broad sense memristor G
mcomposition.Wherein, the memristor G of negative resistance G, diode bridge cascaded RC filters formation
mwith electric capacity C
1parallel connection, its positive and negative a, b of being extremely denoted as respectively holds; Inductance L and electric capacity C
2parallel connection, its positive and negative c, d of being extremely denoted as respectively holds; Coupling resistance R
1be connected across between a, c two ends.
Containing four dynamic elements in circuit shown in Fig. 3, correspond respectively to one of four states variable: electric capacity C
1both end voltage v
1, electric capacity C
2both end voltage v
2, flow through inductance L current i
3, memristor G
minternal state variable and electric capacity C both end voltage v
c.Then Cai Shi shown in Fig. 3 recall resistance chaos circuit can be represented by following first order differential equation system:
Wherein, ρ=1/ (2nV
t), I
s, n and V
trepresent the reverse saturation current of diode, emission ratio and cut-ff voltage respectively.Four diodes adopt 1N4148, and parameter is I
s=6.891nA, n=1.827, and V
t=25mV.
Utilize MATLAB simulation software, Numerical Simulation Analysis can be carried out to the circuit described by formula (1).When circuit parameter is set to typical circuit parameter 1:R
1=1.5k Ω, G=0.71mS, L=17.2mH, C
1=10nF, C
2=100nF, R=1k Ω, C=10nF; Circuit initial condition is set to: v
1(0)=0.01V, v
2(0)=1V, i
3(0)=0A and v
c(0) circuit shown in=0V, Fig. 3 can produce two snail volume chaos attractor.Produce the phase rail figure Numerical Simulation Results of chaos attractor in different phase plane as shown in Figure 4: Fig. 4 (a) is at v
1(t)-v
2t the projection in () plane, Fig. 4 (b) is at v
1(t)-i
3t the projection in () plane, Fig. 4 (c) is at v
1(t)-v
ct the projection in () plane, Fig. 4 (d) is at v
1projection in (t)-i (t) plane.
According to the circuit parameter of setting, three given balance points of this circuit can be calculated:
S
0＝(0,0,0,0)
S
1＝(0.8300,0,0.00055,0.0395) (2)
S
2＝(-0.8300,0,-0.00055,0.0395)
The characteristic root at three balance point places obtains by the latent root equation solving corresponding Jacobian matrix:
S
0:λ
1＝11074.3,λ
2,3＝-6497.1±j14387.3,λ
4＝-100015.1
(3)
S
1,2:λ
1,2＝1319.7±j20377.5,λ
3＝-31904.4,λ
4＝-159084.1
Can find out, S
0for the saddle point of instability, S
1,2the saddle being a pair instability is burnt.Therefore, under this group parameter, it is self-excitation attractor that Cai Shi recalls the chaos attractor that produces of resistance chaos circuit.Fig. 5 gives R
1when 1.4k Ω ~ 1.65k Ω regional change, v
1the bifurcation graphs of (t) and Lyapunov exponential spectrum.Wherein, remittance can be observed, limit cycle, chaos, fork coexist and the phenomenon such as cycle window.The phenomenon disclosed can be verified by numerical simulation and experiment test, and accordingly result respectively as shown in figs. 6 and 10.
When circuit parameter is set to typical circuit parameter 2:R
1=1.96k Ω, G=0.56mS, L=80mH, C
1=22nF, C
2=113nF, R=0.8k Ω, C=1 μ F, circuit initial condition is identical with Fig. 4, and this circuit still can produce two snail volume chaos attractor.Produce chaos attractor MATLAB numerical simulation phase rail figure corresponding in different phase plane as shown in Figure 7: Fig. 7 (a) is at v
1(t)-v
2t the projection in () plane, Fig. 7 (b) is at v
1projection in (t)-i (t) plane.
This circuit has three given balance points:
S′
0＝(0,0,0,0)
S′
1＝(0.8324,0,0.00042,0.0343) (4)
S′
2＝(-0.8324,0,-0.00042,0.0343)
The characteristic root at three balance point places is respectively:
S′
0:λ
1＝5939.07,λ
2＝-1250.15,λ
3,4＝-1250.15±j5204.42(5)
(5)
S′
1,2:λ
1,2＝-109.55±j9394.59,λ
3＝-1191.43,λ
4＝-23830.48
Can find out, S
0for the saddle point of instability, S
1,2be that a pair stable saddle is burnt, self-excitation attractor can not be produced.Therefore, under this group parameter, Cai Shi shown in Fig. 3 recalls resistance chaos circuit and is operated in hidden attractor oscillatory regime.
Work as R
1for variable element, when 1.9k Ω ~ 2.0k Ω regional change, two non-zero balance points are at R
1it is burnt that=1.94741k Ω place changes stable saddle by saddle Jiao of instability.Illustrate that hidden attractor phenomenon only may produce at R
1the region of >1.94741k Ω.V shown in composition graphs 8 again
1t bifurcation graphs and the Lyapunov exponential spectrum of () can be found out, hidden attractor phenomenon mainly produces at 1.94741k Ω <R
1<1.9885k Ω region, and can observe in this regional extent that two snail rolls up hidden attractor, single snail that coexists rolls up hidden attractor, recessive chaos phenomenon that the hidden limit cycle that coexists etc. is abundant.The phenomenon disclosed can be verified by numerical simulation and experiment test, and accordingly result respectively as shown in figures 9 and 11.
As can be seen from above-mentioned analysis: the novel Cai Shi constructed by the present invention recalls resistance Generation of Chaotic Signals, the self-excitation attractor and hidden attractor with complicated chaos phenomenon can be produced by regulating circuit parameter, and the hidden attractor phenomenon found does not rely on the initial condition of circuit, can observe in experimental circuit.Therefore, this circuit has theoretical foundation and the realizability physically of science, can be applicable to the engineering field such as secure communication, chaos encryption.
Accompanying drawing explanation
In order to make content of the present invention be more likely to be clearly understood, below basis specific embodiment and by reference to the accompanying drawings, the present invention is further detailed explanation, wherein:
What Fig. 1 diode bridge cascade single order RC filter was formed recalls resistance equivalent electric circuit;
Fig. 2 negative resistance equivalent electric circuit;
The Cai Shi that Fig. 3 has self-excitation attractor and hidden attractor recalls resistance chaos circuit main circuit;
Under Fig. 4 (a) typical circuit parameter 1 state, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t self-excitation chaos attractor that () plane projects;
Under Fig. 4 (b) typical circuit parameter 1 state, Cai Shi recalls resistance chaos circuit at v
1(t)-i
3the self-excitation chaos attractor of the projection in (t) plane;
Under Fig. 4 (c) typical circuit parameter 1 state, Cai Shi recalls resistance chaos circuit at v
1(t)-v
cthe self-excitation chaos attractor of the projection in (t) plane;
Under Fig. 4 (d) typical circuit parameter 1 state, Cai Shi recalls resistance chaos circuit at v
1the self-excitation chaos attractor of the projection in (t)-i (t) plane;
Under Fig. 5 (a) typical circuit parameter 1 state, R
1during change, state variable v
1the bifurcation graphs of (t);
Under Fig. 5 (b) typical circuit parameter 1 state, R
1lyapunov exponents during change;
Under Fig. 6 (a) typical circuit parameter 1 state, R
1during=1.42k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t fork that () plane projects coexists limit cycle (simulation result);
Under Fig. 6 (b) typical circuit parameter 1 state, R
1during=1.47k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t multicycle limit cycle (simulation result) that () plane projects;
Under Fig. 6 (c) typical circuit parameter 1 state, R
1during=1.5k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t two snails volume chaos attractor (simulation result) that () plane projects;
Under Fig. 6 (d) typical circuit parameter 1 state, R
1during=1.57k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t fork that () plane projects coexist single snail volume chaos attractor (simulation result)
Under Fig. 6 (e) typical circuit parameter 1 state, R
1during=1.593k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t fork that () plane projects coexists limit cycle (simulation result);
Under Fig. 6 (f) typical circuit parameter 1 state, R
1during=1.615k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t fork that () plane projects coexists limit cycle (simulation result);
Under Fig. 7 (a) typical circuit parameter 2 state, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t hidden attractor that () plane projects;
Under Fig. 7 (b) typical circuit parameter 2 state, Cai Shi recalls resistance chaos circuit at v
1the hidden attractor of the projection in (t)-i (t) plane;
Under Fig. 8 (a) typical circuit parameter 2 state, 1.9k Ω≤R
1during≤2.0k Ω, state variable v
1the bifurcation graphs of (t);
Under Fig. 8 (b) typical circuit parameter 2 state, R
1lyapunov exponents during change;
Under Fig. 8 (c) typical circuit parameter 2 state, 1.915k Ω≤R
1≤ 1.98k Ω, state variable v
1the bifurcation graphs of (t);
Under Fig. 9 (a) typical circuit parameter 2 state, R
1during=1.95k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t multicycle limit cycle (simulation result) that () plane projects;
Under Fig. 9 (b) typical circuit parameter 2 state, R
1during=1.961k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t two snails volume chaos attractor (simulation result) that () plane projects;
Under Fig. 9 (c) typical circuit parameter 2 state, R
1during=1.967k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t fork that () plane projects coexist single snail volume chaos attractor (simulation result);
Under Fig. 9 (d) typical circuit parameter 2 state, R
1during=1.98k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t fork that () plane projects coexists limit cycle (simulation result);
Under Figure 10 (a) typical circuit parameter 1 state, R
1during=1.428k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t fork that () plane projects coexists limit cycle (test result);
Under Figure 10 (b) typical circuit parameter 1 state, R
1during=1.474k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t multicycle limit cycle (test result) that () plane projects;
Under Figure 10 (c) typical circuit parameter 1 state, R
1during=1.50k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t two snails volume chaos attractor (test result) that () plane projects;
Under Figure 10 (d) typical circuit parameter 1 state, R
1during=1.542k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t fork that () plane projects coexist single snail volume chaos attractor (test result);
Under Figure 10 (e) typical circuit parameter 1 state, R
1during=1.593k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t fork that () plane projects coexists limit cycle (test result);
Under Figure 10 (f) typical circuit parameter 1 state, R
1during=1.588k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t fork that () plane projects coexists limit cycle (test result);
Under Figure 11 (a) typical circuit parameter 2 state, R
1during=1.95k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t multicycle limit cycle (test result) that () plane projects;
Under Figure 11 (b) typical circuit parameter 2 state, R
1during=1.96k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t two snails volume chaos attractor (test result) that () plane projects;
Under Figure 11 (c) typical circuit parameter 2 state, R
1during=1.965k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t fork that () plane projects coexist single snail volume chaos attractor (test result);
Under Figure 11 (d) typical circuit parameter 2 state, R
1during=1.977k Ω, Cai Shi recalls resistance chaos circuit at v
1(t)-v
2t fork that () plane projects coexists limit cycle (test result).
Embodiment
A kind of Cai Shi with self-excitation attractor and hidden attractor of the present invention recalls resistance Generation of Chaotic Signals primarily of negative resistance G, coupling resistance R
1, electric capacity C
1, electric capacity C
2, inductance L, diode bridge cascade single order RC filter form broad sense memristor G
mcomposition.The broad sense memristor G that diode bridge cascaded RC filters is formed
mcircuit structure as shown in Figure 1, the realizing circuit of negative resistance G as shown in Figure 2, Cai Shi recall resistance chaos circuit main circuit structure figure as shown in Figure 3.Wherein, the memristor G of negative resistance G, diode bridge cascaded RC filters formation
mwith electric capacity C
1parallel connection, its positive and negative extreme being connected is denoted as a, b end respectively; Inductance L and electric capacity C
2parallel connection, its positive and negative extreme being connected is denoted as c, d end respectively; Coupling resistance R
1be connected across a, c two ends.
Mathematical modeling: the diode D shown in Fig. 1 described in circuit
kconstitutive relation can be described as
Wherein, k=1,2,3,4, ρ=1/ (2nV
t), v
kand i
krepresent respectively by diode bridge D
kvoltage and current, I
s, n and V
trepresent diode reverse saturation current, emission ratio and thermal voltage respectively.Four diodes adopt 1N4148, and parameter is I
s=6.891nA, n=1.827, and V
t=25mV.
If broad sense memristor two ends input voltage and electric current are respectively v and i, electric capacity C both end voltage is v
c, its Mathematical Modeling can be expressed as:
Containing four dynamic elements in circuit shown in Fig. 3, correspond respectively to one of four states variable: electric capacity C
1both end voltage v
1, electric capacity C
2both end voltage v
2, flow through inductance L current i
3, memristor G
minternal state variable and electric capacity C both end voltage v
c.Constitutive relation according to kirchhoffs law and circuit element can obtain:
Formula (2) is substituted into formula (4), the kinetics equation group that Cai Shi recalls resistance chaos circuit can be obtained, be described below:
Self-excitation attractor numerical simulation: utilize MATLAB simulation software to carry out Numerical Simulation Analysis to the circuit described by formula (5).Adopt Runge-Kutta (ODE45) algorithm to solve system equation, the phase rail figure of this circuit state variable can be obtained.When circuit parameter is set to typical circuit parameter 1:R
1=1.5k Ω, G=0.71mS, L=17.2mH, C
1=10nF, C
2=100nF, R=1k Ω, C=10nF; Circuit initial condition is set to: v
1(0)=0.01V, v
2(0)=0.01V, i
3(0)=0A and v
c(0)=0V, circuit can generate two snail volume chaos attractor shown in Fig. 4, and wherein, Fig. 4 (a) is at v
1(t)-v
2t phase rail figure that () plane projects, Fig. 4 (b) is at v
1(t)-i
3t phase rail figure that () plane projects, Fig. 4 (c) is at v
1(t)-v
ct phase rail figure that () plane projects, Fig. 4 (d) is at v
1t phase rail figure that ()-i (t) plane projects.
The balance point of circuit shown in Fig. 3 can obtain by solving following equation group:
Under typical circuit parameter 1 state, this circuit has three given balance points:
S
0＝(0,0,0,0)
S
1＝(0.8300,0,0.00055,0.0395) (7)
S
2＝(-0.8300,0,-0.00055,0.0395)
Solve the latent root equation of corresponding Jacobian matrix, three balance point characteristic of correspondence roots in formula (7) can be obtained and be respectively:
S
0:λ
1＝11074.3,λ
2,3＝-6497.1±j14387.3,λ
4＝-100015.1
(8)
S
1,2:λ
1,2＝1319.7±j20377.5,λ
3＝-31904.4,λ
4＝-159084.1
Can find out, S
0have a positive real root, a pair real part is negative Conjugate complex roots and a negative real root, is a unstable saddle point; S
1,2having a pair real part is positive Conjugate complex roots and two negative real roots, is that the saddle of a pair instability is burnt.Therefore, under this group parameter, it is self-excitation attractor that Cai Shi recalls the chaos attractor that produces of resistance chaos circuit.
In order to the dynamic behavior of further analysis circuit, selection circuit parameter R
1for variable element, remaining circuit parameter constant, utilizes MATLAB to emulate the bifurcation graphs of circuit state variable and Lyapunov exponential spectrum, with this analysis circuit parameter R
1cai Shi during change recalls the dynamics of resistance chaos circuit.Work as R
1when changing within the scope of 1.4k Ω ~ 1.65k Ω, state variable v
1bifurcation graphs as shown in Fig. 5 (a); Adopt Wolf algorithm to calculate corresponding Lyapunov exponential spectrum, result is as shown in Fig. 5 (b) simultaneously.Wherein intactly give front 3 Lyapunov indexes, be labeled as LE respectively
1, LE
2and LE
3.
Observe Fig. 5 (a) known, along with parameter R
1progressively increase, first the track of the variable of circuit state shown in Fig. 3 becomes cycle 1 limit cycle from remittance, then by period doubling bifurcation road transition to chaos, gets back to the cycle 1 finally by reverse period doubling bifurcation road by chaos.Meanwhile, there is multiple narrow cycle window in chaotic region, cycle window plays an important role in the dynamic behavior of chaos system develops.Lyapunov exponential spectrum shown in bifurcation graphs and Fig. 5 (b) shown in comparative analysis Fig. 5 (a), the system dynamics behavior that both map is consistent.
It should be noted that at 1.4k Ω≤R
1≤ 1.42k Ω and 1.564k Ω≤R
1, there is fork and to coexist phenomenon in≤1.65k Ω region, namely selects different variable initial values to define in two non-zero balance point domains of attraction independently attractor, makes system there are two kinds of different Bifurcation Patterns.When initial condition is set to v
1(0)=0.01V, v
2(0)=0.01V, i
3(0)=0A and v
c(0)=0V, produce attractor and concentrate on v
1t the region of () >0, is called right attractor; When initial condition is set to v
1(0)=-0.01V, v
2(0)=-0.01V, i
3(0)=0A and v
c(0)=0V, produce attractor and concentrate on v
1t the region of () <0, is called left attractor.These two kinds of Bifurcation Patterns coexisted are existed caused by two different non-zero balance points by the described system of formula (5).
Under Fig. 6 gives typical circuit parameter 1 state, R
1when getting different value, Cai Shi recalls the self-excitation attractor of resistance chaos circuit generation at v
1(t)-v
2the simulation result of t phase rail figure that () plane projects.Wherein, Fig. 6 (a) to coexist limit cycle (R for fork
1=1.42k Ω); Fig. 6 (b) is multicycle limit cycle (R
b=1.47k Ω); Fig. 6 (c) is two snail volume chaos attractor (R
b=1.5k Ω); Fig. 6 (d) is single scroll chaotic attractor (R for fork coexists
b=1.57k Ω); Fig. 6 (e) to coexist limit cycle (R for fork
b=1.593k Ω); Fig. 6 (d) to coexist limit cycle (R for fork
b=1.615k Ω).
Hidden attractor numerical simulation: when circuit parameter is set to typical circuit parameter 2:R
1=1.96k Ω, G=0.56mS, L=80mH, C
1=22nF, C
2=113nF, R=0.8k Ω, C=1 μ F, the initial value of circuit state variable is identical with Fig. 4, and this circuit still can produce two snail volume chaos attractor.Its phase rail figure simulation result at different phase plane inner projection as shown in Figure 7.Wherein, Fig. 7 (a) is at v
1(t)-v
2t phase rail figure that () plane projects, Fig. 7 (b) is at v
1t phase rail figure that ()-i (t) plane projects.
Under typical circuit parameter 2 state, solve formula (6) and three given balance points can be obtained:
S′
0＝(0,0,0,0)
S′
1＝(0.8324,0,0.00042,0.0343) (9)
S′
2＝(-0.8324,0,-0.00042,0.0343)
The characteristic root at three balance point places is respectively
S′
0:λ
1＝5939.07,λ
2＝-1250.15,λ
3,4＝-1250.15±j5204.42
(10)
S′
1,2:λ
1,2＝-109.55±j9394.59,λ
3＝-1191.43,λ
4＝-23830.48
Can find out, S '
0have a positive real root, a pair real part is negative Conjugate complex roots and a negative real root, is a unstable saddle point; S '
1,2having a pair real part is negative Conjugate complex roots and two negative real roots, is that a pair stable saddle is burnt.Therefore, under this group parameter, Cai Shi shown in Fig. 3 recalls resistance chaos circuit can not produce self-excitation attractor, and this circuit working is in hidden attractor oscillatory regime.
Work as R
1for variable element, when 1.9k Ω ~ 2.0k Ω regional change, state variable v
1bifurcation graphs as shown in Fig. 8 (a); In computational process, initial condition set-up mode is identical with Fig. 5.The Lyapunov exponential spectrum that employing Wolf algorithm calculates is as shown in Fig. 8 (b).By calculating different R
1time balance point characteristic root, can find: S
0be always unstable saddle point; S
1,2at R
1it is burnt that stable saddle is changed into by unstable saddle Jiao in=1.94741k Ω place.V shown in composition graphs 8
1t bifurcation graphs and the Lyapunov exponential spectrum of () can be found out, hidden attractor phenomenon mainly produces at 1.94741k Ω <R
1<1.9885k Ω region.Fig. 8 (c) gives 1.915k Ω <R
1v in <1.98k Ω region
1the bifurcation graphs of (t), the recessive chaos phenomenon that can find out hidden attractor region existence two snail volume chaos attractor, single snail that coexists entrainments introduction, the limit cycle that coexists etc. enriches.
Under Fig. 9 gives typical circuit parameter 2 state, in hidden attractor region, R
1when getting different value, Cai Shi recalls the hidden attractor of resistance chaos circuit generation at v
1(t)-v
2the simulation result of t phase rail figure that () plane projects.Wherein, Fig. 9 (a) is multicycle limit cycle (R
1=1.95k Ω); Fig. 6 (b) is two snail volume chaos attractor (R
1=1.961k Ω); Fig. 6 (c) is single scroll chaotic attractor (R for fork coexists
1=1.967k Ω); Fig. 6 (d) to coexist limit cycle (R for fork
1=1.98k Ω).
Experiment test: in order to verify that Cai Shi recalls science and the physical realizability of resistance chaos circuit, the present invention has carried out experimental verification to above-mentioned theory analysis and Numerical Simulation Results.In experimental circuit, diode bridge adopts 1N4148 diode composition; Negative resistance equivalent electric circuit core devices selects OP07CP operational amplifier, and operating voltage is ± 15V; Resistance adopts accurate adjustable resistance, and electric capacity is leaded multilayer ceramic capacitor, and inductance is manual coiling inductance.Theory analysis and numerical simulation show, the self-excitation attractor that this circuit produces and hidden attractor insensitive to initial condition, impact signal when utilizing experimental circuit to power up, be easy to realize required for state variable initial value.
Adopt Tektronix DPO3034 digital storage oscilloscope to catch measured waveform, carried out experimental verification respectively to the self-excitation attractor shown in Fig. 6 and Fig. 9 and hidden attractor phase rail figure, experimental result respectively as shown in Figure 10 and Figure 11.
Test the self-excitation attractor that records at v shown in Figure 10
1(t)-v
2projection in (t) plane; Figure 10 (a) is the limit cycle (R that coexists that diverges
1=1.428k Ω); Figure 10 (b) is multicycle limit cycle (R
1=1.474k Ω); Figure 10 (c) is two snail volume chaos attractor (R
1=1.50k Ω); Figure 10 (d) is the single scroll chaotic attractor (R that coexists that diverges
1=1.542k Ω); Figure 10 (e) is the limit cycle (R that coexists that diverges
1=1.593k Ω); Figure 10 (f) is the limit cycle (R that coexists that diverges
1=1.588k Ω).
Test the hidden attractor that records at v shown in Figure 11
1(t)-v
2projection in (t) plane; Figure 11 (a) is multicycle limit cycle (R
1=1.95k Ω); Figure 11 (b) is two snail volume chaos attractor (R
1=1.96k Ω); Figure 11 (c) is the single scroll chaotic attractor (R that coexists that diverges
1=1.965k Ω); Figure 11 (d) is the limit cycle (R that coexists that diverges
1=1.977k Ω).
Due to the impact of parasitic parameter, value and the simulation result of actual measurement resistance have trickle deviation.But the non-linear phenomena observed in experimental circuit and development law thereof and simulation result fit like a glove, can the correctness of proof theory analysis and numerical simulation.Comparing result can illustrate: the Cai Shi constructed by the present invention recalls resistance chaos circuit can produce self-excitation attractor and hidden attractor, and hidden attractor phenomenon determines primarily of circuit parameter, can observe in experimental circuit easily.Therefore, the Cai Shi with self-excitation attractor and hidden attractor constructed by the present invention recalls theoretical foundation and the realizability physically that resistance chaos circuit has science, can play positive impetus to the engineer applied of chaos circuit.
Above-described embodiment is only for example of the present invention is clearly described, and is not the restriction to embodiments of the present invention.For others skilled in the art, other multi-form variation or improvement can also be made on the basis of the above description.Here exhaustive without the need to also giving all execution modes.
Claims (3)
1. what have self-excitation attractor and hidden attractor recalls a resistance Generation of Chaotic Signals, it is characterized in that: comprise negative resistance G, resistance R
1, electric capacity C
1, electric capacity C
2, inductance L, diode bridge cascade single order RC filter form broad sense memristor G
m; Wherein, the broad sense memristor G of negative resistance G, diode bridge cascade single order RC filter formation
mwith electric capacity C
1parallel connection, its positive and negative a, b of being extremely denoted as respectively holds; Inductance L and electric capacity C
2parallel connection, its positive and negative c, d of being extremely denoted as respectively holds; Coupling resistance R
1be connected across between a, c two ends.
2. according to claim 1 a kind of there is self-excitation attractor and hidden attractor recall resistance Generation of Chaotic Signals, it is characterized in that: work as R
1=1.5k Ω, G=0.71mS, L=17.2mH, C
1=10nF, C
2=100nF, R=1k Ω, during C=10nF, circuit can produce self-excitation attractor, wherein can observe double scroll chaos, multicycle limit cycle, fork coexists single scroll chaotic attractor, fork coexists the complex nonlinear phenomenons such as limit cycle.
3. according to claim 1 a kind of there is self-excitation attractor and hidden attractor recall resistance Generation of Chaotic Signals, it is characterized in that: work as R
1=1.96k Ω, G=0.56mS, L=80mH, C
1=22nF, C
2during=113nF, R=0.8k Ω, C=1 μ F, circuit can produce hidden attractor, wherein can observe double scroll chaos, multicycle limit cycle, fork coexists single scrollwork attractor, fork coexists the complex nonlinear phenomenons such as limit cycle.
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