论文集 2018_Symmetric periodic bursting behavior and bifurcation mechanism in a third-order memristive diode bridge-based oscillator

Chaos Solitons and Fractals 109 2018 146 153 Contents lists available at ScienceDirect Chaos Solitons and Fractals Nonlinear Science and Nonequilibrium and Complex Phenomena journal homepage Review Symmetric periodic bursting behavior and bifurcation mechanism in a third order memristive diode bridge based oscillator B C Bao P Y Wu H Bao H G Wu X Zhang M Chen School of Information Science and Engineering Changzhou University Changzhou 213164 China a r t i c l e i n f o Article history Received 22 January 2018 Revised 20 February 2018 Accepted 23 February 2018 Keywords Symmetric periodic bursting Memristive oscillator Bifurcation sets Bifurcation mechanism a b s t r a c t This paper presents a novel third order autonomous memristive diode bridge based oscillator with fast slow effect Based on the modeling of the presented memristive oscillator stability of the equilibrium point is analyzed by using the eigenvalues of the characteristic polynomial and then symmetric periodic bursting behavior is revealed through bifurcation diagrams phase plane plots time sequences and 0 1 test Furthermore bifurcation mechanism of the symmetric periodic bursting behavior is explored by con structing the fold and Hopf bifurcation sets of the fast scale subsystem with the variations of the system parameter and slow scale variable Consequently the presented memristive oscillator is always unstable and exhibits complex dynamical behavior of symmetric periodic bursting oscillations with a symmetric fold Hopf cycle cycle burster In addition experimental measurements are performed by hardware circuit to confi rm the numerical simulations 2018 Elsevier Ltd All rights reserved 1 Introduction Based on multifarious memristor emulators using commercially available components various kinds of memristor based chaotic circuits have been physically implemented for breadboard experi mental studies in recent years 1 10 which effectively promotes the development of the fundamentals of memristive circuits Gen erally the memristor emulators mainly have two types of the implementation circuits with the off the shelf discrete compo nents namely the op amps and analog multipliers based equiv alent circuits 1 4 and memristive diode bridges cascaded with L RC or LC components 5 10 Like as the constructing method of memristor based chaotic circuits reported in 1 10 this pa per presents a novel third order autonomous memristive oscilla tor which is extended from the memristive Wien bridge oscillator 8 or memristive diode bridge coupled Sallen Key low pass fi lter 9 10 However in most of the previous reports except Ref 8 all the state variables of the memristive circuits are specifi ed on the same time scale In this paper the presented third order memris tive oscillator can be considered as two subsystems to have or der gap between the two time scales related to the two subsys tems 11 12 Particularly it should be clarifi ed that the newly pre sented autonomous memristive oscillator is only third order but Corresponding author E mail address mchen M Chen the memristive Wien bridge oscillator reported in 8 is fourth order resulting in the briefness and easiness of the mathematical model and quantitative analyses Recently based on some special memristor emulators or pure mathematical memristor models several memristor based spiking and bursting neuron circuits have been developed 13 15 which denote that these memristor based application circuits can exhibit the spiking and bursting fi ring behaviors with a biologically plausi ble spike shapes 13 16 However the hardware experimental level on a breadboard is hard to fabricate due to the complexity of these memristor emulators or pure mathematical memristor models The spiking and bursting oscillations are often encountered in a large class of neuronal models 16 21 and nonlinear dynamical systems 11 12 22 26 These neuronal models and nonlinear dynamical sys tems usually involve two time scales leading to the occurrence of the bursting fi ring behaviors since the fast scale variables are mod ulated by the slow scale variables Similarly with the appropriate parameters the presented third order memristive oscillator refer ring to two time scales also can exhibit the novel phenomenon of symmetric periodic bursting oscillations as well which to the au thors knowledge is rarely appeared in the previously published literatures 8 In particular the symmetric periodic bursting phe nomenon emerging in such a third order memristive oscillator is very interesting which can be taken as a paradigm in mathemati cal and experimental demonstrations of some special nonlinear dy namics https doi org 10 1016 j chaos 2018 02 031 0960 0779 2018 Elsevier Ltd All rights reserved B C Bao et al Chaos Solitons and Fractals 109 2018 146 153 147 Fig 1 Third order autonomous memristive oscillator a memristive diode bridge based oscillator b memristive diode bridge cascaded with an inductor The paper content is constructed as follows In Sect 2 based on a simplifi ed fi rst order memristive diode bridge a novel third order autonomous memristive oscillator is presented and its cor responding three dimensional system model is then built More over with the eigenvalues of the characteristic polynomial stabil ity of the equilibrium point is analyzed In Sect 3 by using sev eral conventional dynamical methods period oscillations and sym metric periodic bursting behaviors are revealed In the following Sect 4 based on the fast scale and slow scale subsystems divided from the three dimensional system model the evolutions of the equilibrium point and its stability of the fast scale subsystem are discussed and its bifurcation mechanism is explored through con structing the fold and Hopf bifurcation sets of the system parame ter and slow variable In Sect 5 hardware circuit experiments are performed to confi rm the numerical simulations The conclusions are summarized in the last section 2 Third order autonomous memristive oscillator A novel third order autonomous memristive oscillator is pre sented as shown in Fig 1 a by introducing a fi rst order mem ristive diode bridge emulator only cascaded with an inductor as shown in Fig 1 b to substitute the resistor of the parallel RC network in Wien bridge oscillator 27 Also the proposed mem ristive oscillator can be derived from the fourth order memris tive Wien bridge oscillator reported in 8 through simplifying the LC network in second order memristive diode bridge emulator by an inductor or the third order memristive diode bridge coupled Sallen Key low pass fi lter reported in 9 10 through removing the grounded resistor It is remarked that the third order memristive diode bridge based oscillator has the same topological structure as the traditional Wien bridge oscillator 2 1 State equations and normalized model For the emulator input voltage v 1 and current i along with the inductor L current i 0 as shown in Fig 1 b the memristive diode bridge emulator is modeled by 9 10 i W i 0 v 1 v i 0 2 I S tanh v 1 d i 0 d t 1 L ln 2 I S cosh v 1 1 L ln i 0 2 I S 1 where 1 2 nV T Three considered model parameters of 1N4148 diode are assigned as I S 5 84 nA n 1 94 and V T 25 mV respec tively With the two capacitor voltages v 1 and v 2 and one inductor current i 0 in Fig 1 the circuit equations of the third order au tonomous memristive diode bridge based oscillator are established as d i 0 d t 1 L ln 2 I S cosh v 1 1 L ln i 0 2 I S d v 1 d t k v 1 R C 1 i 0 2 I S tanh v 1 C 1 v 2 R C 1 d v 2 d t k v 1 R C 2 v 2 R C 2 2 where k R 2 R 1 By utilizing three dimensionless variables and four dimensionless parameters as x R i 0 y v 1 z v 2 t R C 1 a R 2 C 1 L b 2 R I S C 1 C 2 k R 2 R 1 3 A normalized system model is thus built from 2 as x a ln b cosh y a ln x b y ky x b tanh y z z ky z 4 Therefore the newly presented autonomous memristive oscilla tor is a three dimensional nonlinear dynamical system which has much simpler algebraic equations than the fourth order memris tive Wien bridge oscillator reported in 8 more suitable for the quantitative analyses The desired circuit parameters of linear elements in Fig 1 are chosen as L 20 mH C 1 1 nF C 2 33 nF R 30 k R 1 2 k and R 2 6 k and the considered system parameters of 4 are then calculated by 3 as a 45 b 0 0036 0 0303 and k 3 5 Note that the inductor L and capacitors C 1 and C 2 considered in Fig 1 are three differentiable elements which are completely different from the non differentiable elements reported in 28 2 2 Stability for the equilibrium point The system 4 has only one original equilibrium point 0 0 0 The characteristic polynomial of the system Jacobian at 0 0 0 is given by P a b 2 k b b 0 6 Thus three eigenvalues are yielded as 1 0 5 k b 0 5 k b 2 4 b 2 0 5 k b 0 5 k b 2 4 b 3 a b 7 As k is considerably larger than b and the roots of 1 and 2 are positive real constants whereas the root of 3 is negative real constant which indicate that the original equilibrium point 0 0 0 of 4 is an unstable saddle point Considering the typical system parameters given in 5 as an example three eigenvalues in 7 are calculated as 1 2 9661 2 0 0 0 0 04 and 3 12 50 0 Con sequently the third order autonomous memristive diode bridge based oscillator is always unstable 3 Symmetric periodic bursting behavior By utilizing MATLAB software tool ODE23S algorithm is con sidered to draw bifurcation diagrams phase plane plots time se quences and 0 1 test in our next works Due to the existence of complex nonlinearities in system 4 pure mathematical anal ysis methods for instance integral transform methods proposed in 29 32 are inapplicable to the solutions of system 4 leading to that some computer aided simulation tools should be utilized 148 B C Bao et al Chaos Solitons and Fractals 109 2018 146 153 Fig 2 Dynamics versus the resistance R a bifurcation diagram b partial enlarged view of a where 0 0303 and k 3 are fi xed while a and b are synchronously adjusted with the variation of the resistance R Fig 3 Numerically periodic behaviors by using time sequences of y and z for different values of R a R 23 k b R 25 k c R 25 92 k d R 25 96 k Taking the resistance R as a bifurcation parameter i e the sys tem parameters a and b being synchronously adjustable the bifur cation diagrams of the local maxima X of the state variable x of system 4 are plotted in Figs 2 a and 2 b respectively where Fig 2 b is the partial enlarged view of Fig 2 a It should be mentioned that the fi rst Lyapunov exponent is always less than or equal to zero indicating the inexistence of chaotic behavior in the third order memristive diode bridge based oscillator As the adjusting resistance R is changed from 20 k to 50 k the system orbit starting from the period 1 oscillation successively enters into multiple types of periodic oscillations with period 4 6 and 8 at R 24 06 k 25 90 k and 25 94 k respectively Be yond the region of the periodic oscillations the system orbit sud denly shifts into symmetric periodic bursting oscillations with un countable periodicities at R 26 01 k and then the periodicities and dynamic amplitudes of the system orbit will gradually shrink with the increase of the resistance R in a wider parameter region In the region of 20 k R 19 2455 or z 19 2455 i e G 1 y 1 0 only one cross ing point exists when 19 2455 z 19 2455 i e G 1 y 2 0 150 B C Bao et al Chaos Solitons and Fractals 109 2018 146 153 z R k FB1 R 30 k H1 FB2 F1H2 HB2 HB1 F2 H4F3H3F4 R 45 k Fig 6 Fold and Hopf bifurcation sets of the fast scale subsystem distributed in the z R phase plane G 1 y 1 three crossing points appear while when z 19 2455 or z 19 2455 i e G 1 y 1 0 or G 1 y 2 0 one tangent point along with one crossing point occur leading to the generation of fold bi furcation 11 To estimate the stability of the equilibrium point E FS the Jaco bian matrix at E FS is derived and simplifi ed as J FS a b sech y a tanh y tanh y k b sech y 14 The corresponding characteristic polynomial is deduces as P 2 0 15 where b a b sech y k and a ka b sech y With the characteristic polynomial 15 bifurcation analyses of the fast scale subsystem can be performed It should be remarkable that the coeffi cients and of the characteristic polynomial 15 are only determined by the solutions of the equilibrium point Eq 9 whose values are closely associ ated with the slow variable z Therefore the stability of the fast scale subsystem is dependent to the slow scale subsystem 4 2 Fold and Hopf bifurcation sets Observed from Fig 5 the equilibrium points of the fast scale subsystem have a transition from three to one with the varia tion of the slow variable z At the critical condition of G 1 y 1 0 or G 1 y 2 0 a small perturbation of z may cause the tangent point to disappear or to split into two crossing points resulting in the occurrence of fold bifurcation Based on the critical condition the fold bifurcation set can be easily derived Alternatively in consideration of that the fold bifurcation is re ferred to one zero eigenvalue 14 can be thereby rewritten as P FB 0 16 GND Memristive diode bridge 15 V 15 V R R2 R1 U C2 C1 D1 D2D3 D4 L Second order autonomous oscillator Fig 8 Screen capture of the experimental prototype for the presented third order autonomous memristive diode bridge based oscillator the left is a global graph of the oscilloscope linking to the circuit breadboard and the right is the partial en larged drawing of the circuit breadboard i e 0 Therefore with the Eq 9 the fold bifurcation set can be deduced as FB k y b sinh y z 0 a ka b sech y 0 17 and simplifi ed as FB k arccosh k b b sinh arccosh k b z 0 18 The Hopf bifurcation is associated with the appearance of a pair of pure imaginary eigenvalues Then 15 can be expressed as P HB 2 0 19 i e 0 Therefore with the Eq 9 the Hopf bifurcation set can be gotten as HB k y b sinh y z 0 b a b sech y k 0 20 and simplifi ed as HB k arccosh a b b k b sinh arccosh a b b k z 0 21 Consider that the resistance R is adjusted in the region 20 k 50 k By MATLAB numerical simulations the fold bifurcation Fig 7 Bifurcation mechanism of periodic bursters at a R 30 k and b R 45 k where F 1 F 2 F 3 and F 4 stand for the fold bifurcation points H 1 H 2 H 3 and H 4 represent the Hopf bifurcation points and EP 1 and EP 2 are the equilibrium point curves of the fast scale subsystem with the z variation B C Bao et al Chaos Solitons and Fractals 109 2018 146 153 151 Fig 9 Experimentally periodic behaviors by using time sequences of v 1 t and v 2 t for different R a R 21 5 k b R 21 8 k Fig 10 Symmetric periodic bursting behaviors by hardware circuit experiments a1 phase plane plot in the v 1 t v 2 t plane and a2 time sequences of v 1 t and v 2 t at R 26 6 k b1 phase plane plot in the v 1 t v 2 t plane and b2 time sequences of v 1 t and v 2 t at R 41 5 k set of 18 marked as FB 1 and FB 2 and Hopf bifurcation set of 21 marked as HB 1 and HB 2 in the z R phase plane can be to gether drawn as shown in Fig 6 The bifurcation sets in Fig 6 are the possible bifurcations of the fast scale subsystem which may be correspondence to the turning points between the quiescent state and spiking state when periodic bursting oscillation takes place in the system 4 4 3 Bifurcation mechanism of periodic bursters Let R 30 k and R 45 k be two examples using for ex plaining the mechanism of the periodic bursters Referring to Fig 6 when R 30 k two fold bifurcation points locate at z 19 2455 V marked as F 2 and F 1 whereas two Hopf bifurcation points are at z 12 0738 V marked as H 2 and H 1 The phase plane plot of system 4 and the equilibrium point curve EP 1 152 B C Bao et al Chaos Solitons and Fractals 109 2018 146 153 of the fast scale subsystem dotted by the four bifurcation points are depicted in Fig 7 a The moving trajectory of the system 4 beginning at H 1 grad ually enters into the quiescent state with the decrease of the os cillating amplitude The quiescent state ends once the trajectory moves at the fold bifurcation point F 1 leading to that the trajec tory rises to the upper and the system 4 goes into the spiking stat The spiking oscillation around the equilibrium point curve EP 1 is maintained with the large oscillating amplitude After the Hopf bifurcation occurs at H 2 the moving trajectory re enters into the quiescent state and dives to the lower via the fold bifurcation route at F 2 The spiking state appears again and ends once the trajectory moves at the Hopf bifurcation point H 1 which initiates next pe riod of the periodic burster Due to the small oscillation of the qui escent state the periodic burster can be regarded as a symmetric fold Hopf cycle cycle burster 16 Similarly when R 45 k two fold bifurcation points locate at z 18 0291 marked as F 4 and F 3 whereas two Hopf bifurca tion points are at z 5 4598 marked as H 3 and H 4 The phase plane plot of the system 4 and the equilibrium point curve EP 2 of the fast scale subsystem dotted by the four bifurcation points are demonstrated in Fig 7 b By utilizing the bifurcation analysis in the above section analogous bifurcation details of EP 2 are en countered for this case i e the system 4 has a transition from the quiescent state to spiking state and back via the fold and Hopf bifurcation routes It should be addressed that the spanning time length of the quiescent oscillation increases with the resistance R 5 Hardware experimental confi rmations According to the circuit designed in Fig 1 an experimental pro totype of the presented third order autonomous memristive diode bridge based oscillator is photographed and its screen capture is shown in Fig 8 Three precision potentiometers two monolithic ceramic capacitors four 1N4148 diodes a manually winding induc tor and an AD711KN op amp with 15 V power supply are chosen The linear element parameters used during numerical simulations are selected in hardware experiments and the experimental phase plane orbits and time sequences are measured by a 4 channel dig ital oscilloscope By turning the precision poten