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Figure from:Dissipative Toda-Rayleigh lattice and its oscillatory modes

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FIG. 2. Block scheme of the Toda-Rayleigh lattice: (DC) is a double capacitor, (NR) accounts for the nonlinear dissipative func- tion G(P) in the model Eq. (1); (VA) provides variable shift of the nonlinear function G(P) to compensate voltage drift; R is an addi- tional stabilizing high value resistor (R=120 kQ,). Diodes are 1n4148.

Figure 2 Block scheme of the Toda-Rayleigh lattice: (DC) is a double capacitor, (NR) accounts for the nonlinear dissipative func- tion G(P) in the model Eq. (1); (VA) provides variable shift of the nonlinear function G(P) to compensate voltage drift; R is an addi- tional stabilizing high value resistor (R=120 kQ,). Diodes are 1n4148.

Related Figures (18)

FIG. 1. Circuit implementations of the Toda lattice: (a) nonlin- ear LC circuit by Hirota and Suzuki [35], (b) diode-double capaci- tor circuit by Singer and Oppenheim [36]. The lattice system Eq. (3) has two independent parameters: (i) @q which is the base frequency of linear oscillations in the original Toda lattice [34], and (ii) w which is the linear pumping constant accounting for the dissipative Rayleigh part of the system. Their values and relation define the spatio-temporal dynamics of the chain. Further, for conve- nience, we shall use nearest-neighbor relative coordinates along the lattice, i.e.
FIG. 1. Circuit implementations of the Toda lattice: (a) nonlin- ear LC circuit by Hirota and Suzuki [35], (b) diode-double capaci- tor circuit by Singer and Oppenheim [36]. The lattice system Eq. (3) has two independent parameters: (i) @q which is the base frequency of linear oscillations in the original Toda lattice [34], and (ii) w which is the linear pumping constant accounting for the dissipative Rayleigh part of the system. Their values and relation define the spatio-temporal dynamics of the chain. Further, for conve- nience, we shall use nearest-neighbor relative coordinates along the lattice, i.e.
The direct measurement of dV,,/dt (which corresponds to the momentum in the mechanical analog) is not possible. The alternative is to obtain these values from voltage differences FIG. 3. Particular circuit implementation of the black boxes in the scheme shown in Fig. 2: (a) double capacitor, (b) nonlinear resistor, and (c) voltage adder. Diodes are 1n4148. Unless otherwise specified we take the following parameter values C,=Cy, =0.1 uF, R,}=R2=10 kQ.
The direct measurement of dV,,/dt (which corresponds to the momentum in the mechanical analog) is not possible. The alternative is to obtain these values from voltage differences FIG. 3. Particular circuit implementation of the black boxes in the scheme shown in Fig. 2: (a) double capacitor, (b) nonlinear resistor, and (c) voltage adder. Diodes are 1n4148. Unless otherwise specified we take the following parameter values C,=Cy, =0.1 uF, R,}=R2=10 kQ.
FIG. 4. Experimental current-voltage relation for the nonlinear resistor, /,,(AV), where AV=V,,— V,— Vex; is voltage applied to its terminals.
FIG. 4. Experimental current-voltage relation for the nonlinear resistor, /,,(AV), where AV=V,,— V,— Vex; is voltage applied to its terminals.
Introducing in Eq. (13) new dimensionless variables, p, =(d/B)P,, y,=BY,,, with Comparing the ‘mechanical Eq. (1) and electrical Eq. (13) implementations of the Toda-Rayleigh lattice we get the fol- lowing relations between parameters and variables:
Introducing in Eq. (13) new dimensionless variables, p, =(d/B)P,, y,=BY,,, with Comparing the ‘mechanical Eq. (1) and electrical Eq. (13) implementations of the Toda-Rayleigh lattice we get the fol- lowing relations between parameters and variables:
FIG. 5. Dynamics of a single unit: (a) phase plane portrait of a single unit in the case “>, (single/double arrows on the limit cycle correspond to slow/fast motions; the dashed line corresponds to the curve v=a sinh '[(u—w°*)/a]). (b) Experimental measure- ment of voltage at the unit V, (V«u) and voltage between upper and left-side pins of the double capacitor AV«dV/dtxu (Ra. =681 QO, w~0.16, wy~9X 1074).
FIG. 5. Dynamics of a single unit: (a) phase plane portrait of a single unit in the case “>, (single/double arrows on the limit cycle correspond to slow/fast motions; the dashed line corresponds to the curve v=a sinh '[(u—w°*)/a]). (b) Experimental measure- ment of voltage at the unit V, (V«u) and voltage between upper and left-side pins of the double capacitor AV«dV/dtxu (Ra. =681 QO, w~0.16, wy~9X 1074).
FIG. 6. Oscillation period as a function of the ratio Ay/Rq,. for two different values of the capacitance C, [Fig. 3(a)]. Squares cor- respond to C,=0.105 mF and triangles correspond to C, =0.053 uF. The straight line is the theoretical estimate of the period Eq. (21).
FIG. 6. Oscillation period as a function of the ratio Ay/Rq,. for two different values of the capacitance C, [Fig. 3(a)]. Squares cor- respond to C,=0.105 mF and triangles correspond to C, =0.053 uF. The straight line is the theoretical estimate of the period Eq. (21).
FIG. 7. Ring with two units. Experimental voltages at the units for different values of V,,,, corresponding to two nonoscillatory modes (top and bottom) and the optical mode (center region) (R4, = 13681 0, w~3.75X 1074, wy~ 10°).
FIG. 7. Ring with two units. Experimental voltages at the units for different values of V,,,, corresponding to two nonoscillatory modes (top and bottom) and the optical mode (center region) (R4, = 13681 0, w~3.75X 1074, wy~ 10°).
FIG. 8. Average voltage on the units V vs current through the nonlinear resistor /,, for the optical mode (squares) and for the homogeneous solutions (up/down triangles correspond to “‘left’’/ “right” rotations).
FIG. 8. Average voltage on the units V vs current through the nonlinear resistor /,, for the optical mode (squares) and for the homogeneous solutions (up/down triangles correspond to “‘left’’/ “right” rotations).
FIG. 9. Schematic diagram of mode amplitudes Eq. (43) as a function of bifurcation parameter for different mode number m (N=6).
FIG. 9. Schematic diagram of mode amplitudes Eq. (43) as a function of bifurcation parameter for different mode number m (N=6).
FIG. 11. Oscilloscope traces of the voltages taken from all six units of the ring for three different modes. All parameter values are the same as for Fig. 10. Figure 11 shows experimental traces for three different modes taken for the same external voltage. The m=1 wave has the largest amplitude while waves with m=2,3 have about the same amplitude as theoretically predicted (Fig. 9). Besides, the shape and the mean values of voltages are clearly different for all modes. Figure 12 shows voltage dif- ferences, V,—V,,, taken from all units (Fig. 2). These volt- age differences are proportional to the time derivatives of voltages dV,,/dt [see Eq. (11)]. Due to the designed left- right symmetry of the system we have two similar waves propagating clockwise and counterclockwise. The clockwise [Fig. 11(a)]/counterclockwise [Fig. 11(b)] waves propagate over some negative/positive base levels and have positive/
FIG. 11. Oscilloscope traces of the voltages taken from all six units of the ring for three different modes. All parameter values are the same as for Fig. 10. Figure 11 shows experimental traces for three different modes taken for the same external voltage. The m=1 wave has the largest amplitude while waves with m=2,3 have about the same amplitude as theoretically predicted (Fig. 9). Besides, the shape and the mean values of voltages are clearly different for all modes. Figure 12 shows voltage dif- ferences, V,—V,,, taken from all units (Fig. 2). These volt- age differences are proportional to the time derivatives of voltages dV,,/dt [see Eq. (11)]. Due to the designed left- right symmetry of the system we have two similar waves propagating clockwise and counterclockwise. The clockwise [Fig. 11(a)]/counterclockwise [Fig. 11(b)] waves propagate over some negative/positive base levels and have positive/
FIG. 10. Oscilloscope traces of current through a diode for dif- ferent modes in the ring of six units. For each mode the first peak has been positioned at the same instant of time (V.,,=0.45 V, Roc= 13681 0, w~3.75X 1074, wo~ 10°).
FIG. 10. Oscilloscope traces of current through a diode for dif- ferent modes in the ring of six units. For each mode the first peak has been positioned at the same instant of time (V.,,=0.45 V, Roc= 13681 0, w~3.75X 1074, wo~ 10°).
FIG. 12. Oscilloscope traces showing propagation of the two longest waves: (a) wave running clockwise, m=1 (Vex =0.45 V) and (b) wave running counterclockwise, m=—1 (Vox =0.45 V). The Y axes provide voltage differences between voltage on the unit and left side pin of the double capacitor (Fig. 2) that is proportional to the time derivative of V,,(t) Eq. (11). The total time interval corresponds to the oscillation period. All parameter values (excluding V,,,) are the same as on Figs. 10 and 11.
FIG. 12. Oscilloscope traces showing propagation of the two longest waves: (a) wave running clockwise, m=1 (Vex =0.45 V) and (b) wave running counterclockwise, m=—1 (Vox =0.45 V). The Y axes provide voltage differences between voltage on the unit and left side pin of the double capacitor (Fig. 2) that is proportional to the time derivative of V,,(t) Eq. (11). The total time interval corresponds to the oscillation period. All parameter values (excluding V,,,) are the same as on Figs. 10 and 11.
FIG. 14. Jumping sequence between modes for low values of the resistor R=60 kO, (Fig. 2). The remaining parameter values are the same as for Fig. 13.
FIG. 14. Jumping sequence between modes for low values of the resistor R=60 kO, (Fig. 2). The remaining parameter values are the same as for Fig. 13.
FIG. 13. Average voltage on units for different modes and di- rection of jumps between them vs the external voltage V,,, in the ring of six units. Each curve corresponds to a certain mode with mode number marked m on its corresponding top (R=120 kQ).
FIG. 13. Average voltage on units for different modes and di- rection of jumps between them vs the external voltage V,,, in the ring of six units. Each curve corresponds to a certain mode with mode number marked m on its corresponding top (R=120 kQ).
=(/u/2a9)w, r= mu, s= Vu. » tnew=2@otoia, and ex- pand sinh[(\/4/2@))w]. Then from Eq. (24) we get with e=/2w,). Now introducing amplitude-phase (a,@) variables, w=2acos(t+¢), u=—2a sin(t+ @), and averag- ing over the period of oscillations we obtain
=(/u/2a9)w, r= mu, s= Vu. » tnew=2@otoia, and ex- pand sinh[(\/4/2@))w]. Then from Eq. (24) we get with e=/2w,). Now introducing amplitude-phase (a,@) variables, w=2acos(t+¢), u=—2a sin(t+ @), and averag- ing over the period of oscillations we obtain
FIG. 15. Phase plane of the system Eq. (A3). The derivative of W(a,z) at W=h along the solution curves
FIG. 15. Phase plane of the system Eq. (A3). The derivative of W(a,z) at W=h along the solution curves
The first two equations, Eq. (A3), are independent of the third one, Eq. (A4), and we can consider them separately.
The first two equations, Eq. (A3), are independent of the third one, Eq. (A4), and we can consider them separately.