(astable multivibrator) (monostable multivibrator) (bistable multivibrator) (excitable vector field) (a) (b) 2

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1 Slow-fast Hybrid (a) (b)

2 (astable multivibrator) (monostable multivibrator) (bistable multivibrator) (excitable vector field) (a) (b) 2

3 2 v 0 = a(v d ) Slow-fast 5. Hybrid 2. 2 v 0 = a(v d ), v d = v p v n () 3 3(c) 3

4 3 v 0 = a(v d ) v p + v p Kirchhoff v 0 v p R 3 + v p R 4 + E v p R 5 = 0 (2) p = v p = v 0 R 3 + E R 5 R 3 + R 4 + = pv 0 + qe (3) R 5 R 3 R 3 + R 4 +, q = R 5 R 5 R 3 + R 4 + R v n v v n = v R 2 C dv dt + v = v 0 (4) v d (3) (4) v d = v p v n = pv 0 + qe v (5) 4

5 4 2.3 () (5) v d v 0 v (4) v v 0 v 4 (a) () (5) 0 v 0 E v 0 v v 0 4 (b) v v 0 (4) 2.4 Slow-fast 2.4. v 0 []IT Application Report [2] µ dv 0 dt + v 0 = a(v d ) = a(pv 0 + qe v) (6) µ R 2 C * : µ * v0 5

6 5 (7) (a) v (b) v v 0. (6) 2. (6) v 0 = E v 0 = 0 0 < v 0 < E 3. (v, v 0 ) = ((p + q)e, E) (v, v 0 ) = (qe, 0) * (b) v (6) Slow-fast v (4) (6) slow-fast R 2 C dv dt + v = v 0 µ dv 0 dt + v 0 = a(pv 0 + qe v) (7) 5. v 0 = a(pv 0 + qe v) null cline *2 (tangent bifurcation) (saddle-node bifurcation) 6

7 v (slow manifold) 2. v v = (p + q)e, v 0 = E b v v v = qe, v 0 = d b c d a 3. v v 0 = Ev 0 = 0 v 0 = p (v qe) v = v 0 = (v qe) q 4. abcd 5. T 6 ab R 2 C dv dt + v = E a v = qe b (p + q)e T v(0) = qe v(t) v(t) = (q )Ee R 2 C t + E t = T v(t ) = (p + q)e T T = R 2 C ln q p + q (8) cd T 2 T 2 = R 2 C ln p + q q (9) T T = R 2 C ln q p + q + R 2C ln p + q q = R 2 C ln (q )(p + q) (p + q )q (0) 2.5 Hybrid (FSM: Finite State Machine) hybrid 7 7

8 6. v 0 M = [qe, (p + q)e] v M 2. v 0 = 0 R 2 C dv dt + v = 0 () v 0 = 3. R 2 C dv dt + v = E (2) v = qe v v = qe v = (p + q)e v v = (p + q)e. v 0 = 0 v(0) v(0) M = [qe, (p + q)e] 8

9 mode 0 State : v 0 = 0 v(t) Dynamics : R 2 C v + v = 0 v(t) < qe transitions v(t) > (p+q)e mode State : v 0 = v(t) Dynamics : R 2 C v + v = E 7 hybrid 8 2. v(t) () v = qe * M 8 v 0 M, M 0, M 6 *3 slow-fast hybrid 9

10 9 (a), (b) R 3. (a) 3.. C dv C dt = E v + v 0 v R R 2 v ( dv C dt + + ) v = E + v 0 (3) R R 2 R R 2 0

11 (6) C dv dt µ dv 0 dt + ( R + R 2 ) v = E R + v 0 R 2 + v 0 = a(pv 0 + qe v) v 0 v v 0 = E v = E v 0 = 0 v = (4) R 2 R + R 2 E q = R 5 R 3 + R 4 + < R 5 R 2 R + R 2 (5) v R 2 = E R + R 2 0(a) (b) v R 2 r 2 R 2 v qe 0(a) a bb c c d v * (v, v 0 ) = (v, 0) 2. (v, v 0 ) = (qe, 0) slow-fast 0(a) a,b,c,d *4 R2

12 0 (a) (b) (spiking) (excitable vector field) E. M. Izhikevich [3] 3..3 LED (b) C C dv dt + v R = v 0 v R 2 v ( dv C dt + + ) v = v 0 (6) R R 2 R 2 2

13 VCC V CC 2OUT 2IN- 2IN+ OUT IN- IN+ GND GND LED 2 LED (6) C dv dt µ dv 0 dt + ( R + R 2 ) v = v 0 R 2 + v 0 = a(pv 0 + qe v) (7) v = R R + R 2 E, v 0 = E (8) R R + R 2 < p + q = R 3 + R 5 R 3 + R 4 + (9) R 5 3

14 3 (a) (b) 3 9(a) v 0 = 0 (b) (stable high) (stable low) 4 4(a) 9 4(a) 4(b) 4. - Kirchhoff ( dv C dt ) v = E + v 0 (20) R r R 2 R R 2 v 0 = 0 v = r R + r + E (2) R 2 4

15 4 v 0 = E v = r + R 2 R + r + E (22) R 2 4(b) r + R 2 R + r + < p + q, p < R 2 r R + r + (23) R 2 R 3 = R 4 = R 5 = 330k, R = r = 00k, R 2 = M R 2 v = r R + E = R r R + r E (24) p < R R + r < p + q (25) 4.2 v p v n 5(a)(b) 5

16 (a), (b), (b2), (c) 7(a), (b), (b2), (c). v p null cline 2. v n 3. 6(a) 6

17 6 (a) (b, b2) (c) 7 (a) (b, b2) (c) 7

18 4. 6(b, b2) 5. 6(c) 5.2. A.A. Andronov et al. [4] slow-fast slow-fast 3..3 LED 4. slow-fast stiff 8

19 [] [2] Texas Instruments Application Report SLOA03A: Effect of parasitic capacitance in Op Amp circuits, [3] E. M. Izhikevich Dynamical Systems in Neuroscience The Geometry of Excitability and Bursting, MIT Press, Excitable [4] A.A. Andropov, A.A. Pitt and S.E. Khaikin: Theory of Oscillators, Pergamon Press, slow-fast 9