Stabilizing unknown unstable periodic orbits of chaotic spiking oscillators

Tadashi Tsubone

Nagaoka University of Technology

Chaos control : Stabilizing Unstable Periodic Obit (Chaos control : Stabilizing Unstable Periodic Obit (Chaos control : Stabilizing Unstable Periodic Obit (Chaos control : Stabilizing Unstable Periodic Obit (OttOttOttOtt, , , , GrebogiGrebogiGrebogiGrebogi, , , , YorkeYorkeYorkeYorke, 1990), 1990), 1990), 1990)

--------Delayed Feedback Control (DFC) (Delayed Feedback Control (DFC) (Delayed Feedback Control (DFC) (Delayed Feedback Control (DFC) (PyragasPyragasPyragasPyragas, 1992) , 1992) , 1992) , 1992)

Stabilizing Unknown Steady StatesStabilizing Unknown Steady StatesStabilizing Unknown Steady StatesStabilizing Unknown Steady States : ( without location information ) : ( without location information ) : ( without location information ) : ( without location information )

--------Stabilizing and tracking unknown steady statesStabilizing and tracking unknown steady statesStabilizing and tracking unknown steady statesStabilizing and tracking unknown steady states( for Continuous time ( for Continuous time ( for Continuous time ( for Continuous time system) (system) (system) (system) (PyragasPyragasPyragasPyragas, 2002) , 2002) , 2002) , 2002)

--------Generalized Washout filterGeneralized Washout filterGeneralized Washout filterGeneralized Washout filter----aided control aided control aided control aided control (for Continuous and discrete time system(for Continuous and discrete time system(for Continuous and discrete time system(for Continuous and discrete time system)))) ((((HassounehHassounehHassounehHassouneh, et al. 2004), et al. 2004), et al. 2004), et al. 2004)

Introduction (1/2) 1

Spiking neuron model:Spiking neuron model:Spiking neuron model:Spiking neuron model:

--------IntegratedIntegratedIntegratedIntegrated----andandandand----fire neuronfire neuronfire neuronfire neuron

--------ResonantResonantResonantResonant----andandandand----fire neuron (fire neuron (fire neuron (fire neuron (IzhikevichIzhikevichIzhikevichIzhikevich, 2001), 2001), 2001), 2001)

Chaotic Spiking OscillatorChaotic Spiking OscillatorChaotic Spiking OscillatorChaotic Spiking Oscillator : (with impulsive switch and jump operation) : (with impulsive switch and jump operation) : (with impulsive switch and jump operation) : (with impulsive switch and jump operation)

--------2222----D Piecewise linear model (D Piecewise linear model (D Piecewise linear model (D Piecewise linear model (MitsuboriMitsuboriMitsuboriMitsubori and T.Saito, 1997)and T.Saito, 1997)and T.Saito, 1997)and T.Saito, 1997)

--------2222----D Piecewise constant model (D Piecewise constant model (D Piecewise constant model (D Piecewise constant model (TsuboneTsuboneTsuboneTsubone, W. Schwarz and T.Saito, 1999), W. Schwarz and T.Saito, 1999), W. Schwarz and T.Saito, 1999), W. Schwarz and T.Saito, 1999)

Introduction (2/2) 2

Contents

1.1.1.1. 3333----D Simple Chaos Spiking OscillatorD Simple Chaos Spiking OscillatorD Simple Chaos Spiking OscillatorD Simple Chaos Spiking Oscillator (2D + switch)(2D + switch)(2D + switch)(2D + switch)-------- Piecewise linear 1Piecewise linear 1Piecewise linear 1Piecewise linear 1----D return map D return map D return map D return map

2.2.2.2. Controlled CSOControlled CSOControlled CSOControlled CSO-------- Generation of ``Stabilized Unstable Periodic ObitsGeneration of ``Stabilized Unstable Periodic ObitsGeneration of ``Stabilized Unstable Periodic ObitsGeneration of ``Stabilized Unstable Periodic Obits’’ of CSOof CSOof CSOof CSO

-------- Almost completely analysis for stability and existence Almost completely analysis for stability and existence Almost completely analysis for stability and existence Almost completely analysis for stability and existence

3.3.3.3. 4444----D Simple Chaos Spiking OscillatorD Simple Chaos Spiking OscillatorD Simple Chaos Spiking OscillatorD Simple Chaos Spiking Oscillator (3D + switch)(3D + switch)(3D + switch)(3D + switch)-------- from 3from 3from 3from 3----D CSOD CSOD CSOD CSO-------- ````````resonant and integratedresonant and integratedresonant and integratedresonant and integrated’’ and fire model?and fire model?and fire model?and fire model?

TodayTodayTodayToday’s talk :s talk :s talk :s talk :

Controlled Chaotic Spiking Oscillator Controlled Chaotic Spiking Oscillator Controlled Chaotic Spiking Oscillator Controlled Chaotic Spiking Oscillator with ``Stabilized Uncertainwith ``Stabilized Uncertainwith ``Stabilized Uncertainwith ``Stabilized Uncertain----Unstable Periodic ObitsUnstable Periodic ObitsUnstable Periodic ObitsUnstable Periodic Obits’’

3

Simple chaotic spiking oscillator

2g1g

2v1v1c

2c I

..MM

1

1S

S

2i1i

1Cc

AND

Ivvgt

vc

vgt

vc

+−=

=

)(d

dd

d

1222

2

211

1

Dynamics:

1v

2v

0

0

V/div.][2

V/div.][2

If and , then the voltage is reset to the inverse voltage instantaneously.

02 =v 01 <v 1v

1v−

2v

1v

V/div.][2ms/div.][2t

4

Simple chaotic spiking oscillator

2g1g

2v1v1c

2c I

..MM

1

1S

S

2i1i

1Cc

AND

If and , then the voltage is reset to the inverse voltage instantaneously.

0=y 0<x x

x−

),,10(

12

21 vyvx

xyy

yx

∝∝<<+−=

=

δδɺ

ɺ

Dynamics:

x

x

y

0 12+

12+

12−−12−−

01−

x

y

10

)( nTx

)( +nTx

5

Simple chaotic spiking oscillator

),(

12

21 vyvx

xyy

yx

∝∝+−=

=δɺ

ɺ

Dynamics:

1v

2v

0

0

V/div.][2

V/div.][2

If and , then jumps to . 0=y 0<x x x−

6

{ }0,01|),( =≤≤−= yxyxl

)2()(

))exp((,:

1 <=

=→

+ axfx

allf

nn

ωδπ

−≤<−

++−

−≤<−

++−

≤<−

++−

=

−

+

⋮

⋮

11

11

for

1)1(

11

11

for

1)1(

011

for

1)1(

1

12

2

1

1

1

kkk

kk

n

k

n

n

n

ax

a

xa

ax

a

xa

xa

xa

x

x

y

nx 10

1+nx

0

0

nx

1+nx

1−1−

0

0

nx

1+nx

1−1−

1+−A

1px

2px

3px

8.1=a 2→a)(:PointFixed pxpx xfx =

Embedded Return Map 7

Controlled Oscillator

2g1g

2v1v1c

2c

1

1S

K

1S

1

S

0Cv

1Cv

2i1i

1Cc

0Cc

..MM

AND

If and ,

then the voltageis reset to the

inverse voltage instantaneously.

( is a control parameter. )

02 =v 01 <v

1v

1Cv−

K

KvvKv CC 011 )1( +−=

),(

12

21 vyvx

xyy

yx

∝∝+−=

=δɺ

ɺ

0

1 )1(

Cn

nnn

vz

KzxKz

∝+−=+

8

Controlled Return Map

Return map:

nnn KzzfKz +−=+ )()1(1

).( assuch ofpoint fixed a is

1)(

)1(

pkpkpk

xzn

n

xfxfx

Kz

zfK

pkn

=

<+∂

∂−=

Condition for stability:

x

y

x

y

kz

1+kz

3

2=K

0

0

1−1−

0

0

1−1− x

y

0 12+

12+

12−−12−−

01−

0=K

kz

1+kz

x

y

1

0

nx

nz

1+nz

1+nz

nznnn KzxKz +−=+ )1(1

)(:PointFixed pxpx xfx =

)( nn zfx =

9

Laboratory measurements

1v

2v

0

0

V/div.][2

V/div.][2

2v

1v

V/div.][2

ms/div.][2t

1v

2v

0

0

V/div.][2

V/div.][2

2v

1v

V/div.][2

ms/div.][2t

1v

2v

0

0

V/div.][2

V/div.][2

2v

1v

V/div.][2

ms/div.][2t

0

0

nz

1+nz

1−1−

0

0

nz

1+nz

1−1−

0

0

nz

1+nz

1−1−

0

0

nz

1+nz

1−1−

8.1=a

0=K 4.0=K

65.0=K 8.0=K

10

Bifurcation Diagram

0

0

nz

1+nz

1−1−

0

0

nz

1+nz

1−1−

0

0

nz

1+nz

1−1−

0

0

nz

1+nz

1−1−

8.1=a

0=K 4.0=K

65.0=K 8.0=K

1

1

20

a

K

1I

2I

3I 4I

KA

A =+−

1

1

AA

A −=+−

11

13

3

KA

A =+−

1

12

2

(d)

(c)

(b)

(a)

KA

A =+−

1

13

3

(a) (b)

(c) (d)

chaos

11

4444----D Simple Chaos Spiking Oscillator (3D + switch)D Simple Chaos Spiking Oscillator (3D + switch)D Simple Chaos Spiking Oscillator (3D + switch)D Simple Chaos Spiking Oscillator (3D + switch)

12

4-D chaotic spiking oscillator

If and , then the voltage is reset to the inverse voltage instantaneously.

02 =v 13 vv > ),( 31 vv

),( 31 vv −−

2g1g

2v1v1c 2c

I1

1S

S

1xv

2i1i

1xc

..MM

AND

3g

3v3c

1

1S

S

1zv1zc KI

),,(

12

321 vzvyvx

Kzz

xyy

yx

∝∝∝+=

+−==

λδ

ɺ

ɺ

ɺ

Dynamics:

13

4-D chaotic spiking oscillator

If and , then the voltage is reset to the inverse voltage instantaneously.

0=y xz > ),( zx

),( zx −−

),( nn zx −−

),( nn zx

),( 00 zx

y

x

z

),0,1(λK

Jump

),,(

12

321 vzvyvx

Kzz

xyy

yx

∝∝∝+=

+−==

λδ

ɺ

ɺ

ɺ

Dynamics:

14

4-D chaotic spiking oscillator

If and , then the voltage is reset to the inverse voltage instantaneously.

0=y xz > ),( zx

),( zx −−

),,(

12

321 vzvyvx

Kzz

xyy

yx

∝∝∝+=

+−==

λδ

ɺ

ɺ

ɺ

Dynamics:

x

x

y

0 7

7

7−7−

0

x

z

0 7

7

7−7−

0

z

τSpike

15

4-D chaotic spiking oscillator

If and , then the voltage is reset to the inverse voltage instantaneously.

0=y xz > ),( zx

),( zx −−

1v

2v

0

0

1v

3v

0

0

τ

1v

3v

),,(

12

321 vzvyvx

Kzz

xyy

yx

∝∝∝+=

+−==

λδ

ɺ

ɺ

ɺ

Dynamics:

16

),(),(

),exp(

)exp(,:

11 nnnn zxfzx

Kcb

af

=

==

=→

++

λωλπ

ωδπ

{ } ccxab

zCczbz

xaxn

kkn

nk

n

nk

n ++−+−<++=

+−−=

+

+ )1()1()(1

for)(

1)1(

1

1

Embedded Return Map

),( nn zx −−

y

x

z

),0,1(λK

Jump

),( 11 ++ nn zx

x

z

0 5

5

5−5−

0

x

z

0 5

5

5−5−

0( )),(),(:PointFixed pxpxpxpx zxfzx =

),( nn zx

17

Controlled Oscillator

2g1g

2v1v1c 2c I

1

1

K

1S

1

S

0xv

1xv

2i1i

1xc

0xc

..MM

AND

3g

3v3c

1

1S

1zv1zc

KI

K

1S

1

S

0zv0zc

1v

2v

1v

3v

τ

1v

3v

-- Controlling Hyperchaos

-- Stabilizing Unknown UPO of 4-D system

18

1.1.1.1. 3333----D Simple Chaos Spiking OscillatorD Simple Chaos Spiking OscillatorD Simple Chaos Spiking OscillatorD Simple Chaos Spiking Oscillator (2D + switch)(2D + switch)(2D + switch)(2D + switch)-------- Piecewise linear 1Piecewise linear 1Piecewise linear 1Piecewise linear 1----D return map D return map D return map D return map

2.2.2.2. Controlled CSOControlled CSOControlled CSOControlled CSO-------- Generation of ``Stabilized Unstable Periodic ObitsGeneration of ``Stabilized Unstable Periodic ObitsGeneration of ``Stabilized Unstable Periodic ObitsGeneration of ``Stabilized Unstable Periodic Obits’’ of CSOof CSOof CSOof CSO

-------- Almost completely analysis for stability and existence Almost completely analysis for stability and existence Almost completely analysis for stability and existence Almost completely analysis for stability and existence

3.3.3.3. 4444----D Simple Chaos Spiking OscillatorD Simple Chaos Spiking OscillatorD Simple Chaos Spiking OscillatorD Simple Chaos Spiking Oscillator (3D + switch)(3D + switch)(3D + switch)(3D + switch)-------- from 3from 3from 3from 3----D CSO to higher order CSOD CSO to higher order CSOD CSO to higher order CSOD CSO to higher order CSO-------- ````````resonant and integratedresonant and integratedresonant and integratedresonant and integrated’’ and fire model?and fire model?and fire model?and fire model?

Conclusions 19

Stabilizing Unknown Steady StateStabilizing Unknown Steady StateStabilizing Unknown Steady StateStabilizing Unknown Steady State of continuous-time systems(Pyragas, 2002)

Introduction (2/4)

),0(

)(

f

f

x

xxx

>

−=

λλɺ

Object (1-D linear unstable system):

)(

)()(

xww

xwkxxxC

f

−=

−+−=

λ

λ

ɺ

ɺ

Controlled system:

),(),( ff xxwx =

is unknown

is a fixed point.

Washout filterWashout filterWashout filterWashout filter----aided control for Stabilizing Unknown Steady Stateaided control for Stabilizing Unknown Steady Stateaided control for Stabilizing Unknown Steady Stateaided control for Stabilizing Unknown Steady Stateof continuous- and discrete-time systems

(Hassouneh, et al. 2004)

Introduction (4/4)

) : (

)denotes(

)(

)(

f

f

x

xxx

zxKu

zxPz

BuAxx

−−=−=

+=ɺ

ɺ

Continuous-Time System

equilibrium point ) : (

)denotes(

)(

)(1

1

f

fkk

kkk

kkk

kkk

x

xxx

zxGu

zPIPxz

BuAxx

−−=

−+=+=

+

+

Discrete-Time System

fixed point

Stabilizing Unknown Steady StateStabilizing Unknown Steady StateStabilizing Unknown Steady StateStabilizing Unknown Steady State of discrete-time systems(Tsubone, 2003)

Introduction (3/4)

)(1 kk xFx =+

)( ff xFx = kkk

kkk

KyxKy

KyxKFx

+−=+−=

+

+

)1(

))1((

1

1

Object ( n-D nonlinear system):

( is unknown )

Controlled system:

is a fixed point. ),(),( ffnn xxyx =

)(1 kk xFx =+

)( ff xFx =

−−

=

+

+

k

k

k

k

y

x

KK

AKKA

y

x

ˆ

ˆ

1

)1(

ˆ

ˆ

1

1

fxxxF

xA n

=∂∂≡ )(

fkk

fkk

xyy

xxx

−≡−≡

ˆ

ˆ

Local linearization

Stabilizing Unstable Fixed Point

kkk

kkk

KyxKy

KyxKFx

+−=+−=

+

+

)1(

))1((

1

1

Object ( n-D nonlinear system):

is unknown

Controlled system:

is a fixed point. ),(),( ffkk xxyx =kx

F K

1−Z

1−Zky1+ky

),,(

]][[ 1

nnRIAK

AIAIK×

−

∈

−−=

λ

λ

Control Gain :