Calculation of the Isocline for theFixed Point with a Specified
Argument of Complex Multipliers
Tetsushi Ueta, Shigeki Tsuji, Tetsuya Yoshinaga and Hiroshi Kawakami
tetsushi@is.tokushima-u.ac.jp

Faculty of Engineering, Tokushima University, Tokushima, 770-8506 JapanT. Ueta, for ISCAS 2001 – p.1/??

Background
Analysis of nonlinear dynamical system described
by ODE or DE
Equilibria or fixed/periodic points:• location• stability → bifurcation
Calculation of periodic points: solving boundaryvalue problem• shooting method; Newton’s method• collocation method• automatic differentiation

T. Ueta, for ISCAS 2001 – p.2/??

Proposal
1. Calculation of isocline: a fixed argument of thecomplex multipliers• simple shooting algorithm• classify topological property of periodicpoints
2. an application of calculation: calculation of
Neimark-Sacker bifurcation.

T. Ueta, for ISCAS 2001 – p.3/??

Description of the Problem
A discrete map T :T : Rn → Rn ;u 7→ T (u),u ∈ Rn

Parameters: λ1 ∈ R, λ2 ∈ R• T is C ∞ for u, λ1 , and λ2 .
The fixed point u0 :•T (u0 ) = u0 .

T. Ueta, for ISCAS 2001 – p.4/??

Complex multipliers
A pair of complex conjugate multipliers:µ, µ̄ = re±iθ = r(cos θ ± i sin θ).
r: the radius, θ: the argument.






r and θ determine stability and angular velocityof the local linear space about u0 .

T. Ueta, for ISCAS 2001 – p.5/??

Complex multipliers
limit cycle

1.5

NS Bifurcation
10.5

0-0.5

-1

quasi-periodic

-1.5
-1.5

-1-0.500.5

11.5

T. Ueta, for ISCAS 2001 – p.6/??

Calculation of the fixed point
J : the Jacobian matrix of T at u0
The characteristic equation:
χ = det[J − reiθ ] = <χ + i=χ = 0
Condition of the fixed point with θ
T (u) − u
=0
F =
<χ=χ
Variables: (u, λ1 , r) ∈ Rn+2 .

T. Ueta, for ISCAS 2001 – p.7/??

Jacobian matrix
Then the Jacobian matrix for Newton’s method: ∂T∂T∂T − In ∂u0∂λ1∂r  ∂<χ∂<χ∂<χ
DF =  ∂u∂λ∂r01 ∂=χ∂=χ ∂=χ ∂u0∂λ1∂r

T. Ueta, for ISCAS 2001 – p.8/??

The variational equation of TFor ODE:∂f ∂ϕd ∂ϕ=dt ∂x0∂x ∂x0∂f ∂ϕ∂fd ∂ϕ=+dt ∂λ1∂x ∂x0 ∂λ1
can be obtained by numerical numerical integration
like Runge-Kutta method.

T. Ueta, for ISCAS 2001 – p.9/??

Isocline

k-Sa

cker

For the fixed θ ∈ [0, π], (u0 , λ1, r) forms an isocline
in the parameter plane λ1 -λ2 .
r=0
StableλUnstable
r=8r=1Neimar

2

λ1
T. Ueta, for ISCAS 2001 – p.10/??

Properties
θ = 0 or θ = π: critical line between real andcomplex multipliers. (nodal or rotational linear
space about u0 .)• any rational ratio with π of θ: deeply related to
the NS bifurcation and frequency entrainment
regions.
•T. Ueta, for ISCAS 2001 – p.11/??

Calculation of NS bifurcation
Exchange variables and parameters: r  θLet r = 1, then solve
T (u) − u
=0F =<χ=χ
for (u, λ1, λ2 ) ∈ Rn+2

T. Ueta, for ISCAS 2001 – p.12/??

The Jacobian matrix ∂T∂T∂T−I ∂u0∂λ1 ∂λ2  ∂<χ∂<χ∂<χDF =  ∂u
∂λ∂λ012  ∂=χ∂=χ ∂=χ ∂u0
∂λ1 ∂λ2

T. Ueta, for ISCAS 2001 – p.13/??

Features
Neimark-Sacker bifurcation — explicitlyparameterized by θ.
Very simple formulation to calculate NSbifurcation.
• Codimension-two bifurcation parameter values
can be obtained, θ = 0 and θ = π.
• No predictor-corrector method is needed.
• Not sensitive for stability of the periodic point•

T. Ueta, for ISCAS 2001 – p.14/??

Example 1
A Discrete Chaotic Mapx1 (k + 1) = x2 (k) + ax1 (k)x2 (k + 1) = x21 (k) + b

k = 0, 1, 2, · · ·The characteristic equation :<χ = r2 (cos2 θ − sin2 θ) − r cos θ tr X + det X=χ = 2r cos θ − tr X
X : the fundamental solution matrix of the variationalequation.

T. Ueta, for ISCAS 2001 – p.15/??

Bifurcation diagram
0.5I10G1π0Oπ/6b→π/32π/5-0.5π/2
3π/5-15π/6G6 G6G5G4G42π/3G5NS-1.5-2-1.5-1-0.5

0a→

0.5

11.5

2T. Ueta, for ISCAS 2001 – p.16/??

Bifurcation diagram

T. Ueta, for ISCAS 2001 – p.17/??

NS bifurcation: b = 0.5a − 0.75
0.5I1G100O

ππ/6π/3b→2π/5-0.5π/2

G6 G6G53π/5-1G45π/6G42π/3G5NS-1.5-2-1.5-1-0.5

0a→

0.5

11.5

2T. Ueta, for ISCAS 2001 – p.18/??

Period-6, θ = π/3
0.03

0.02y→0.010-0.01

-0.02

-0.03
-0.52

-0.5

x→-0.48

a = 1.0, b = −0.25, u0 = (−0.5, 0.0).
T. Ueta, for ISCAS 2001 – p.19/??

period-5, θ = 2π/5
-0.16

-0.17

y→

-0.18

-0.19

-0.2

-0.21

-0.22
-0.54

-0.52

-0.5

x→

-0.48

-0.46

a = 0.618033, b = −0.440983,
T. Ueta, for ISCAS 2001 – p.20/??

A couple of neurons
A system described by ODE:ẋ1 = 10f (x1 ) − 10f (x2 ) − 2.5 + B cos ωtẋ2 = 10f (x1 ) + 2f (x2 ) − 9.0
1f (x) =1 + e−x• quasi-periodic solutions• chaotic motion• frequency entrainment

T. Ueta, for ISCAS 2001 – p.21/??

Bifurcation diagram
2.5
NSI1

B→

2.45
I1G5

2.4
I7
G7

2.35

4π/ 5I5π/ 3

2.3
0.81

0.82

0.83

π

0.84

6 π/ 7

0.85

0.86

ω→
T. Ueta, for ISCAS 2001 – p.22/??

Period-5, θ = 2π/5
-0.71
-0.72

y→

-0.73
-0.74
-0.75
-0.76
-0.77
-0.78
2.18

2.2

2.22

2.24

x→

2.26

2.28T. Ueta, for ISCAS 2001 – p.23/??

Period-6, θ = 2π/7.
-0.69

-0.7

y→-0.71

-0.72

-0.73

-0.74

-0.75
2.22

2.24

2.26

2.28

x→

2.3

2.32

T. Ueta, for ISCAS 2001 – p.24/??

Forced BVP oscillator
Another neuron model:ẋ = y + 1.6x − x3ẏ = −x − 0.77y + B0 + B cos tquasi-periodic solutions• chaotic motion• frequency entrainment•

T. Ueta, for ISCAS 2001 – p.25/??

Bifurcation diagram
0.2

0.15

I1G4

B→πG4

2π/3

0.1

G4π/2π/30.05

π/4

G4π/6NS00.35

0.4

0.45

0.5

B0 →

0.55

0.6

T. Ueta, for ISCAS 2001 – p.26/??

Bifurcation diagram20.1
G4π/2

0.08
G5π/3

B→

0.06

π/4

4G

0.04

G5π/6

0.02

00.46

0.48

0.5

G6NS

0.52

0.54

B0 →

0.56

T. Ueta, for ISCAS 2001 – p.27/??

Bifurcation diagram

NSB→

0.056
0.05550.055
G60.0545
0.054
0.05350.053
0.05250.052
0.505

π/6G4G60.51

B0 →

0.515

0.52

T. Ueta, for ISCAS 2001 – p.28/??

Conclusions
Calculation of isocline with fixedargument for thefixedpoint
1. several isoclines corresponding to the special
value of arguments
2. stability and instantaneous phase
3. a new calculation method of NS bifurcation
4. cross points of a cusp in a periodic entrainment
region and the NS bifurcation curve

T. Ueta, for ISCAS 2001 – p.29/??