Towards the analysis of codim 2 bifurcations in planar ...

Towards the analysis of codim 2bifurcations in planar Filippov

systemsYu.A. Kuznetsov

Department of Mathematics, Utrecht University

Bifurcations of Filippov Systems – p. 1/41

Contents

1. Filippov systems

2. Codim 1 local bifurcations

3. Codim 1 global bifurcations

4. Examples of codim 2 bifurcationsDegenerate boundary focus bifurcationBoundary Hopf bifurcationSliding-grazing of a nonhyperbolic cycle

5. Example: Harvesting a prey-predator community

6. Open problems


References

Yu.A. Kuznetsov, S. Rinaldi, and A. Gragnani. One-parameterbifurcations in planar Filippov systems, Int. J. Bifuraction & Chaos

13(2003), 2157-2188

F. Dercole and Yu.A. Kuznetsov. SlideCont: An AUTO97 driver forsliding bifurcation analysis. ACM Trans. Math. Software 31 (2005),95-119

P. Kowalczyk, M. di Bernardo, A.R. Champneys, S.J. Hogan, M.Homer, P.T. Piiroinen, Yu.A. Kuznetsov, and A. Nordmark.Two-parameter discontinuity-induced bifurcations of limit cycles:classification and open problems. Int. J. Bifurcation & Chaos 16

(2006), 601-629


1. Filippov systems

S2

S1Σ

f (2)

f (1)


1. Filippov systems

S2

S1Σ

f (2)

f (1)

x =

f (1)(x), x ∈ S1,

f (2)(x), x ∈ S2.


1. Filippov systems

S2

S1Σ

f (2)

f (1)

x =

f (1)(x), x ∈ S1,

f (2)(x), x ∈ S2.

S1 = {x ∈ R2 : H(x) < 0},

S2 = {x ∈ R2 : H(x) > 0},

where H : R2 → R has nonvanishing gradient Hx(x) on

Σ = {x ∈ R2 : H(x) = 0}.

For x ∈ Σ, define σ(x) = 〈Hx(x), f (1)(x)〉〈Hx(x), f (2)(x)〉.


Crossing orbits

On Σc = {x ∈ Σ : σ(x) > 0} :

Σc

S2

x

S1

Hx(x)


Crossing orbits

On Σc = {x ∈ Σ : σ(x) > 0} :

Σc

S2

f (1)(x)

x

S1

Hx(x)


Crossing orbits

On Σc = {x ∈ Σ : σ(x) > 0} :

Hx(x)

Σc

S2

f (1)(x)

x

S1

f (2)(x)


Filippov sliding orbits

On Σs = {x ∈ Σ : σ(x) ≤ 0} :

Hx(x)

Σs

S1

S2

x



On Σs = {x ∈ Σ : σ(x) ≤ 0} :

f (1)(x)

Hx(x)

Σs

S1

S2

x



On Σs = {x ∈ Σ : σ(x) ≤ 0} :

x

f (1)(x)

Hx(x)

Σs

f (2)(x)S1

S2



On Σs = {x ∈ Σ : σ(x) ≤ 0} :

S2

x

f (1)(x)

Hx(x)

Σs

f (2)(x)

g(x)

S1

x = g(x), x ∈ Σs,

where g(x) = λf (1)(x) + (1 − λ)f (2)(x) with

λ =〈Hx(x), f (2)(x)〉

〈Hx(x), f (2)(x) − f (1)(x)〉.


Special sliding points

Singular sliding points:

x ∈ Σs, 〈Hx(x), f (2)(x) − f (1)(x)〉 = 0, f (1,2)(x) 6= 0.

Pseudo-equilibria: x ∈ Σs, g(x) = 0, f (1,2)(x) 6= 0.

Boundary equilibria: x ∈ Σs, f(1)(x) = 0 or f (2)(x) = 0.

Tangent points: x ∈ Σs, 〈Hx(x), f (i)(x)〉 = 0, f (1,2)(x) 6= 0.

S2 S2

S1S1

(invisible)(visible)

TTΣ

Σ


Quadratic tangent point T

Tε K(1)(ε)

x1

x2

x2 =1

2νx2

1 +O(x31)

If f (1)(x) =

p+ ax1 + bx2 + · · ·

cx1 + dx2 + 12qx

21 + rx1x2 + 1

2sx22 + · · ·

,

then ν =c

pand

K(1)(ε) = −ε+ k(1)2 ε2 +O(ε3), k

(1)2 =

2

3

(

a+ c

p−

q

2c

)

.

[Filippov, 1988]


Fused focus (singular sliding point)

When two invisible tangent points coincide, define the Poincaré map:

P (ε) = ε+ k2ε2 +O(ε3), k2 = k

(1)2 − k

(2)2 .

unstable (k2 > 0) stable (k2 < 0)

S1

S2 S2

S1

T0

T0

Σ Σ


2. Codim 1 local bifurcations

Collisions of

standard equilibria with Σ

tangent points

pseudo-equilibria


Boundary focus: BF1

Σ

S2

S1

α < 0

Xα

Tα Pα

Lα


Boundary focus: BF1

X0

S1

S2

α = 0

Σ

S2

S1

α < 0

Xα

Tα Pα Σ

Lα


Boundary focus: BF1

X0

S1

S2

α = 0

Σ

S2

S1

α < 0

Xα

Σ

S2

α > 0

Tα Pα Σ

S1Lα

Tα


Boundary focus: BF2

Σ

S2

S1

α < 0

Tα

Xα

Pα


Boundary focus: BF2

X0

S1

S2

α = 0

Σ

S2

S1

α < 0

Tα Σ

Xα

Pα


Boundary focus: BF2

X0

S1

S2

α = 0

Σ

S2

S1

α < 0

Σ

S2

α > 0

Tα TαΣ

S1Xα

Pα


Degenerate boundary focus: BDF

T

x1

x2

R

x2 = 0

x1 = 0

x1 = ax1 + bx2,

x2 = cx1 + dx2,

T =

(

−d

c, 1

)

, R =

(

−b

a, 1

)

.

d− a

2ωtg

[

ω

a+ dln

(

−bc

a2

)]

= 1,

where ω = 12

√

−(a− d)2 − 4bc.


Boundary focus: BF3

S1

α < 0

S2

Tα Σ

LαXα


Boundary focus: BF3

S2

S1

S1

α < 0 α = 0

S2

Tα X0Σ Σ

LαXα


Boundary focus: BF3

S2S2

S1

S1

α < 0 α = 0 α > 0

Σ

S1

S2

Tα X0Σ Σ Tα

Pα

LαXα


Boundary focus: BF4

Σ

S2

S1

Tα

α < 0

Xα

X0

S2

α = 0 α > 0

Σ

S1

S2

Tα

S1

PαΣ


Boundary focus: BF5

S2S2

S1

α < 0 α = 0 α > 0

S2

Σ

S1

TαX0 ΣPα Tα

S1

Xα

Σ


Double tangency: DT1

T0

S2

T2α

S2

Σ Σ

S1

S2

Σ

S1S1

α < 0 α = 0 α > 0

T1α


Double tangency: DT2

T0

S2S2

Σ

S2

Σ

T1α

T2α

S1S1

S1

Σ

α < 0 α = 0 α > 0


Collision of two invisible tangencies: II1

α < 0

S2

S1

Σ

T(2)α T

(1)α



α < 0 α = 0

S2

S1

Σ

T(2)α T0

Σ

S1

S2

T(1)α



α < 0 α = 0 α > 0

Σ

S2

S1

T(2)α

S2

S1

Σ

T(2)α T0

Σ

S1

S2

T(1)αT

(1)α



Σ

S1

T(1)α

T(2)α

Pα

S2

α < 0

[Gubar’, 1971; Filippov, 1988]



Σ Σ

T0

S1

T(1)α

T(2)α

Pα

S1

S2 S2

α = 0α < 0




Σ

Σ Σ

T0

S1

T(1)α

T(2)α

Pα

S1 S1

T(1)α

Lα

Pα T(2)α

S2 S2 S2

α > 0α = 0α < 0



Pseudo-saddle-node: PSN

α < 0

P 1α

P 2α

S2

Σ

S1



α = 0α < 0

P 1α

P 2α

S2

P0

S2

ΣΣ

S1 S1



α > 0α = 0α < 0

P 1α

P 2α

S2

P0

S2 S2

ΣΣ

S1 S1 S1

Σ


3. Codim 1 global bifurcations

Bifurcations of sliding cycles:Grazing-slidingAdding-slidingSwitching-slidingCrossing-sliding

Pseudo-homoclinic bifurcations:Homoclinic orbit to a pseudo-saddle-nodeHomoclinic orbit to a pseudo-saddle

Sliding homoclinic orbit to a saddle

Pseudo-heteroclinic bifurcations


Grazing-sliding: TC1

Σ

S2

S1

Tα

Lα

α < 0



L0

Σ

S2

S1

T0

α = 0

Σ

S2

S1

Tα

Lα

α < 0



L0

Σ

S2

S1

T0

α > 0α = 0

Σ

S2

S1

Tα

Lα

Σ

S2

Lα

S1

Tα

α < 0



α < 0

Σ

S2

Lsα

Tα

Luα

S1



Σ

S2

S1

α = 0α < 0

Σ

S2

Lsα

Tα

Luα

T0

L0

S1



Σ

S2

S1

α > 0α = 0α < 0

Σ

S2

Lsα

Tα

Luα

T0

L0

S1

S2

S1

ΣTα


Adding-sliding: DT2 with global reinjection

α < 0

S2

S1

Σ

Lα



S1

S2

L0

Σ

T0

α < 0

S2

S1

Σ

Lα

α = 0



S1

S2

L0

Σ

α > 0

T0

α < 0

S2

S1

T 1α

Lα

S2

S1

Σ

Lα

α = 0

T 2α Σ


Switching-sliding

α < 0

Lα

Σ

S2

S1

T(1)α


Switching-sliding

α < 0 α = 0

Lα

Σ

S2

L0

Σ

S2

S1S1

T(1)0

T(1)α


Switching-sliding

α < 0 α = 0 α > 0

Lα

Σ

S2

L0

Σ

S2

Lα

Σ

S2

S1S1

S1

T(1)0

T(1)αT

(1)α


Crossing-sliding

Σ

S1

S2

Lα

α < 0

T(1)α


Crossing-sliding

α = 0

Σ

S1

S2

S1

S2

Lα

T(1)0

Σ

L0

α < 0

T(1)α


Crossing-sliding

α = 0

Σ

S1

S2

S1

S2

S1

S2

Lα

T(1)0 T

(1)α

Σ

L0

Σ

α < 0

Lα

α > 0

T(1)α


Homoclinic orbit to a pseudo-saddle-node

α < 0

S2

Σ

Lα

S1



α = 0α < 0

Σ

S2

L0

P0

S2

Σ

Lα

S1 S1



α = 0 α > 0α < 0

Σ

S2

Σ

L0

P1α

S2

P2α

P0

S2

Σ

Lα

S1 S1 S1


Homoclinic orbit to a pseudo-saddle: TGP

α < 0

Lα

S2

S1

ΣPα



α < 0 α = 0

Lα H0

S2

S1

S2

S1

Σ ΣPαP0



α < 0 α = 0 α > 0

Lα H0

S2

Σ

S2

S1S1

S2

S1

Σ ΣPαP0 Pα



α < 0

Σ

Tα

S1 Xα

S2

Lα



α < 0 α = 0

Σ

T0

S2

S1X0

Γ0

Σ

Tα

S1 Xα

S2

Lα



α < 0α > 0α = 0

Σ

T0

S2

S1X0

S2

S1Xα

Tα

Γ0

ΣΣ

Tα

S1 Xα

S2

Lα


4. Examples of codim 2 bifurcations

Local bifurcations:Degenerate boundary focusBoundary Hopf

Global bifurcations:Sliding-grazing of a nonhyperbolic cycle (fold-grazing)


Degenerate boundary focus: DBF

1

02

1

20

β2

Oβ1

BF1

BF2

TGP

S1 S1

TGP

S1

S1

S2

Σ

Σ

S2

Σ

S2

Σ

S2


Boundary Hopf: BHP

5

1

25

4

3

3

1 2

4

S1

S1 S1

β1

S1

S1

S1

PSNBF1

BF4

O

β2

S1

ΣS2S2

H

Σ

S2

Σ

S2

Σ

Σ

S2TCH1

TCH1

S2

Σ

Σ

PSN

S2


Fold-grazing: FG1

1

2

1

0

2

0

S1

S1

S1

O

LPC

S1

S1

TCH2

S1

β2

β1O

LPC

TCH2

S1

S2

Σ

Σ

TCH1

Σ

S2

S2

TCH1

Σ

S2

S2

Σ

Σ

S2

S2

Σ


Fold-grazing: FG2

1

1

0

0

2

2

S1

β2

LPC

TCH2

S1S1

LPC

S1

TCH2

O

S1

β1

S1

S1

S1

O

S2

Σ

S2

TCH1

Σ

S2

TCH1

Σ

Σ

Σ

S2

Σ

S2

S2

Σ

S2


5. Example: Harvesting a prey-predator community

Rosenzweig-MacArthur model

Harvesing control

Two-parameter bifurcation diagram


Rosenzweig-MacArthur-Holling model

x1 = x1(1 − x1) −ax1x2

b+ x1

x2 =ax1x2

b+ x1− cx2

Nontrivial zero-isoclines:

x2 =1

a(b+ x1)(1 − x1), x1 =

bc

a− c.



x1 = x1(1 − x1) −ax1x2

b+ x1

x2 =ax1x2

b+ x1− cx2


x2 =1

a(b+ x1)(1 − x1), x1 =

bc

a− c.

(a)

x1 = 0

x2 = 0x2

x1



x1 = x1(1 − x1) −ax1x2

b+ x1

x2 =ax1x2

b+ x1− cx2


x2 =1

a(b+ x1)(1 − x1), x1 =

bc

a− c.

(a) (b)

x2 = 0

x1

x1 = 0

x2

x1

x2



x1 = x1(1 − x1) −ax1x2

b+ x1

x2 =ax1x2

b+ x1− cx2


x2 =1

a(b+ x1)(1 − x1), x1 =

bc

a− c.

(a) (b) (c)

x2 = 0

x1x1x1

x1 = 0

x2x2 x2


Harvesting control

Assume that the predator population is harvested at constant effort e > 0

only when abundant (x2 > α5). This leads to a planar Filippov system:

x =

f (1)(x), x2 > α5,

f (2)(x), x2 < α5,

where

f (1) =

x1(1 − x1) − ψ(x1)x2

ψ(x1)x2 − dx2

, f (2) =

x1(1 − x1) − ψ(x1)x2

ψ(x1)x2 − dx2 − ex2

,

ψ(x1) =ax1

α2 + x1.

Fix a = 0.3556, d = 0.0444, e = 0.2067.


Two-parameter bifurcation diagram

0 1 2 30

0.5

1

1

6

4

8

9

7

23

5

α5

B

A

H1

TTD1

BEQ2

BEQ1

TCT1

PSN

TCH1

α2

TGP1


Two-parameter bifurcation diagram (zoom in)

0 0.5 10

0.1

0.2

19

21

34

6 24

30

29

14

13

12

15

16

25

2827

17

10

11

9

18

23

7

8

3

33

32

20

22

3126

TCH2TGP2

TCD1TCP1

TTD1

α5

BEQ2

α2

TCT2

TTD2

H2

I

E

D

G

HF

C

TCT1

BEQ1


6. Open problems

1. Complete analysis of codim 2 local bifurcations for n = 2.

2. Bifurcation diagrams of smooth approximations of Filippov systemsand their limits. A codim 1 bifurcation can

disappear: BF3,4,DT1,2, II1, TC1, CC;become a single smooth bifurcation:BF2,5, PSN → LP, II2 → H, TC2 → LPC, TGP → HOM;split into a several smooth bifurcations: BF1 → H + LP.

What happens to codim 2 bifurcations?

3. Grazing of nonhyperbolic cycles when n = 3 (codim 2).

4. Other codim 1 and 2 local and global bifurcations when n ≥ 3.


Towards the analysis of codim 2 bifurcations in planar ...

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