Lecture 1. Filippov systems:

Sliding solutions and bifurcations

Yuri A. Kuznetsov

March 2, 2010

References:

• P.T. Piiroinen and Yu.A. Kuznetsov. An event-driven method to

simulate Filippov systems with accurate computing of sliding mo-

tions. ACM TOMS 34 (2008), no.3, Article 13, 24p.

• Yu.A. Kuznetsov, S. Rinaldi, and A. Gragnani. One-parameter bi-

furcations in planar Filippov systems. Int. J. Bifuraction & Chaos

13(2003), 2157-2188

• M. di Bernardo, C.J. Budd, A.R. Champneys, P. Kowalczyk. Piecewise-

smooth Dynamical Systems: Theory and Applications. Springer-

Verlag, London, 2008.

• A.F. Filippov. Differential Equations with Discontinuous Right-

Hand Sides. Kluwer Academic, Dordrecht, 1988.

Contents

1. Standard and sliding solutions of Filippov systems.

2. Numerical integration of sliding solutions.

3. Codim 1 bifurcations of 2D Filippov systems.

4. Codim 2 bifurcations of 2D Filippov systems.

5. Example: Controlled harvesting a prey-predator community.

1. Standard and sliding solutions of Filippov systems

Consider a discontinuous system

x =

{

f(1)(x), x ∈ S1,

f(2)(x), x ∈ S2,(1)

where x ∈ Rn,

S1 = {x ∈ Rn : H(x) < 0}, S2 = {x ∈ R

n : H(x) > 0},

H : Rn → R is smooth with Hx(x) 6= 0 on the discontinuity boundary

Σ = {x ∈ Rn : H(x) = 0} ,

and f(i) : Rn → Rn are smooth functions.

Orbits of (1) are defined by concatenation of standard and sliding orbit

segments.

For x ∈ Σ, define

σ(x) = 〈Hx(x), f(1)(x)〉〈Hx(x), f

(2)(x)〉

and introduce the sets of

• crossing points: Σc = {x ∈ Σ : σ(x) > 0}

• sliding points: Σs = {x ∈ Σ : σ(x) ≤ 0}

• regular sliding points:

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

Crossing orbits:

At x ∈ Σc, concatenate the standard orbit of f (1) reaching x from S1

with the standard orbit of f(2) departing from x into S2, or vice versa.

Hx(x)

S2

S1

Σc

x

f (2)(x)

f (1)(x)

Sliding orbits:

For x ∈ Σs define the Filippov vector

g(x) = λ(x)f(1)(x) + (1 − λ(x))f(2)(x),

where

λ(x) =〈Hx(x), f(2)(x)〉

〈Hx(x), f(2)(x) − f(1)(x)〉.

g(x)

x

Hx(x)

f (2)(x)

f (1)(x)

S1

S2

Σs

Utkin’s equivalent control method:

One can write

g(x) =f(1)(x) + f(2)(x)

2+f(2)(x) − f(1)(x)

2µ(x),

where

µ(x) = −〈Hx(x), f(1)(x) + f(2)(x)〉

〈Hx(x), f(2)(x) − f(1)(x)〉.

It follows that

λ(x) =1 − µ(x)

2,

so that g = f(1) if µ = −1 (λ = 1) and g = f(2) if µ = 1 (λ = 0).

This gives the sliding system

x = g(x), x ∈ Σs. (2)

At x ∈ Σs, concatenate the standard orbit of f (i) reaching x from Si

with the maximal sliding orbit of g in Σs departing from x. The sliding

orbit can reach a point x at the boundary of Σs that is composed of

singular sliding points, boundary equilibria, and tangent points.

An equilibrium of (2) satisfies g(X) = 0. We could have

• pseudo-equilibria where f (i)(X) are both transversal to Σs and

anti-collinear;

• boundary equilibria where

f(1)(X) = 0 or f(2)(X) = 0.

If both f(i)(T ) 6= 0 but

〈Hx(T ), f(1)(T )〉 = 0 or 〈Hx(T ), f(2)(T )〉 = 0,

point T ∈ Σs is called a tangent point. Note that µ(T ) = ±1, while

λ(T ) = 0 or 1.

Tangent points are called visible (invisible) if the orbits of f (i) starting

from them at time t = 0 belong to Si (Sj, j 6= i) for all sufficiently small

|t| 6= 0.

Σc

ΣsT

S1

S2

Σc

Σs

T

S1

S2

Quadratic tangent point in 2D

Tε K(1)(ε)

x1

x2

S1

S2

x2 =1

2νx21 +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

)

.

Fused focus in 2D (a singular sliding point)

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

P (ε) = ε+ k2ε2 +O(ε3), k2 = k

(1)2 − k

(2)2 .

S2

T0

T0

Σ

Σ

S1

S2

S1

(a) (b)

2. Numerical integration of sliding solutions

The regular sliding set Σs is a neutral invariant manifold for the Filippov

vector field

g(x) =f(1)(x) + f(2)(x)

2+f(2)(x) − f(1)(x)

2µ(x), x ∈ R

n,

where

µ(x) = −〈Hx(x), f(1)(x) + f(2)(x)〉

〈Hx(x), f(2)(x) − f(1)(x)〉.

Moreover, Σs is an attracting invariant manifold for the modified Fi-

lippov vector field:

G(x) = g(x) −H(x)Hx(x), x ∈ Rn, (3)

so that the sliding orbits on it can be merely integrated forward in time

using x = G(x) with x ∈ Rn and x0 ∈ Σs.

Event functions and variables

e1(t) = H(x(t))

e2(t) = 〈Hx(x(t)), f(1)(x(t))〉

e3(t) = 〈Hx(x(t)), f(2)(x(t))〉

Define domains

M = {x ∈ Rn : |µ(x)| ≥ 1}, M = {x ∈ R

n : |µ(x)| < 1}

x ∈ v e1 e2 e3S1 ∪M (1,−1,−1,1,−1) e−1 = 0 e±2 = 0 e±3 = 0

S1 ∪ M (1,−1,−1,−1,1) e−1 = 0 e±2 = 0 e∓3 = 0

S2 ∪M (−1,1,−1,1,−1) e+1 = 0 e±2 = 0 e±3 = 0

S2 ∪ M (−1,1,−1,−1,1) e+1 = 0 e±2 = 0 e∓3 = 0

Σs (−1,−1,1,−1,1) − e±2 = 0 e∓3 = 0

Use f(1)(x) if v1 = 1, f(2)(x) if v2 = 1, and G(x) if v3 = 1.

3. Codim 1 bifurcations of 2D Filippov systems

x =

{

f(1)(x, α), H(x, α) < 0,

f(2)(x, α), H(x, α) > 0,(4)

where H : Rn×Rm → R is smooth with Hx(x, α) 6= 0 on the discontinuity

boundary

Σ(α) = {x ∈ Rn : H(x, α) = 0} ,

and f(i) : Rn × R

m → Rn are smooth functions.

Two systems (4) corresponding to different parameter values are called

topologically equivalent if there is a homeomorphism of Rn that maps

any standard/sliding orbit segment of the first system onto the stan-

dard/sliding orbit segment of the second system, preserving the direction

of time.

Bifurcations are changes of the topological equivalence class under

parameter variations.

3.1 Codim 1 local bifurcations in 2D

Collisions of

• standard equilibria with Σ

• tangent points

• pseudo-equilibria

Boundary focus cases

BF1

BF2

X0

S1

S2

α = 0

Σ

S2

S1

α < 0

Xα

Σ

S2

α > 0

Tα Pα TαΣ

S1

X0

S1

S2

α = 0

Σ

S2

S1

α < 0

Σ

S2

α > 0

Tα Pα Σ

S1Xα

Lα

Tα

Degenerate boundary focus: DBF

T

x1

R

x2 = 0

x1 = 0

x2

{

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 cases (continue)

BF3

BF4

BF5

α = 0α < 0 α > 0

Xα

S2S2

S1

S1

α < 0 α = 0 α > 0

Σ

S1

S2

Tα Σ Σ Tα

Pα

Σ

S2

S1

Tα

α < 0

Xα

X0 Σ

S2

α = 0 α > 0

Σ

S1

S2

Tα

S1

S2

Σ

S2

S1

α < 0 α = 0 α > 0

S2

Σ

S1

Tα ΣPα Tα

S1

Lα

Xα

Pα

X0

X0

Double tangency

DT1

DT2

T0

S2

T2

α

S2

Σ Σ

S1

S2

Σ

S1S1

α < 0 α = 0 α > 0

T0

S2S2

Σ

S2

Σ

α = 0 α > 0

T1

α

T2

α

S1S1

S1

Σ

α < 0

T1

α

Collision of two invisible tangencies

II1

II2

α < 0 α = 0 α > 0

Σ

S2

S1

S2

ΣΣ

S1

S2

S1

Σ

S2

T(2)α T0 T

(2)α

T(1)α

T(1)α

α = 0 α > 0α < 0

S1

S2

Σ T0

S2

Σ

PαT(2)α

T(1)α

Lα

S1

S1

T(2)α

Pα T(1)α

Pseudo-saddle-node: PSN

S2

S1

Σ

α > 0α = 0α < 0

P 1α

P 2α

S2

P0

S2

Σ

S1 S1

Σ

3.2 Codim 1 global bifurcations in 2D

• Bifurcations of sliding cycles:

– Grazing-sliding

– Adding-sliding

– Switching-sliding

– Crossing-sliding

• Pseudo-homoclinic bifurcations:

– Homoclinic orbit to a pseudo-saddle-node

– Homoclinic orbit to a pseudo-saddle

• Sliding homoclinic orbit to a saddle

• Pseudo-heteroclinic bifurcations

Grazing-sliding cases

Σ

S2

S1

T0

L0

α > 0α = 0

Σ

S2

S1

Tα

Lα

Σ

S2

Lα

S1

Tα

α < 0

Σ

S2

S1

α > 0α = 0α < 0

Σ

S2

Ls

α

Tα

Lu

α

T0

L0

S1

S2

S1

Tα Σ

GR1

GR2

Adding-sliding: DT2 with global reinjection

S1

S2

L0

Σ

α = 0 α > 0

T0

α < 0

S2

S1

T1α

Σ

Lα

S2

S1

Σ

Lα

T2α

Switching-sliding

α < 0 α = 0 α > 0

Lα

Σ

S2

L0

Σ

S2

Lα

Σ

S2

S1

S1

S1

T(1)αT

(1)0

T(1)α

Crossing-sliding cases

CC

SC

α = 0α < 0 α > 0

S1 S1

Lα

Σ

Lα

S2

Σ

S2

S1

S2

T(1)0 T

(1)α

T(1)α

α = 0

Σ

S1

S2

S1

S2

S1

S2

Lα

Σ Σ

α < 0

Lα

α > 0

T(1)0 T

(1)α

T(1)α

Σ

L0

L0

Homoclinic orbit to a pseudo-saddle-node

S2

P 1α

S2

P 2α

P0

S2

Σ Σ Σ

Lα L0

S1S1S1

α < 0 α > 0α = 0

Homoclinic orbit to a pseudo-saddle: HPS

α < 0 α = 0 α > 0

Lα H0

S2

Σ

S2

S1

S1

S2

S1

Σ ΣPαP0 Pα

Sliding homoclinic orbit to a saddle

α < 0α > 0α = 0

Σ

Tα

S1

Σ

T0

S2

S1X0

S2

S1Xα

Xα

Tα Σ

S2

Lα Γ0

4. Codim 2 bifurcations in 2D Filippov systems

• Local bifurcations:

– Degenerate boundary focus

– Boundary Hopf

• Global bifurcations:

– Sliding-grazing of a nonhyperbolic cycle (fold-grazing)

Degenerate boundary focus: DBF

2

0 2

1

0

1

β1

BF1

BF2

HPS

S1 S1

HPS

S1

S1 β2

O

Σ

Σ

S2

Σ

S2

S2

Σ

S2

Boundary Hopf: BHP

1

25

4

5

21

4

3

3

β2

O

BF4

BF1

S1

β1

S1

PSNS1

S1S1S1

S1GR1

GR1

ΣH

S2

Σ

S2

S2

S2

PSN

S2

ΣΣ

S2

Σ

S2

Σ

Σ

Fold-grazing: FG1

0

1

10

2

2

S1

S1

S1

O

LPC

S1

S1

GR2

S1

β2

β1O

LPC

S1

GR2

Σ

S2

Σ

Σ

S2

S2

Σ

S2

Σ

Σ

S2

S2

S2

Σ

GR1

GR1

Fold-grazing: FG2

1

0

2

2

1

0

S1

β2

LPC

GR2

S1S1

LPC

S1

O

S1

β1

S1

S1

GR2

S1

O

S2

Σ

GR1

S2

Σ

S2

Σ

S2

Σ

S2

Σ

S2

Σ

Σ

S2

GR1

5. Example: Controlled harvesting a prey-predator community

Rosenzweig-MacArthur-Holling model:

x1 = x1(1 − x1) −ax1x2

α2 + x1x2 =

ax1x2

α2 + x1− cx2

Nontrivial zero-isoclines:

x2 =1

a(α2 + x1)(1 − x1), x1 =

α2c

a− c.

(c)(b)(a)

x2

x1x1

x2 x2

x1

Relay control by harvesting

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 < 0,

f(2)(x), x2 − α5 > 0,

where

f(1) =

(

x1(1 − x1) − ψ(x1)x2ψ(x1)x2 − dx2

)

, f(2) =

(

x1(1 − x1) − ψ(x1)x2ψ(x1)x2 − dx2 − ex2

)

,

ψ(x1) =ax1

α2 + x1.

Fix a = 0.3556, d = 0.0444, e = 0.2067 and α2 = 0.33

Stable standard cycle: α5 = 2.75

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

x

y

fig.1 − alpha=2.75

Grazing-sliding GR1: α5 ≈ 2.440

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

x

y

fig.2 − alpha=2.44005978786827

Stable sliding cycle: α5 = 1.625

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

x

y

fig.3 − alpha=1.625

Pseudo-saddle-node PSN: α5 ≈ 1.2437

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

x

y

fig.4 − alpha=1.24375781250000

Stable sliding cycle and pseudo-node: α5 ≈ 1.2375

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

x

y

fig.13 − alpha=1.2375

Homoclinic orbit to pseudo-saddle HPS: α5 ≈ 1.2277

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

x

y

fig.14 − alpha=1.2277

Stable pseudo-node: α5 ≈ 1.175

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

x

y

fig.5 − alpha=1.175

Homoclinic orbit to pseudo-saddle HPS: α5 ≈ 1.03

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

x

y

fig.6 − alpha=1.03

Stable sliding cycle (small) and pseudo-node: α5 = 1.02

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

x

y

fig.7 − alpha=1.02

Boundary focus BF1: α5 ≈ 1.01017

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

x

y

fig.8 − alpha=1.01070918367347

Stable pseudo-node: α5 = 0.9

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

x

y

fig.9 − alpha=0.9

Boundary node BN1: α5 ≈ 0.6527

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

x

y

fig.10 − alpha=0.65275464010865

Stable node: α5 = 0.5

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

x

y

fig.11 − alpha=0.5

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

A - degenerate boundary focus (DBF ); B - boundary Hopf (BHP ).