Post on 11-Jul-2020
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 ).