2. piecewise-smooth maps chris budd. maps key idea … the functions or one of their nth...
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2. Piecewise-smooth maps
Chris Budd
0)()(
0)()()(
2
1
xHifxF
xHifxFxfx
Maps
Key idea …
The functions or one of their nth derivatives, differ when
0)(: xHxx
Discontinuity set
)(),( 21 xFxF
Interesting discontinuity induced bifurcations occur when limit sets of the map intersect the discontinuity set
Piecewise-Linear, continuous
0,
0,)(
2
1
xx
xxxfx
Origin of piecewise-linear continous maps:
Direct:
•Electronic switches [Hogan, Homer, di Bernardo, Feigin, Banerjee]
Poincare maps of flow:
• Corner bifurcations
• DC-DC convertors
0,1
0,)(
2
1
xx
xxxfx
Piecewise-Linear, discontinuous
Origin of piecewise-linear discontinuous maps:
Direct:
• Neuron dynamics [Keener, Stark, Bressloff]
• Heart beats [Keener]
• Electronic switches [Hogan, Banerjee]
• Circle maps [Glendinning, Arnold]
Poincare maps of flow:
• Impact oscillators [B, Pring]
• Cam dynamics [B, Piiroinen]
• Pin-ball machines [Pring]
.,
,,)(
xx
xxxfx
Square-root-Linear, continuous
Origin of Square-root-Linear, continuous maps
• Local behaviour of the Poincare maps of hybrid systems close to grazing impacts
[Budd, Nordmark, Whiston]
• Quasi-local behaviour of the Poincare maps of piecewise-smooth flows close to grazing
(The very local behaviour of such flows leads to maps with a piecewise linear map coupled to a map with a 3/2 power law)
All maps have fixed points over certain ranges of
Border collision bifurcations occur when for certain parameter values the fixed points intersect with the discontinuity set
Get exotic dynamics close to these parameter values
0,
0,)(
2
1
xx
xxxfx
I: Dynamics of the piecewise-linear-continuous map
[Feigin, Hogan. Homer, di Bernardo]
0)1(
,0)1( 2
21
1
xx
Fixed points
Not all fixed points are admissible!
101 21
1x
Persistence of a stable
Fixed point
2x
21 101
21 101 Non-smooth
Fold Bifurcation1x
2x
101 21
0
1
)1(,0
1
)1(
21
22
21
11
yy
121
Non-smooth
Period-doubling2x
12 ,4.0,1
Period-adding
Robust chaos
doublingPeriod n2
1
1
collisionBorder
1
22
21
)1(
1n
n
orbitPeriodicnLR
II: Dynamics of the piecewise-linear-discontinous map
[Glendinning, Keener, Arnold, Stark, Shantz,..]
0,1
0,)(
2
1
xx
xxxfx
Two fixed points
)1(
)1(
)1(
22
11
x
x
R admissible if x > 0
L admissible if x < 0
Co-existing periodic orbits
1,0,1 21
Fixed point
1,0,1 21
Fixed point
Region of non-existence … expect exotic dynamics here
High period periodic orbits mu-1
Period 2 LR periodic orbit
10,0 21 yxxy
21
11
21
2
1
)1(0,
1
1)1(0
yx
21
1
1
1
1
42
1
42
11,121
Admissible range:
Separate from L or R if ,overlaps with L if 01 01
Period n: Ln-1
R periodic orbit
jjn xxyxxy 112111 0,10,0
112
112
11
1
)1/()(1
1
nx
112
1121
12
1
12
1
)1/()(
1
n
n
y
n
n
n
n
1
)1(
1
)1( 21
21
1
22
1
)1(
1
)1(
n
n
n
n
If:
Period n: Ln-1
R periodic orbit exists when
Period 2n-1: Ln-1
RLn-2
R periodic orbit exists for certain parameter values
In the interval
Period-adding 11, nnn if
10 21
1
Parameter range for simple periodic orbits
Fractions 1/n Fractions (n-1)/n
Period adding Farey sequence
Fixed point
Homoclinic orbit
1,0,1 21
Fixed point
Winding number
Dynamics of the piecewise-linear map
21 10
Period adding Farey sequence
Chaotic
Dynamics of the piecewise-linear map
101 21
Period incrementing sequence
x
.,
,,)(
xx
xxxfx
III. Piecewise Square-root-linear maps
Map arises in the study of grazing bifurcations of flows and hybrid systems
Infinite stretching when
Fixed point at 00 ifx
4
10
3
2
4
1
Chaos
Period adding
13
2
Immediate jump to robust chaos
Partial period adding
2 )log(
)log(
Period adding
Period adding windows scale geometrically with lambda squared.
)log(
Stable fixed point x=0 if
Unstable fixed point if
10,0
0
Typical dynamics
Trapping region: xxV :
VVxxF
VxVx
xm
xx
x
xxfxVx
m
mm
nnn
:),(
,
),(
)(,
0
1
0
01
1
0010
Typically
Induced map
Maximal value
]1,[/ xz
Map shows a strong degree of self-similarity
1,/)(/)(,2 mmzFzF
F is piecewise-smooth, m is piecewise-constant
Implies geometric scaling of period-adding windows
F/mu m
,4,3,2,1,0,1
1),(
]1,[),(/)(
))(/(
1
2
kz
zG
zGzF
F
k
With
Then
If
• G has an infinite number of fixed points
• All unstable if
• First stable if
• First and second stable if
3/23/24/1
4/1
Get similar behaviour in higher-dimensional square-root maps
.)(
,0)(,
,0)()(,
),(
xHy
xHByMAx
xHNMAxCMAx
xfx
Map [Nordmark] also arises naturally in the study of grazing in flows and hybrid systems.
Behaviour depends on the eigenvalues of A
If A has complex eigenvalues we see discontinuous transitions between periodic orbits similar to the piecewise-linear case.
If A has real eigenvalues we see similar behaviour to the 1D map
Next lecture .. See how this allows us to explain the dynamics of hybrid and related systems