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