2 Piecewisesmooth maps Chris Budd Maps Key idea

2. Piecewise-smooth maps Chris Budd

Maps

Key idea … The functions derivatives, differ when or one of their nth Discontinuity set Interesting discontinuity induced bifurcations occur when limit sets of the map intersect the discontinuity set

Piecewise-Linear, continuous

Origin of piecewise-linear continous maps: Direct: • Electronic switches [Hogan, Homer, di Bernardo, Feigin, Banerjee] Poincare maps of flow: • Corner bifurcations • DC-DC convertors

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]

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

I: Dynamics of the piecewise-linear-continuous map [Feigin, Hogan. Homer, di Bernardo] Fixed points Not all fixed points are admissible!

Persistence of a stable Fixed point

Non-smooth Fold Bifurcation

Non-smooth Period-doubling

Period-adding Robust chaos

II: Dynamics of the piecewise-linear-discontinous map [Glendinning, Keener, Arnold, Stark, Shantz, . . ] Two fixed points L admissible if x < 0 R admissible if x > 0

Co-existing periodic orbits

Fixed point Region of non-existence … expect exotic dynamics here

High periodic orbits mu-1 Period 2 LR periodic orbit Admissible range: Separate from L or R if , overlaps with L if

Period n: L n-1 R periodic orbit

If: Period n: L n-1 R periodic orbit exists when Period-adding In the interval n-1 n-2 Period 2 n-1: L RL R periodic orbit exists for certain parameter values

Parameter range for simple periodic orbits Fractions 1/n Fractions (n-1)/n

Homoclinic orbit Fixed point Period adding Farey sequence

Winding number

Dynamics of the piecewise-linear map Period adding Farey sequence Chaotic

Dynamics of the piecewise-linear map Period incrementing sequence

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

Period adding Chaos

Immediate jump to robust chaos Partial period adding

Period adding windows scale geometrically with lambda squared.

Stable fixed point x=0 if Unstable fixed point if Typical dynamics

Trapping region: Maximal value Induced map

F is piecewise-smooth, m is piecewise-constant F/mu m Map shows a strong degree of self-similarity Implies geometric scaling of period-adding windows

• G has an infinite number of fixed points • All unstable if • First and second stable if

Get similar behaviour in higher-dimensional square-root maps 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 1 D map

Next lecture. . See how this allows us to explain the dynamics of hybrid and related systems