Canard cycles in generic slow-fast systems on the two-torus How - - PowerPoint PPT Presentation

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Canard cycles in generic slow-fast systems on the two-torus How - - PowerPoint PPT Presentation

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Canard cycles in generic slow-fast systems on the two-torus How many ducks can dance on the torus? Ilya V. Schurov ilya at schurov.com Department of Mathematics and Mechanics Moscow State University January 15, 2010 Topology, Geometry, and

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Canard cycles in generic slow-fast systems

  • n the two-torus

How many ducks can dance on the torus? Ilya V. Schurov ilya at schurov.com

Department of Mathematics and Mechanics Moscow State University

January 15, 2010 Topology, Geometry, and Dynamics: Rokhlin Memorial Saint Petersburg, Russia

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 1 / 11

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Slow-fast systems: definitions

Definition

The slow-fast system is a system of the following form:

  • ˙

x = f (x, y, ε), ˙ y = εg(x, y, ε), ε ∈ (R, 0). (1) Variables: x is a fast variable, and y is a slow one, ε is a small parameter. Slow curve is a set M := {(x, y) | f (x, y, 0) = 0}.

Remark

Outside of any fixed neighborhood of the slow curve M, for ε small enough, the fast variable x changes much more rapidly than the slow variable y.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 2 / 11

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Slow-fast systems: definitions

Definition

The slow-fast system is a system of the following form:

  • ˙

x = f (x, y, ε), ˙ y = εg(x, y, ε), ε ∈ (R, 0). (1) Variables: x is a fast variable, and y is a slow one, ε is a small parameter. Slow curve is a set M := {(x, y) | f (x, y, 0) = 0}.

Remark

Outside of any fixed neighborhood of the slow curve M, for ε small enough, the fast variable x changes much more rapidly than the slow variable y.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 2 / 11

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SLIDE 4

Slow-fast systems: definitions

Definition

The slow-fast system is a system of the following form:

  • ˙

x = f (x, y, ε), ˙ y = εg(x, y, ε), ε ∈ (R, 0). (1) Variables: x is a fast variable, and y is a slow one, ε is a small parameter. Slow curve is a set M := {(x, y) | f (x, y, 0) = 0}.

Remark

Outside of any fixed neighborhood of the slow curve M, for ε small enough, the fast variable x changes much more rapidly than the slow variable y.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 2 / 11

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SLIDE 5

Slow-fast systems: definitions

Definition

The slow-fast system is a system of the following form:

  • ˙

x = f (x, y, ε), ˙ y = εg(x, y, ε), ε ∈ (R, 0). (1) Variables: x is a fast variable, and y is a slow one, ε is a small parameter. Slow curve is a set M := {(x, y) | f (x, y, 0) = 0}.

Remark

Outside of any fixed neighborhood of the slow curve M, for ε small enough, the fast variable x changes much more rapidly than the slow variable y.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 2 / 11

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Fast system

Fast dynamics for ε = 0: slow variable y is a constant. Attracting part of the slow curve M consist of stable fixed points. Repelling part of the slow curve M consist of unstable fixed points. Folds are neutral fixed points.

Figure: Fast system and its fixed points

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 3 / 11

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Fast system

Fast dynamics for ε = 0: slow variable y is a constant. Attracting part of the slow curve M consist of stable fixed points. Repelling part of the slow curve M consist of unstable fixed points. Folds are neutral fixed points.

Figure: Fast system and its fixed points

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 3 / 11

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Fast system

Fast dynamics for ε = 0: slow variable y is a constant. Attracting part of the slow curve M consist of stable fixed points. Repelling part of the slow curve M consist of unstable fixed points. Folds are neutral fixed points.

Figure: Fast system and its fixed points

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 3 / 11

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Fast system

Fast dynamics for ε = 0: slow variable y is a constant. Attracting part of the slow curve M consist of stable fixed points. Repelling part of the slow curve M consist of unstable fixed points. Folds are neutral fixed points.

Figure: Fast system and its fixed points

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 3 / 11

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Slow-fast dynamics: generic planar case

Pick a point far from M it quickly falls on attracting segment of M than slowly moves along M than jumps near the fold point than falls on attracting segment of M, and so on.

Figure: Relaxation oscillation: slow and fast motions

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 4 / 11

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Slow-fast dynamics: generic planar case

Pick a point far from M it quickly falls on attracting segment of M than slowly moves along M than jumps near the fold point than falls on attracting segment of M, and so on.

Figure: Relaxation oscillation: slow and fast motions

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 4 / 11

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SLIDE 12

Slow-fast dynamics: generic planar case

Pick a point far from M it quickly falls on attracting segment of M than slowly moves along M than jumps near the fold point than falls on attracting segment of M, and so on.

Figure: Relaxation oscillation: slow and fast motions

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 4 / 11

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SLIDE 13

Slow-fast dynamics: generic planar case

Pick a point far from M it quickly falls on attracting segment of M than slowly moves along M than jumps near the fold point than falls on attracting segment of M, and so on.

Figure: Relaxation oscillation: slow and fast motions

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 4 / 11

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SLIDE 14

Slow-fast dynamics: generic planar case

Pick a point far from M it quickly falls on attracting segment of M than slowly moves along M than jumps near the fold point than falls on attracting segment of M, and so on.

Figure: Relaxation oscillation: slow and fast motions

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 4 / 11

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SLIDE 15

Slow-fast dynamics: generic planar case

Pick a point far from M it quickly falls on attracting segment of M than slowly moves along M than jumps near the fold point than falls on attracting segment of M, and so on.

Figure: Relaxation oscillation: slow and fast motions

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 4 / 11

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SLIDE 16

Slow-fast dynamics: generic planar case

Pick a point far from M it quickly falls on attracting segment of M than slowly moves along M than jumps near the fold point than falls on attracting segment of M, and so on.

Figure: Relaxation oscillation: slow and fast motions

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 4 / 11

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Canard solutions

Definition 1

Duck (or canard) solutions are solutions, whose phase curves contain an arc of length bounded away from 0 uniformly in ε, that keeps close to the unstable part

  • f the slow curve

Definition 2

Canard cycle is a limit cycle which is a canard.

Figure: Canards

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 5 / 11

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Canard solutions

Definition 1

Duck (or canard) solutions are solutions, whose phase curves contain an arc of length bounded away from 0 uniformly in ε, that keeps close to the unstable part

  • f the slow curve

Definition 2

Canard cycle is a limit cycle which is a canard.

Figure: Canards

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 5 / 11

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Canard solutions

Definition 1

Duck (or canard) solutions are solutions, whose phase curves contain an arc of length bounded away from 0 uniformly in ε, that keeps close to the unstable part

  • f the slow curve

Remark

There’s no attracting canard cycles in generic planar systems.

Figure: Canards

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 5 / 11

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Ducks on the torus: introduction

Consider slow-fast system on the two-torus Pick a point far from M Consider its trajectory in forward time Reverse the time When ε decreases, L moves down, and R moves up. For some ε, we’ve got canard cycle.

Figure: Ducks on the torus (Yu. S. Ilyashenko, J. Guckenheimer, 2001)

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 6 / 11

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Ducks on the torus: introduction

Consider slow-fast system on the two-torus Pick a point far from M Consider its trajectory in forward time Reverse the time When ε decreases, L moves down, and R moves up. For some ε, we’ve got canard cycle.

Figure: Ducks on the torus (Yu. S. Ilyashenko, J. Guckenheimer, 2001)

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 6 / 11

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Ducks on the torus: introduction

Consider slow-fast system on the two-torus Pick a point far from M Consider its trajectory in forward time Reverse the time When ε decreases, L moves down, and R moves up. For some ε, we’ve got canard cycle.

Figure: Ducks on the torus (Yu. S. Ilyashenko, J. Guckenheimer, 2001)

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 6 / 11

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SLIDE 23

Ducks on the torus: introduction

Consider slow-fast system on the two-torus Pick a point far from M Consider its trajectory in forward time Reverse the time When ε decreases, L moves down, and R moves up. For some ε, we’ve got canard cycle.

Figure: Ducks on the torus (Yu. S. Ilyashenko, J. Guckenheimer, 2001)

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 6 / 11

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SLIDE 24

Ducks on the torus: introduction

Consider slow-fast system on the two-torus Pick a point far from M Consider its trajectory in forward time Reverse the time When ε decreases, L moves down, and R moves up. For some ε, we’ve got canard cycle.

Figure: Ducks on the torus (Yu. S. Ilyashenko, J. Guckenheimer, 2001)

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 6 / 11

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Ducks on the torus: introduction

Consider slow-fast system on the two-torus Pick a point far from M Consider its trajectory in forward time Reverse the time When ε decreases, L moves down, and R moves up. For some ε, we’ve got canard cycle.

Figure: Ducks on the torus (Yu. S. Ilyashenko, J. Guckenheimer, 2001)

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 6 / 11

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Main results: the structure of ducky area

Figure: Intervals Cn: the ducks live here

There exists a sequence of intervals {Cn}∞

n=1 such that for every

ε ∈ Cn the system has attracting canard cycles. Intervals Cn are exponentially small. They accumulate to 0. Their density is 0 near ε = 0.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 7 / 11

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SLIDE 27

Main results: the structure of ducky area

Figure: Intervals Cn: the ducks live here

There exists a sequence of intervals {Cn}∞

n=1 such that for every

ε ∈ Cn the system has attracting canard cycles. Intervals Cn are exponentially small. They accumulate to 0. Their density is 0 near ε = 0.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 7 / 11

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SLIDE 28

Main results: the structure of ducky area

Figure: Intervals Cn: the ducks live here

There exists a sequence of intervals {Cn}∞

n=1 such that for every

ε ∈ Cn the system has attracting canard cycles. Intervals Cn are exponentially small. They accumulate to 0. Their density is 0 near ε = 0.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 7 / 11

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SLIDE 29

Main results: the structure of ducky area

Figure: Intervals Cn: the ducks live here

There exists a sequence of intervals {Cn}∞

n=1 such that for every

ε ∈ Cn the system has attracting canard cycles. Intervals Cn are exponentially small. They accumulate to 0. Their density is 0 near ε = 0.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 7 / 11

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How many ducks can dance on the torus?

Theorem 1 (Upper estimate for the number of canards)

Consider slow-fast system on the two-torus, i.e. (x, y) ∈ T2, and the speed of the slow motion is bounded away from zero (g > 0). Assume M is connected nondegenerate curve with 2N fold points, N < ∞, and some additional nondegenericity assumptions hold. Then there exists number 0 < K ≤ N, such that the following assertions hold: There exists a sequence {Cn}∞

n=1 of intervals on the ray {ε > 0},

accumulating to 0, such that for every ε ∈ Cn, the system has exactly 2K canard cycles (K attracting and K repelling). For any ε > 0 small enough, the number of limit cycles that make one rotation along y-axis is bounded by 2K. Their basins have bounded away from 0 measure.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 8 / 11

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How many ducks can dance on the torus?

Theorem 1 (Upper estimate for the number of canards)

Consider slow-fast system on the two-torus, i.e. (x, y) ∈ T2, and the speed of the slow motion is bounded away from zero (g > 0). Assume M is connected nondegenerate curve with 2N fold points, N < ∞, and some additional nondegenericity assumptions hold. Then there exists number 0 < K ≤ N, such that the following assertions hold: There exists a sequence {Cn}∞

n=1 of intervals on the ray {ε > 0},

accumulating to 0, such that for every ε ∈ Cn, the system has exactly 2K canard cycles (K attracting and K repelling). For any ε > 0 small enough, the number of limit cycles that make one rotation along y-axis is bounded by 2K. Their basins have bounded away from 0 measure.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 8 / 11

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How many ducks can dance on the torus?

Theorem 1 (Upper estimate for the number of canards)

Consider slow-fast system on the two-torus, i.e. (x, y) ∈ T2, and the speed of the slow motion is bounded away from zero (g > 0). Assume M is connected nondegenerate curve with 2N fold points, N < ∞, and some additional nondegenericity assumptions hold. Then there exists number 0 < K ≤ N, such that the following assertions hold: There exists a sequence {Cn}∞

n=1 of intervals on the ray {ε > 0},

accumulating to 0, such that for every ε ∈ Cn, the system has exactly 2K canard cycles (K attracting and K repelling). For any ε > 0 small enough, the number of limit cycles that make one rotation along y-axis is bounded by 2K. Their basins have bounded away from 0 measure.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 8 / 11

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SLIDE 33

How many ducks can dance on the torus?

Theorem 1 (Upper estimate for the number of canards)

Consider slow-fast system on the two-torus, i.e. (x, y) ∈ T2, and the speed of the slow motion is bounded away from zero (g > 0). Assume M is connected nondegenerate curve with 2N fold points, N < ∞, and some additional nondegenericity assumptions hold. Then there exists number 0 < K ≤ N, such that the following assertions hold: There exists a sequence {Cn}∞

n=1 of intervals on the ray {ε > 0},

accumulating to 0, such that for every ε ∈ Cn, the system has exactly 2K canard cycles (K attracting and K repelling). For any ε > 0 small enough, the number of limit cycles that make one rotation along y-axis is bounded by 2K. Their basins have bounded away from 0 measure.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 8 / 11

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Illustrations for main result: N = K = 2

Figure: Simple example: two pair of folds, two attracting ducks (two repelling ducks are not shown)

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 9 / 11

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How many ducks can dance on the torus? (2)

Remark

It follows from theorem 1, that for convex slow curve, there exists exactly one pair of canard cycles

Remark

The number K of canard cycles can be effectively computed without intergration of the system.

Theorem 2 (Sharp estimate for K)

For every N > 0 there exists an open set in the space of slow-fast systems on the two-torus for which the number of canard cycles reaches its maximum: K = N.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 10 / 11

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SLIDE 36

How many ducks can dance on the torus? (2)

Remark

It follows from theorem 1, that for convex slow curve, there exists exactly one pair of canard cycles

Remark

The number K of canard cycles can be effectively computed without intergration of the system.

Theorem 2 (Sharp estimate for K)

For every N > 0 there exists an open set in the space of slow-fast systems on the two-torus for which the number of canard cycles reaches its maximum: K = N.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 10 / 11

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SLIDE 37

How many ducks can dance on the torus? (2)

Remark

It follows from theorem 1, that for convex slow curve, there exists exactly one pair of canard cycles

Remark

The number K of canard cycles can be effectively computed without intergration of the system.

Theorem 2 (Sharp estimate for K)

For every N > 0 there exists an open set in the space of slow-fast systems on the two-torus for which the number of canard cycles reaches its maximum: K = N.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 10 / 11

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The duck farm

Figure: The construction of open set of slow-fast systems with maximal number of canard cycles. E.g. K = N = 4 on the figure.

Ilya V. Schurov (MSU) Canards on the two-torus January 15, 2010 11 / 11