Investigating a heterodimensional cycle in the flow of a four-dimensional differential equation

Transcription

1 Investigating a heterodimensional cycle in the flow of a four-dimensional differential equation Gemma Mason December 2016 Andy Hammerlindl Bernd Krauskopf Hinke Osinga

2 Introduction This talk is about a 4D differential equation model for intracellular calcium.

3 Introduction This talk is about a 4D differential equation model for intracellular calcium. For some parameter values, the flow of the equations includes a heterodimensional cycle.

4 Introduction This talk is about a 4D differential equation model for intracellular calcium. For some parameter values, the flow of the equations includes a heterodimensional cycle. Heterodimensional cycles are interesting:

5 Introduction This talk is about a 4D differential equation model for intracellular calcium. For some parameter values, the flow of the equations includes a heterodimensional cycle. Heterodimensional cycles are interesting: They can only happen in four or more dimensions.

6 Introduction This talk is about a 4D differential equation model for intracellular calcium. For some parameter values, the flow of the equations includes a heterodimensional cycle. Heterodimensional cycles are interesting: They can only happen in four or more dimensions. Always breakable by changing the parameters just a little.

7 Introduction This talk is about a 4D differential equation model for intracellular calcium. For some parameter values, the flow of the equations includes a heterodimensional cycle. Heterodimensional cycles are interesting: They can only happen in four or more dimensions. Always breakable by changing the parameters just a little. But heterodimensional cycles are always arbitrarily close to robust heterodimensional cycles!(bonatti & Diaz, 2006)

8 Introduction This talk is about a 4D differential equation model for intracellular calcium. For some parameter values, the flow of the equations includes a heterodimensional cycle. Heterodimensional cycles are interesting: They can only happen in four or more dimensions. Always breakable by changing the parameters just a little. But heterodimensional cycles are always arbitrarily close to robust heterodimensional cycles!(bonatti & Diaz, 2006) We will look at what the invariant manifolds of the system are doing when this heterodimensional cycle exists.

9 Introduction This talk is about a 4D differential equation model for intracellular calcium. For some parameter values, the flow of the equations includes a heterodimensional cycle. Heterodimensional cycles are interesting: They can only happen in four or more dimensions. Always breakable by changing the parameters just a little. But heterodimensional cycles are always arbitrarily close to robust heterodimensional cycles!(bonatti & Diaz, 2006) We will look at what the invariant manifolds of the system are doing when this heterodimensional cycle exists. Because the system is in 4D, we will use two different ways of reducing it to a 3D picture: projection, and Poincaré maps.

10 Introduction This talk is about a 4D differential equation model for intracellular calcium. For some parameter values, the flow of the equations includes a heterodimensional cycle. Heterodimensional cycles are interesting: They can only happen in four or more dimensions. Always breakable by changing the parameters just a little. But heterodimensional cycles are always arbitrarily close to robust heterodimensional cycles!(bonatti & Diaz, 2006) We will look at what the invariant manifolds of the system are doing when this heterodimensional cycle exists. Because the system is in 4D, we will use two different ways of reducing it to a 3D picture: projection, and Poincaré maps. By the end, we should be able to see at least a hint of how a heterodimensional cycle could ever be robust.

11 4D Atri model for intracellular calcium dc dt = v D dv dt = sv dc t dt = ɛ (J k pc) s dn dt = 1 2 (α k f c 2 ( φ2 φ 2 + c n ) ( γ(ct + Dv sc) c 2 φ 2 n 1 s ) ) c + k s c ɛ(j k p c) (variables in blue, parameters in red)

12 A heterodimensional cycle

13 A heterodimensional cycle (J = , s = 9) two periodic orbits See also How to find a codimension-one heteroclinic cycle between two periodic orbits, Zhang et al, 2012

14 A heterodimensional cycle (J = , s = 9) two periodic orbits saddles, with stable manifolds of different dimensions in this case, the top one has 2D stable and 3D unstable other has 2D unstable and 3D stable See also How to find a codimension-one heteroclinic cycle between two periodic orbits, Zhang et al, 2012

15 A heterodimensional cycle (J = , s = 9) two periodic orbits saddles, with stable manifolds of different dimensions in this case, the top one has 2D stable and 3D unstable other has 2D unstable and 3D stable The two 2D manifolds intersect in a codimension-one orbit from one periodic orbit to the other See also How to find a codimension-one heteroclinic cycle between two periodic orbits, Zhang et al, 2012

16 A heterodimensional cycle (J = , s = 9) two periodic orbits saddles, with stable manifolds of different dimensions in this case, the top one has 2D stable and 3D unstable other has 2D unstable and 3D stable The two 2D manifolds intersect in a codimension-one orbit from one periodic orbit to the other The 3D manifolds also intersect to give some codimension-zero orbits back. See also How to find a codimension-one heteroclinic cycle between two periodic orbits, Zhang et al, 2012

17 A heterodimensional cycle (J = , s = 9) two periodic orbits saddles, with stable manifolds of different dimensions in this case, the top one has 2D stable and 3D unstable other has 2D unstable and 3D stable The two 2D manifolds intersect in a codimension-one orbit from one periodic orbit to the other The 3D manifolds also intersect to give some codimension-zero orbits back. See also How to find a codimension-one heteroclinic cycle between two periodic orbits, Zhang et al, 2012

18 3D projection and Poincaré section

19 3D projection and Poincaré section

20 3D projection and Poincaré section

21 3D projection and Poincaré section

22 3D projection and Poincaré section

23 Stable manifolds in flows and maps Fig. 2: Stable manifold of a fixed point of a map. The stable manifold of an object is the set of trajectories which approach the object as t. Fig. 1: Stable manifold of a periodic orbit in a flow.

24 Stable manifolds in flows and maps Fig. 2: Stable manifold of a fixed point of a map. Fig. 1: Stable manifold of a periodic orbit in a flow. The stable manifold of an object is the set of trajectories which approach the object as t. The unstable manifold of an object is the set of trajectories which approach the object as t.

25 Calculating stable and unstable manifolds The tool we use is pseudoarclength continuation of boundary value problems. We can follow a solution to a BVP as we change a parameter.

26 Calculating stable and unstable manifolds The tool we use is pseudoarclength continuation of boundary value problems. We can follow a solution to a BVP as we change a parameter. The definition of parameter is pretty general, and we take full advantage of this.

27 Calculating stable and unstable manifolds We use the existing heterodimensional cycle as starting data.

28 Calculating stable and unstable manifolds We use the existing heterodimensional cycle as starting data. We insert a section parallel to the one we want into the codimension-one connection, and then set the position of the section as a parameter. We can now move the section to the place we want it.

29 Calculating stable and unstable manifolds We use the existing heterodimensional cycle as starting data. We insert a section parallel to the one we want into the codimension-one connection, and then set the position of the section as a parameter. We can now move the section to the place we want it. Now we stop allowing the position of the section to vary. We let δ vary instead.

30 Calculating stable and unstable manifolds We use the existing heterodimensional cycle as starting data. We insert a section parallel to the one we want into the codimension-one connection, and then set the position of the section as a parameter. We can now move the section to the place we want it. Now we stop allowing the position of the section to vary. We let δ vary instead. This traces out the intersections of the manifold with the section. We also get extra data about the manifold itself, which we can use to create 3D projections.

31 Stable manifold of the smaller periodic orbit The manifold passes very close to the second, larger periodic orbit. Its behaviour here is governed by the lambda lemma.

32 The lambda lemma As we get closer to the equlibrium point, each forward iteration of the red manifold gets stretched by the orange unstable manifold until it lies nearly parallel to it.

33 The lambda lemma As we get closer to the equlibrium point, each forward iteration of the red manifold gets stretched by the orange unstable manifold until it lies nearly parallel to it.

34 The lambda lemma As we get closer to the equlibrium point, each forward iteration of the red manifold gets stretched by the orange unstable manifold until it lies nearly parallel to it.

35 The lambda lemma As we get closer to the equlibrium point, each forward iteration of the red manifold gets stretched by the orange unstable manifold until it lies nearly parallel to it.

36 The lambda lemma As we get closer to the equlibrium point, each forward iteration of the red manifold gets stretched by the orange unstable manifold until it lies nearly parallel to it. A similar property applies with the stable and unstable manifolds switched, going backward rather than forward with our iterations.

37 The lambda lemma in 3D In 3D, stretching occurs along the direction of strongest expansion (forward time) or contraction (backward time).

38 The lambda lemma in 3D In 3D, stretching occurs along the direction of strongest expansion (forward time) or contraction (backward time). The strongly stable direction is plotted here in light blue.

39 The lambda lemma in 3D In 3D, stretching occurs along the direction of strongest expansion (forward time) or contraction (backward time). The strongly stable direction is plotted here in light blue. The dark blue (stable) manifold lies along the light blue strongly stable manifold when we are close to the fixed point.

40 The lambda lemma in 3D In 3D, stretching occurs along the direction of strongest expansion (forward time) or contraction (backward time). The strongly stable direction is plotted here in light blue. The dark blue (stable) manifold lies along the light blue strongly stable manifold when we are close to the fixed point. Closer to the fixed point, there are more branches of the stable manifold, which we can find using the heterodimensional cycle.

41 The lambda lemma in 3D In 3D, stretching occurs along the direction of strongest expansion (forward time) or contraction (backward time). The strongly stable direction is plotted here in light blue. The dark blue (stable) manifold lies along the light blue strongly stable manifold when we are close to the fixed point. Closer to the fixed point, there are more branches of the stable manifold, which we can find using the heterodimensional cycle.

42 Unstable manifold of the larger periodic orbit

43 Unstable manifold of the larger periodic orbit

44 Unstable manifold of the larger periodic orbit

45 Unstable manifold of the larger periodic orbit

46 Unstable manifold of the larger periodic orbit

47 Zooming in Accumulation on the orange strongly unstable manifold is consistent with the lambda lemma.

48 The codimension-one connection: an overview

49 The codimension-one connection: an overview

50 The codimension-one connection: an overview

51 The codimension-one connection: an overview

52 The codimension-one connection: an overview

53 The codimension-one connection: an overview

54 The unstable manifold has one more trailing edge...

55 The unstable manifold has one more trailing edge...

56 The unstable manifold has one more trailing edge...

57 Future Work Create pictures of 3D projections of this new data.

58 Future Work Create pictures of 3D projections of this new data. Find more branches of the unstable manifold.

59 Future Work Create pictures of 3D projections of this new data. Find more branches of the unstable manifold. We can follow existing trajectories back in time to new intersections with the section.

60 Future Work Create pictures of 3D projections of this new data. Find more branches of the unstable manifold. We can follow existing trajectories back in time to new intersections with the section. Then we can trace out that new intersection between the manifold and the section.

61 Future Work Create pictures of 3D projections of this new data. Find more branches of the unstable manifold. We can follow existing trajectories back in time to new intersections with the section. Then we can trace out that new intersection between the manifold and the section. There are good theoretical reasons to believe that the unstable manifold may create what is known as a blender. Blenders look solid when viewed from the correct angle.

62 Future Work Create pictures of 3D projections of this new data. Find more branches of the unstable manifold. We can follow existing trajectories back in time to new intersections with the section. Then we can trace out that new intersection between the manifold and the section. There are good theoretical reasons to believe that the unstable manifold may create what is known as a blender. Blenders look solid when viewed from the correct angle. Calculate intersections with the Poincaré section of the 3-dimensional stable and unstable manifolds of the second and first periodic orbits, respectively.

63 Future Work Create pictures of 3D projections of this new data. Find more branches of the unstable manifold. We can follow existing trajectories back in time to new intersections with the section. Then we can trace out that new intersection between the manifold and the section. There are good theoretical reasons to believe that the unstable manifold may create what is known as a blender. Blenders look solid when viewed from the correct angle. Calculate intersections with the Poincaré section of the 3-dimensional stable and unstable manifolds of the second and first periodic orbits, respectively. What happens when we shift system parameters? There should be other heterodimensional cycles nearby. In fact, there should be robust heterodimensional cycles nearby.

64 Conclusion Heterodimensional cycles give rise to interesting structures in the surrounding dynamics. The Poincaré section lets us get a good look at them. We can calculate intersections of invariant manifolds with a Poincaré section in a fast, stable and accurate way by using pseudo-arclength continuation of boundary value problems. Having a heterodimensional cycle gives us a useful collection of intersections with the section from which we can start our continuations. This is work in progress. A more complete picture of the dynamics will be forthcoming!