1. On Nonuniqueness of Cycles in Dissipative Dynamical Systems of Chemical Kinetics A.A.Akinshin, V.P.Golubyatnikov Altai State Technical University, Barnaul Sobolev Institute of Mathematics, Novosibirsk 8 June 2012, Novosibirsk. “ Solitons, Collapses and Turbulence”

2. Our aim is to give a mathematical explanations and predictions of numerical experiments with periodic trajectories of nonlinear dynamical systems of chemical kinetics considered as models of gene networks regulated by simple combinations of negative and positive feedbacks. OUTLINE : 1.Existence; 2. Stability; 3. Non-Uniqueness of cycles. 2/57

3. Contemporary mathematics, v. 553, 2011. Russian Journal Numerical Analysis and Math.Modeling, v.26, N 4, 2011. Novosibirsk State University Herald, v.10, NN 1, 3, 2010; v.12, N 2, 2012. A.N.Kolmogorov, I.G.Petrovskii, N.S.Piskunov Moscow University Herald, 1937.

4. 1. Some simple gene networks models We study odd-dimensional dynamical systems are smooth and monotonically decreasing. This corresponds to the negative feedbacks in the gene networks. Functions Lemma 1. Each system of this type has exactly one stationary point in the positive octant: is an invariant domain of the system. [ (2k+1). +(2k) ] 4/57 Lemma 2.

5. We describe below the case (2k+1) 5/57 Or, in vector form : d X /d t = F ( X ) – X. We Since div(F(X) – X) ≡ – (2k + 1), (2k+1)-dimensional volume of any bounded domain W in Q decreases exponentially: Vol (W(t))=exp(–t(2k+1))·Vol(W(0)). All these domains collapse, but their attractors are not points! In most interesting cases these attractors contain cycles.

6. Non-convex 3D invariant domain in composed by six triangle prisms Q 6/57 0

7. Theorem 1. If the stationary point is hyperbolic then the system (2k+1) has at least one periodic trajectory in the invariant domain Q. The following diagram (D) shows the discrete scheme of some of the trajectories of the system (2k+1). 7/57 We reduce this invariant domain Q to the union of 4k+2 triangle prisms in order to localize the position of this cycle (“first’’cycle). Potential level=1.

8. A trajectory and a limit cycle. 8/57 ! !

9. Below we demonstrate projections of trajectories of symmetric 5-D system onto 2-D and 3-D planes. Theorem 1′. If the dynamical system (2k+1) in the Th.1 is symmetric with respect to the cyclic permutation of the variables then the system has a cycle with corresponding symmetry. 9/57

10. 10 Projections of trajectories of 5-D system onto the 2-D plane corresponding to

11. 11 Projections of trajectories of the 5-D system onto the 3-D plane corresponding to the eigenvalues with positive real parts and the negative eigenvalue of the linearization matrix. The blue spot shows the position of projection of the stationary point.

12. The characteristic polynomial of the linearization of the system (2k+1) at the stationary point has the form Here is the product of all derivatives at the point. We arrange the eigenvalues of this linearization according to the values of their real parts: The eigenvalue is real and negative. So, If the point is hyperbolic then none of these real parts vanishes. If k=2 then 12

13. symmetric dynamical system Eigenvalues of one9-D 13/57

14. Trajectories of 9-D symmetric system projected onto 3D-planes corresponding to different eigenvectors of the linearization of this system near the stationary point. The trajectories are contained in Q. 14/57

15. Similar results can be obtained for the systems of the types, etc. 15/57 M.Hirsch (1987).

16. (VM)=(2k+1) 16/57 The eigenvalues of A can be expressed explicitly : η >0. 2. Stability questions

17. The transfer matrix Let be the Jacobi matrix of, and let i = 1,2,… be its norm. 17/57

18. Theorem 2. If the system (2k+1) satisfies the conditions of the theorem 1 and for some positive η then the invariant domain Q contains a stable cycle of this system. Potential level equals 1. Russel Smith has shown that if then the system (VM) has a stable cycle (1987). Actually, he notes that this is not a sharp estimate!! 18

19. 3. Nonuniqueness of cycles in the system (2k+1) According to the Grobman-Hartmann theorem, each nonlinear dynamical system can be linearized in some neighborhood W of its hyperbolic point. Consider in W 2-D planes corresponding to pairs of the eigenvalues with positive real parts. These planes are composed by unwinding trajectories of the dynamical system (2k+1). Conjecture 1: Outside of W different 2-D planes generate different cycles. 19/57

20. Let (2k+1) = p·q Then the phase portrait of the symmetric system contains p-dimensional and q-dimensional invariant planes. If the stationary point of this system is hyperbolic for its restrictions to these planes, then each of these planes contain a cycle. They are not contained in Q! (Potential level > 1.) If the conditions of Theorem 2 are satisfied, then these two cycles are stable within corresponding planes, not in the ambient phase portrait! If p ≠q, then this system has at least 3 different cycles. 20/57

21. symmetric dynamical system (what happens in 5-D?) Eigenvalues of one9-D 21/57

22. Two cycles and some chaos in 5-D (!!) symmetric system 22/57

23. “two” cycles in 5-D symmetric system

24. Projections of two different cycles of 9-D symmetric system onto 3-D plane The second cycle is not contained in Q. Its potential level equals 3. The stationary point is at the top of the picture. 24/57

25. Projections of two different cycles of 11-D symmetric system onto two different 3-D planes left; right. 25/57

26. Trajectory of 9-D system attracted by the first cycle 26/57 Chaotic oscillations

27. Projections of 3 cycles of 15-dimensional system onto the plane 27/57

28. 3 cycles in symmetric 15-D system 26/57 1 P. level = 3 Potential level = 5

29. Trajectory of 15-D system attracted by the first cycle 29/57 chaotic oscillations first cycle

30. Similar phenomenon 30/57 first cycle chaotic oscillations → ↓

31. 4 cycles in 15-D symmetric system 31/57

32. ;; Smooth monotonically decreasing for 4. Model of 3-D gene network regulated by a simple combination of negative and positive feedbacks system (ffΛ): 32/57 or more general unimodal function.

33. Let be the maximal value of and is the inverse function to Lemma 3. Let and either or for Then the system (ffΛ) has exactly one stationary point in the positive octant. ( see Appendix A. ) Let be defined by 33/57

34. Linearization of system (1) at this point is described by the matrix with one negative eigenvalue. Its other eigenvalues are complex. Consider the case (+)‏(+)‏ Theorem 3. If the condition (+) is satisfied then the system (ffΛ): has at least one periodic trajectory. 33/57

35. The proof is based on existence of an invariant domain of the system (ffΛ). This is the parallelepiped Actually, one can construct essentially smaller invariant domain (see below). Now, existence of periodic trajectories follows from the Brower fixed point theorem, as usual. Recall that 35/57

36. Trajectories of the system (ffΛ) right: left: 36/57

37. ; ; for 5. More complicated gene networks models regulated by combinations of positive and negative feedbacks system (fΛΛ): 37/57 or other unimodal functions.

38. Stationary points of the system (fΛΛ). I II III IV V 38/57

40. Stationary points I and II of the system (fΛΛ): I II 40/57

41. 41/57 Analogs of the theorems 1 and 2 about existence of a cycle and existence of a stable cycle hold in the neighborhoods of the stationary points I and III. The stationary point V is stable. The stationary points II and IV have topological index +1.

42. рублей. Stationary points and cycles of the system (fΛΛ): 42

43. Stationary points and cycles of the system (fΛΛ) (same parameters, other trajectories). 43/57

44. 6. Glass-Mackey-type systems Ricker function. Glass-Mackey function. (GM)‏ (w)=r  w  (  –w) Logistic function. (L)‏ 44/57

45. Each of the systems listed here has exactly 7 stationary points in some invariant domain If the parameters of the system are sufficiently large. The origin is also stationary point of each of these systems, but it does not seem to be so interesting. (Λ)‏(Λ)‏ 45/57

46. Positions of the stationary points of the Glass-Mackey system (GM). 46/57 + + + - - - -

47. Topological indices of the stationary points Topological indices of the points marked by “+” equal +1, their Conley indices are :.. Topological indices of the points marked by “-” equal -1, their Conley indices are :. 47/57 The cycles of the Glass-Mackey type systems do appear near the stationary points with negative indices.

48. 48/57 For the stationary point marked by green minus, we have proved analogs of the theorems 1 and 2 about existence of a (stable) cycle. Numerical experiments show existence of cycles near the 1-st, 3-d and the 7-th stationary points marked by blue minuses.

49. Cycles of the system (Λ).  =10, m=5, q=0. 49/57 Similar pictures were observed in the systems (GM), (L).

50. Cycles of the system (GM). α=4.3, γ=17.25. Projections onto the plane Z=0. 50/57 + + + - - - -

51. Higher-dimensional version of Glass-Mackey system. 51/57

52. Our current tasks are connected with: determination of conditions of regular behaviour of trajectories; studies of integral manifolds and non-uniqueness of the cycles, bifurcations of the cycles; their dependence on the variations of the parameters, and connections of these models with other models of the Gene Networks. 52/57

53. APPENDIX: D.P.Furman, T.A.Bukharina The Gene Network Determining Development of Drosophila Melanogaster Mechanoreceptors. Comp. Biol.Chemistry, 2009, v.33, pp. 231 – 234. The scheme of the nonlinear system (DM), see below. A1 CHN=charlatan.

54. More complicated model: A2

55. Dynamics of the above gene network model. x=[AS-C], y=[HAIRY], z=[SENS], u=[SCRT], w=[CHN] concentrations. D=[DA], G=[GRO], E=[EMC] parameters. Sigmoid functions describe the positive feedbacks on the previous slide. (DM) A3

56. Graph of and the stationary points of the system (DM). Stationary points “I” and “III” are stable, the point “II” is unstable. Ind(I)=ind(III)= -1; Ind(II) = + 1. I II III I A4

57. A5

58. Two trajectories of the system (DM) A6

59. Projections of trajectories of one 4-D system onto the plane OXYZ. B1

60. ; ; ; A trajectory convergent to the bifurcation cycle. B2

61. ; A trajectory and a bifurcation cycle. ; ; B3

62. Two trajectories and a bifurcation cycle between them. (in cooperation with K.V.Storozhuk)‏ B4

63. Thank you for your attention

64. Same cycles in (Λ). Similar pictures were observed in the systems (GM), (L). 45/57

65. Two trajectories of the system (DM) A6