On Nonuniqueness of Cycles in Dissipative Dynamical

Systems of Chemical Kinetics  

A.A.Akinshin, V.P.Golubyatnikov

Altai State Technical University, BarnaulSobolev Institute of Mathematics, Novosibirsk

8 June 2012, Novosibirsk.“Solitons , Collapses and Turbulence”

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.

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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.PiskunovMoscow University Herald, 1937.

1. Some simple gene networks models

We study odd-dimensional dynamical systems

...;)( 22122 xmxftd

xd;)( 11121

1 xmxftd

xdk

are smooth and monotonically decreasing. This corresponds to the negative feedbacks

in the gene networks.

.)( 121221212

kkkk

k xmxftd

xd

0)( uf iFunctions

Lemma 1. Each system of this type has exactly one stationary point in the positive octant:

0S

)]0(,0...[)]0(,0[)]0(,0[ 121

1221

211

1

kk fmfmfmQis an invariant domain of the system. [(2k+1). +(2k)]

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)..).((...(( 121

221

2121

12111 xfmfmfmfxm kkkk

Lemma 2.

.0im

We describe below the case (2k+1) ...;)( 212

2 xxftd

xd;)( 1121

1 xxftd

xdk

.)( 1221212

kkk

k xxftd

xd

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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.

Non-convex 3D invariant domain in composed by six triangle prisms

Q

}001{ }011{

}110{

}100{

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}001{}101{}100{}110{}010{}011{}001{

0

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).

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}01...0010{}01...1010{ ...}01...010010{}01...01101{ }100...1010{}0110...1010{

0S

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.

;1

65

xzdt

dx

;

1

37

yxdt

dy

;7 5 ze

dt

dz y

A trajectory and

a limit cycle.

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!

!

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.

,...1

183

1i

i

i xxdt

dx

;5,4,3,2,1i

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).2,2,2,2,2(0 S

10

.0ReRe 54

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

11Projections 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.01

The blue spot shows the position of projection of the stationary point.

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

1

0S

.0)1( 1212 kk 12 k

1/ ii xf0S

.Re...ReRe 12,25,43,21 kk

0S

.0Re 3,21

12

symmetric dynamical system

Eigenvalues of one 9-D13/57

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.

,...1

1306

1i

i

i xxdt

dx

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761 ,, 981 ,,

Similar results can be obtained for the systems of the types

,

etc.

)()( 1311 xgxf

dt

dx ),()( 212

2 xgxfdt

dx

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),,( 1 iiii xxF

dt

dx ,0

i

i

x

F,0

1

i

i

x

F

M.Hirsch (1987).

;)(XXAdt

dX

;

10

01

01

A .

)(

)(

)(

)(

3

2

1

yfy

xfx

zfz

X

...)(Re;1)( 12,21 AA jj

(VM)=(2k+1)

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The eigenvalues of A can be expressed explicitly:

η >0.

2. Stability questions

The transfer matrix

Let

,))1((:)1( 1 AEii

.|)1(|sup)(

i

)(X be the Jacobi matrix of , )(X

and let |)4|(supmax|| iXi

X f

i = 1,2,… be its norm.

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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 if1))((|| X

then the system (VM) has a stable cycle (1987). Actually, he notes that this is not a sharp estimate!!

18

12sin

12

2sin|)(| 1

kkxf ii

).sin2sin1()sin2sin1( if

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.

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Let (2k+1) = p·qThen 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.

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symmetric dynamical system

(what happens in 5-D?)

Eigenvalues of one 9-D21/57

Two cycles and some chaos in 5-D (!!) symmetric system

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“two” cycles in 5-D symmetric system

,...12050

101

ii

i xxdt

dx

Projections of two different cycles of 9-D symmetric system onto 3-D planeThe second cycle is not contained in Q. Its potential level equals 3.

The stationary point is at the top of the picture.

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.,, 761

Projections of two different cycles of 11-D symmetric system onto two different 3-D planes

left; right. 981 ,, 11101 ,,

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Trajectory of 9-D system attracted by the first cycle

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Chaotic oscillations

Projections of 3 cycles of 15-dimensional system onto the plane ., 1110

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3 cycles in symmetric 15-D system26/57

1

P. level = 3

Potential level = 5

Trajectory of 15-D system attracted by the first cycle

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chaotic oscillationsfirst cycle

Similar phenomenon30/57

first cycle

chaotic oscillations →↓

4 cycles in 15-D symmetric system31/57

1311 )( xxf

dt

dx 212

2 )( xxfdt

dx 323

3 )( xxdt

dx; ;

,),0(),0[:)(),( 1231 xfxf Smooth monotonically decreasing 0)( uf i

for .u

4. Model of 3-D gene network regulated by a simple combination of negative and positive feedbacks

system (ffΛ):

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mx

xax

2

223 1)(

or more general unimodal function.

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

)(3 My )(3 yz )(yz )).(( 12 zffy

)),0(())0(( 232 ff

,))0(( 12 Myff )()(,))0(( 312 yyyff M .0 Myy

),( 0,000 zyxS

AAA zyx ,, ),(30 Ayz

)).((),(, 10 AAAAMA yfxyzyyy

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Linearization of system (1) at this point is described by the matrix with one negative eigenvalue. Its other eigenvalues are complex. Consider the case

.0ReRe 32

32 ,

(+)

Theorem 3. If the condition (+) is satisfied then the system (ffΛ): has at least one periodic trajectory.

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),( 0,000 zyxS

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

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.)](,[)]0(,[],0[ 32 MAAA yzfyxQ

),(30 Ayz

)).((),(, 10 AAAAMA yfxyzyyy

Trajectories of the system (ffΛ)

right:

left:

.1

17)(,10)(,

1

10)(

33135.0

231

2

y

yyexf

zzf x

.1

17)(,

1

10)()(

33321 y

yy

zwfwf

36/57

;)( 1311 xxf

dt

dx ;)( 212

2 xxdt

dx 323

3 )( xxdt

dx

;

;

,),0(),0[:)( 1 ii xf 0)( uf i for .u

5. More complicated gene networks models regulated by combinations of positive and negative feedbacks

system (fΛΛ):

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jm

jj

w

waw

1)( or other unimodal functions.

Stationary points of the system (fΛΛ).

I

II

IIIIV

V

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;1

9)(

43

31 xxf

;

1

10)(

41

112 x

xx

.

1

10)(

52

223 x

xx

));(( 1233 xx

);( 311 xfx

1x

3x

Stationary points I and II of the system (fΛΛ):

III

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));(( 1233 xx

);( 311 xfx

3x

1x

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.

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

;1

9)(

43

31 xxf

;

1

10)(

41

112 x

xx

.

1

10)(

32

223 x

xx

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

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6. Glass-Mackey-type systems

Ricker function.

;)( 1311 xx

dt

dx ;)( 212

2 xxdt

dx .)( 323

3 xxdt

dx

w

ww

1

)( Glass-Mackey function. (GM)

(w)=rw(–w) Logistic function. (L)

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)exp()( bwaww m

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.

,

./)2(;

]/)2(,/[;2

,]/,0[;

)(,

wmqq

mqmwwm

mwwm

wqm

(Λ)

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Positions of the stationary points of the Glass-Mackey system (GM).

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+

+

+

-

-

-

-

Topological indices of the stationary points

Topological indices of the points marked by “+”

equal +1, their Conley indices are : .1)( Sh

. Topological indices of the points marked by “-”

equal -1, their Conley indices are : .2)( Sh

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The cycles of the Glass-Mackey type systems do appear near the stationary points

with negative indices.

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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.

Cycles of the system (Λ).

=10, m=5, q=0.

ZC

)(),( VIICVC Z

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Similar pictures were observed in the systems (GM), (L).

C

Cycles of the system (GM). α=4.3, γ=17.25. Projections onto the plane Z=0.

)(,)(),(),( VIICIIICICVC ZYX50/57

+

+

+

-

-

-

-

Higher-dimensional version of Glass-Mackey system.

51/57

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.

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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.

More complicated model: A2

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.

;)1)(1(

)()()(),,,( 531

1 xxEyG

wzxDxwzyxF

td

xd

;1

)( 22 y

u

CyuF

td

yd

,)(3 zxDStd

zd

,)(4 uxDStd

ud

.)(5 wxDStd

wd

;5,4,3),( ixDSi5,3,1, jj

Sigmoid functions

describe the positive feedbacks on the previous slide.

(DM)

A3

Graph ofand 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.

))(),()),((,(:)( 53421 xDSxDSxDSFxFxRf

III

IIII

A4

A5

Two trajectories of the system (DM)

A6

Projections of trajectories of one 4-D systemonto the plane OXYZ.

B1

; ; ;

u1 u2 u3( ) rp1 rp2 rp3( )

xzdt

dx

3

1

1

y

xdt

dy

31

9)1( 3

3 yzdt

dz

8.81 88.23

A trajectory convergent to the bifurcation cycle.

B2

u1 u2 u3( ) rp1 rp2 rp3( )

;)1( 31 zx

dt

dx y

xdt

dy

31

9)1( 3

3 yzdt

dz

1 36.15, 2.4

A trajectory and a bifurcation cycle.

; ;

B3

Two trajectories and a bifurcation cycle between them.

(in cooperation with K.V.Storozhuk)

B4

Thank you for your attention

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

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Two trajectories of the system (DM) A6