November 4, 2015

Master en Bioinformatique Bioinfo rmatique et Mod´ elisation elisati on

Contents 1 Intro duction

4

1.1

Evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2 Deﬁnitions

6

2.1 2.1

Syst System em of diﬀ erential equations . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.3

Phase space, traje ajectories and nullclines . . . . . . . . . . . . . . . . . . . .

7

2.4

Cauchy-Lipschitz theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

3 Linear stability analysis

9

3.1

Case of a 1-variable system . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Case of a 2-variable system . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3

Case of a N-variable system . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Dynamical behaviors

9

15

4.1

Single steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2

Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3

Limit-cycle oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4

Chaos and other complex beh behaviors . . . . . . . . . . . . . . . . . . . . . . 16

5 Hysteresis, bifurcations, and stability diagram

17

5.1

Parameter space and stability diagram . . . . . . . . . . . . . . . . . . . . 17

5.2

Hysteresis and bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Applications

21

6.1 6.1

OneOne-v variabl ariablee sys system tem:: synthe synthesi siss and and deg degrad radati ation on of one compou compound nd . . . . . 21

6.2 6.2

OneOne-v variabl ariablee syst system em:: logist logistic ic syn synthesi thesiss and and line linear ar degra degradat datio ion n . . . . . . . 22

6.3 Tw Two-v o-vari ariabl ablee syst system: em: two two mutua mutually lly activ activated compound compoundss (bist (bistabi abilit lity) y) . . . 23 6.4 6.4

Two-v Tw o-vari ariabl ablee syst system em:: Bruss Brussel elato atorr (lim (limit it-c -cyc ycle le osci oscill llati ation ons) s) . . . . . . . . . 26

7 Appendix 7.1 7.1

31

Detai Detaile led d stud study y of of the the pertur perturba bati tion on evol evoluti ution on in a 2-v 2-vari ariabl ablee sys system tem . . . . 31

7.2 Exampl Examples es of trajectori trajectories es around around the stead steady y state state for for a 3-varia 3-variable ble system system . . 34 7.3 Stabili Stability ty of the steady steady state state for a N-varia N-variable ble syste system: m: Routh-H Routh-Hurw urwitz itz criteri criteriaa 35 2

Contents 1 Intro duction

4

1.1

Evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2 Deﬁnitions

6

2.1 2.1

Syst System em of diﬀ erential equations . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.3

Phase space, traje ajectories and nullclines . . . . . . . . . . . . . . . . . . . .

7

2.4

Cauchy-Lipschitz theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

3 Linear stability analysis

9

3.1

Case of a 1-variable system . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Case of a 2-variable system . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3

Case of a N-variable system . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Dynamical behaviors

9

15

4.1

Single steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2

Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3

Limit-cycle oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4

Chaos and other complex beh behaviors . . . . . . . . . . . . . . . . . . . . . . 16

5 Hysteresis, bifurcations, and stability diagram

17

5.1

Parameter space and stability diagram . . . . . . . . . . . . . . . . . . . . 17

5.2

Hysteresis and bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

6 Applications

21

6.1 6.1

OneOne-v variabl ariablee sys system tem:: synthe synthesi siss and and deg degrad radati ation on of one compou compound nd . . . . . 21

6.2 6.2

OneOne-v variabl ariablee syst system em:: logist logistic ic syn synthesi thesiss and and line linear ar degra degradat datio ion n . . . . . . . 22

6.3 Tw Two-v o-vari ariabl ablee syst system: em: two two mutua mutually lly activ activated compound compoundss (bist (bistabi abilit lity) y) . . . 23 6.4 6.4

Two-v Tw o-vari ariabl ablee syst system em:: Bruss Brussel elato atorr (lim (limit it-c -cyc ycle le osci oscill llati ation ons) s) . . . . . . . . . 26

7 Appendix 7.1 7.1

31

Detai Detaile led d stud study y of of the the pertur perturba bati tion on evol evoluti ution on in a 2-v 2-vari ariabl ablee sys system tem . . . . 31

7.2 Exampl Examples es of trajectori trajectories es around around the stead steady y state state for for a 3-varia 3-variable ble system system . . 34 7.3 Stabili Stability ty of the steady steady state state for a N-varia N-variable ble syste system: m: Routh-H Routh-Hurw urwitz itz criteri criteriaa 35 2

7.4

Cusp and othe ther co-dim dimension 2 bifurcat rcatiions . . . . . . . . . . . . . . . . . 37

7.5

Poincar´e-Bendixson theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7.6

Charac aracte terrizati ation of comp omplex beh behaviors . . . . . . . . . . . . . . . . . . . . 43

8 References

44

3

1 1.1

Intr In trod oduc ucti tio on Evolu Evolutio tion n equat equation ionss

When we modelize the dynamics of cellular phenomena, we are interested into the evolution of concentrations of the products and substrates as a function of time. This variation is described by a time derivative. Such an evolution equation typically involves synthesis, degradation and transport terms: dX = synthesis of X - degradation of X + transpo porrt of X dt

(1 )

Generally, the transport and degradation of X are functions of X but also of other variables of the system. When we modelize a biological system, we are usually faced with a system erential equations because of several coupled equations. These equations are called di ﬀ erential they involv involvee deriv derivative atives. s. Since Since the chemi chemical cal intera interacti ction on laws laws and enzyme enzyme kinetic kineticss are generally generally non-linear (e.g. Michealis Michealis-Men -Menten, ten, Hill), Hill), these equations equations are said non-linear diﬀ erential erential equations .

1.2

An exam exampl ple e

As an example, here is a model proposed to study the dynamics of calcium oscillations observed observed in cells. cells. The model is based on the Calcium-ind Calcium-induced-Ca uced-Calcium lcium release mechanism mechanism through the inositol-trisphosphate receptor (ﬁg. 1) and is used to describe signal-induced Calcium oscillations.

Figure 1: Scheme Scheme of the model proposed for the dynamics of calcium (Dupont G, Berridge Berridge MJ, Goldbeter A, 1991, Signal-induced Ca2+ oscillations: properties of a model based on Ca2+ -induced Ca2+ release, Cell Calcium 12, 12, 73-85). 4

Evolution equations for the cytosolic Calcium (variable Z ) and the intravesicular Calcium (variable Y ) are given by:

dZ Z n Y m Z p = v 0 + v1 β vm2 n + vm3 m + kf Y dt K 2 + Z n K r + Y m K pa + Z p dY Z n Y m Z p = v m2 n vm3 m kf Y dt K 2 + Z n K r + Z m K pa + Z p

− kZ

−

−

(2)

−

The analysis of these equations shows that the model can account for the occurrence of self-sustained oscillations of Calcium and allows to study the eﬀ ect of various parameters, such as the stimulus strength, on the period of these oscillations (ﬁg. 2).

Figure 2: Oscillations of cytosolic Calcium (Z ) and intravesicular Calcium (Y ) obtained by numerical simulation of equations (2), for two diﬀ erent levels of stimulation, measured by the parameter β : (A) β low, (B) β high (Dupont et al , 1991).

Overview of the course In this summary we will expose some notions of the theory of dynamical systems, which are necessary for the analysis of models involving several non-linear diﬀ erential equations (such as eq. (2)). The theory is illustrated by biological applications and some exercises are proposed.

5

2 2.1

Deﬁnitions System of diﬀ erential equations

The evolution of two interacting variables (concentrations of the species) can be described by a system of two diﬀ erential equations:

dX = f (X,Y, δ , µ) dt dY = g(X,Y, δ , µ) dt

(3)

The functions f and g usually depend on variables (concentrations X and Y of compounds, mRNA, proteins,...) and on some kinetics parameters (Michaelis-Menten constant, maximum rate, degree of cooperativity,..., denoted here by δ and µ). Because of their high non-linear degree (due to the presence of terms X 2 , XY , Michaelian, or Hill), ﬁnding an analytical solution of this system is rarely feasable. Consequently, other approaches have to be developped to solve such systems.

2.2

Steady state

First, we can see that if we have

dX =0 dt dY =0 dt

(4)

the concentration of X and Y will not evolve: X and Y are constant. The system (4) is an algebraic system of equations and we denote the solution (X S , Y S ) (where the subscript S stands for steady state ): f (X,Y, δ , µ) = 0 (5) g(X,Y, δ , µ) = 0

The steady state is reached when the synthesis (production) and the degradation (consumption) terms are compensated (see eq. (1)). Note that the equilibrium state is a particular steady state which is speciﬁc to equilibrium reactions. If the initial conditions do not correspond to the steady state (X 0 = X S , Y 0 = Y S ), we expect that the system will evolve until it reaches a steady state satisfying the condition 4 (or, equivalently, 5), as illustrated in ﬁgure 3.

̸

̸

But is it always the case? Will the system always evolve monotonously towards a steady state? Before treating this question, we will ﬁrst deﬁne the notions of phase space, trajectories and nullclines, that are very convenient for our discussion.

6

X

S

e l b a i r a V

X

0

Time

Figure 3: Evolution towards a steady steady state.

2.3

Phase space, trajectories and nullclines

Another way to represent the dynamical behaviors than the temporal evolution is the representation in the phase space. This is a multidimensional space whose coordinates are the variables (concentrations) of the system. Figure 4 exampliﬁes the representation in the phase space (X, Y ) of the evolution toward the steady state. The starting point, represented in white, is called the initial condition (X 0 , Y 0), i.e. the concentations X and Y at time t = 0. The black point is the steady state (X S , Y S ) and the curve is the trajectory of the system. This trajectory is built by calculating at each time step the values of X and Y .

(X ,Y ) S

S

Y e l b a i r a V

(X ,Y ) 0

0

Variable X

Figure 4: Phase space. When the system contains two variables, it is useful to resort to the nullclines representation. These curves are deﬁned by dX/dt = 0 on one hand and dY/dt = 0 on the other hand. Thus, the intersection of these curves correponds to the steady state, for which we have simultaneously dX/dt = 0 and dY/dt = 0. In the diﬀ erent regions delimited by these nullclines, it is possible to determine the direction of the evolution of the system by studying the sign of dX/dt and of dY/dt (ﬁg. 5).

7

dX/dt=0 Y e l b a i r a V

Y e l b a i r a V

dY/dt=0

Variable X

Variable X

Figure 5: Nullclines (left panel) and vector ﬁeld (right panel).

2.4

Cauchy-Lipschitz theorem

Under some continuity and regularity hypotheses (usually satisﬁed in biological ODE models), the theorem of Cauchy-Lipschitz states that from a given initial condition, a system of diﬀ erential equations has one and only one solution, i.e. there is a single trajectory that the system can follow. A direct consequence of this property is that a trajectory can not cross itself or another one. This also implies that, once the initial condition is speciﬁed, the behaviour of the system is completely determined. The behaviour is said deterministic.

Figure 6: Cauchy-Lipschitz theorem and determinism.

8

3

Linear stability analysis

To study the behavior of the system around the steady state, we resort to the linearized stability principle. We describe this method ﬁrst for a one-variable and subsequently for a two-variable system. This theory can be generalized to a N -variable system.

3.1

Case of a 1-variable system

We can write the evolution equation of X as: dX = f (X ) dt

(6)

where f (X ) is a function taking into account both the synthesis and the degradation kinetics of X . By deﬁnition, the steady state satisﬁes the condition: dX S = f (X S ) = 0 dt

(7)

Now, we apply a little perturbation x (with x << X S ) to this steady state: X = X S + x

(8)

Combining equations (6)-(8), we have: dX dX S dx dx = + = = f (X ) dt dt dt dt

(9)

Since the evolution function f (X ) is non-linear, we proceed to a linearization using a Taylor development around X S . The fact that x is small allows us to neglect the terms of order superior or equal to 2 (x2 is very small, x3 is very very small, etc): f (X ) = f (X S ) +

d2f dX 2

df dX

x +

S

x2 + ... =

S

df dX

x

(10)

S

Using (10) for the right-hand side, we see that the equation for the evolution of the perturbation x becomes: dx df = f (X ) = x (11) dt dX S

If we name λ the derivative 1

df dX S

, the solution of equation (11) can be written1

Let us recall the resolution of a di ﬀ erential equation of the ﬁrst order: dx dt

1

= λ x

dx = λ dt x ln (x) = λ t + C

˜eλt x = x

9

x = x˜eλt with:

df dX

λ =

and x˜ = x(0)

(12)

(13)

S

Equation (12) describes the (exponential) evolution with respect to time of the perturbation x: if λ > 0, x will increase and the system will leave its steady state, which is said unstable. On the opposite, if λ < 0, the perturbation will damp in such a way that the system will reach back its steady state; which is said stable.

Figure 7: Stability of the steady state.

3.2

Case of a 2-variable system

Consider now a system of two variables described by the following coupled equations:

dX = f (X, Y ) dt dY = g(X, Y ) dt

(14)

dX S = f (X S , Y S ) = 0 dt dY S = g(X S , Y S ) = 0 dt

(15)

The steady state (X S , Y S ) satisﬁes:

We apply a little perturbation (with x << X S and y << Y S ):

X = X S + x Y = Y S + y

(16)

Combining Eqs. (14)-(16), we have:

dX = dt dY = dt

dX S dx dx + = = f (X, Y ) dt dt dt dY S dy dy + = = g(X, Y ) dt dt dt 10

(17)

Since the system (14) is non-linear, we proceed to a Taylor development (we neglect the terms of order superior or equal to 2):

df f (X, Y ) = f (X S , Y S ) + dX dg g(X, Y ) = g(X S , Y S ) + dX

df x + dY S dg x + dY S

y + ... =

S

y + ... =

S

df dX dg dX

df dY dg dY

x +

S

x +

S

y S

(18)

y

S

We call the Jacobian matrix for the steady state (X S , Y S ) the matrix:

J = where a11 =

df dX

, a12 =

S

a11 a12 a21 a22

df dY

, a21 =

S

dg dX

, a22 =

S

(19)

dg dY

(20) S

Combining (17) and (18), we obtain the system of equations:

dx = a 11x + a12 y dt dy = a 21 x + a22 y dt

(21)

whose solution can be written as:

x = x˜eλt

(22)

y = y˜e λt

Replacing (22) into (21), we obtain :

− λ)˜x + a y˜ = 0 x ˜ + (a − λ)˜ y = 0

(a11 a21

12

(23)

22

This system has a non-trivial solution if: Det

a11 λ a12 a21 a22 λ

−

−

=0

(24)

i.e. λ2

11 + a22 )λ + (a11 a22

− (a

−a

12 a21 )

=0

(25)

We note that a11 + a22 = T = trace of the Jacobian matrix and a11 a22 determinant of the Jacobian matrix

−a

12 a21

=

∆

=

Equation (25) has in general two roots: 1 T ± λ± = 2

√

11

T 2

− 4∆

(26)

By replacing (26) in (22), we obtain the general form of the solution of the system (21):

x(t) = x˜1 eλ+ t + x˜2 eλ− t λ+ t

y(t) = y˜1 e

(27)

λ− t

+ y˜2e

The values of x˜1, x ˜2 , y˜1 and y˜2 can be found given the initial condition x(0) and y(0). The presence of the square root in (22) implies that the λ± (called the eigen values) can be complex. We discuss separately both cases: • If

the eigen values are diﬀ erent and real:

In this case, three situations are possible:

– If λ+ < 0 and λ < 0, the perturbation will decrease with respect to time and the system will come back to its steady state. In this case, the steady state is stable. Because the approach is exponential, this type of steady state is called a node. −

– If λ+ > 0 and λ > 0, the perturbation will increase with respect to time and the system will leave its steady state. In this case, the steady state is unstable. Here also, the steady state is a node. −

– If λ+ > 0 and λ < 0 (or the opposite), the perturbation will either increases or, in a ﬁrst time, decrease, but after a while, it will also increase and the system will leave its steady state. Here, the steady state is unstable and is called a saddle. −

• If

the eigen values are complex conjugates:

We can write these eigen values as λ± = p ± iq (with p = Re λ and q = Im λ) It can be shown that the solution (27) can be written as (see Appendix):

x(t) = Ae pt cos qt + Be pt sin qt y(t) = A e pt cos qt + B e pt sin qt ′

(28)

′

This is the parametric equation of a spiral. Indeed this system can be rewritten as:

x(t) = κ e pt (cos qt + φ) pt

y(t) = κ e (cos qt + φ ) ′

(29)

′

where κ, κ , φ, and φ are function of A, B, A and B . ′

′

′

′

The imaginary part (q ) is responsible of the oscillations around the steady state. In this case, the steady state is called a focus. If p < 0 the oscillations will be damped, while if p > 0 the oscillations will be ampliﬁed. Thus, if p < 0, the steady state is stable, while if p > 0, the steady state is unstable.

12

In summary... In summary, the stability describes the way in which the system reacts to a small perturbation that moves the system slightly away from the steady state. If the system is exactly at its steady state, it will not evolve; X and Y are constant and equal to X S and Y S respectively. If we apply a small perturbation of these values, two situations can arise: either the system comes back on its steady state, or it deﬁnitively leaves this state. In the ﬁrst case, the steady state is stable, in the second case, it is unstable. Stable steady state

X e l b a i r a V

Unstable steady state amplification

perturbation

X perturbation e l b a i r a V

damping

Time

Time

Figure 8: Stability of the steady state. As we have seen, according to the disposition of the eigen values λ , in the plane ( p, q ) (also called the Gauss plane (Re λ, Im λ)), we can obtain diﬀ erent approaches to the steady state. The table and the ﬁgure here below summarize the diﬀ erent possible situations.

3.3

Case of a N-variable system

More generally, for a system involving N variables, the characteristic equation is of degree N and must often be solved numerically. Except in some particular cases, such an equation has N distinct roots that can be real or complex. These values are the eigenvalues of the N x N Jacobian matrix. The general rule is that the steady state is stable if there is no eigen value with a positive real part. It is suﬃcient that one eigenvalue is positive for the steady state to be unstable.

Remark: Very often, because of the complexity of the model (several coupled and nonlinear diﬀ erential equations), the analytical solution (for the trajectories far from the steady state, the limit cycles or chaotic attractor) cannot be obtained. So, we resort to numerical simulations, using a computer.

13

!"#$% '()*$

+,-.!

!

+

/0$(12 30(0$

45(3$ 6)70 2

9:

8$

+;<

! =

<

+,-.! = <

30(?)$ %71$

+=<

! =

<

+,-.! = <

*%30(?)$ %71$

+=< 7* +;<

! ;

<

+,-.! = <

3(11)$ @*%30(?)$A

+;<

! =

<

+,-.! ; <

30(?)$ B7C*3

+=<

! =

<

+,-.! ; <

*%30(?)$ B7C*3

+><

! =

<

+,-.! ; <

E

C$%0$D @:(D#"%() 30(?")"02A

Figure 9: According to the nature of the eigen values, several approaches to the steady state can be observed.

Figure 10: Depending on the sign of the trace T and of the determinant ∆ of the Jacobian matrix and on the sign of T 2 4∆, several approaches to the steady state can be observed.

−

14

4

Dynamical behaviors

What happens if the steady state is unstable? How does the system behave? Where do the trajectories go in the phase plane? Several behaviors are possible (see more examples in section 6). We describe here some of them, which have important consequences in biology.

4.1

Single steady state

If there is a single stable steady state and if the system can not explode, the only possible behavior of the system is to evolve towards this steady state, whatever the initial condition (provided that it is not on an unstable steady state) (ﬁg. 11). X Y

Y , X

Y

Time

X

Figure 11: Evolution towards a steady state.

4.2

Bistability

The system can have several stable steady states. Then depending on the initial condition, the system will evolve towards one or the other steady state (see section 6.3). If two steady states coexist, this is a particular case of multistability, refer to as bistability (ﬁg. 12).

X starting from (X Y starting from (X

Y , X

X starting from (X Y starting from (X

01 01 02 02

,Y ) 01

,Y ) 01

,Y ) 02

,Y ) 02

Time

Y

(X ,Y ) 02

(X ,Y ) 01

02

01

X

Figure 12: Bistability. Bistability plays a crucial role in biology (cell diﬀ erentiation, genetic switch, memory eﬀ ect,...). 15

4.3

Limit-cycle oscillations

The system can exhibit a new type of behavior, such as sustained oscillations. In the phase space, these oscillations correspond to a close curve called a limit cycle (ﬁg. 13). X Y

Y , X

Y

Time

X

Figure 13: Limit cycle oscillations. Limit cycle oscillations plays a crucial role in biology. Indeed, many biological rhythms can be described by a limit cycle (Calcium oscillations, circadian rhythms, cell cycle, glycolytic oscillations, etc).

4.4

Chaos and other complex behaviors

The system can exhibit complex (but still deterministic) behaviors, such as chaos (ﬁg. 14). Such complex behaviors require a phase space with a dimension >2 (i.e. more than 2 variables) and multiple instability mechanisms (obtained via multiple feedback loops in regulatory systems).

Z

X

Y Time

X

Figure 14: Chaos. It is not clear weather chaos play a major role in biology, but several studies show that it might have a role in the heart beat and other constructive eﬀ ets in genetic systems. Other complex behaviors include quasi-periodic oscillations (evolution on a torus in the phase space), bursting (alternance between oscillatory and non-oscillatory phases), synchronization (important in coupled oscillators), or excitability.

16

5 5.1

Hysteresis, bifurcations, and stability diagram Parameter space and stability diagram

For a given system, the linear stability analysis allows us to determine the regions in the phase space where the steady state is stable or unstable. We can represent these domaines graphically in the parameter space (also called two-parameter bifurcation diagram ):

'(%)*+&%,'% -%&.%%, &.( +&"-/% +&%"01 +&"&%+ 2-*+&"-*/*&13 !

# % & % $ " # " !

+&"-/% /*$*& '1'/% 2(+'*//"&*(,+3

+&"-/% +&%"01 +&"&%

!"#"$%&%# !

Figure 15: Stability diagram.

5.2

Hysteresis and bifurcations

A bifurcation diagram shows how a steady state, and more generally the behaviour of the system changes, as a control parameter varies. The following ﬁgures illustrate diﬀ erent types of bifurcations that can be encountered. A transcritical bifurcation (ﬁg. 16) is characterised by the passage from a stable steady state to an unstable steady state. At the same point, an other steady state, switches from unstable to stable.

Figure 16: Transcritical bifurcation. As a function of a certain parameter λ, the curve of the steady state can display a S shape, delimited by two limit points. These points are called saddle node bifurcation points (ﬁg. 17). Between these two critical values, three steady states are present: two are stable and one is unstable. This is a typical case of bistability. 17

Figure 17: Saddle-node bifurcation. If the control parameter λ progressively increases, at the second critical value, the system switches from one stable steady state (X high) to the other stable steady state (X low). When λ progressively decreases, the it is at the ﬁrst critical point that the system switches from the stable steady state with X low to the stable steady state with X high. This behavior is referred to as hysteresis (ﬁg. 18, left). In some particular cases, sometimes due to some biological constraints, only one limit point is present and the transition is irreversible (ﬁg. 18, right).

Figure 18: Hysteresis with reversible (left) and irreversible (right) transitions. A pitchfork bifurcation is characterised by the passage from one stable steady state to an unstable steady state and the simultaneous appearance of two stable steady states (ﬁg. 19). This kind of bifurcation is encountered in particular systems, usually composed of two coupled subsystems. This bifurcation is not very robust: a slight asymetry between the subsystems leads to a symetry breaking: the pitchfork bifurcation is lost and a saddle node (limit point) appears (ﬁg. 20).

Figure 19: Pitchfork bifurcation. 18

Figure 20: Pitchfork destruction (symetry breaking). The most common bifurcation leading to limit cycle oscillations is the Hopf bifurcation (ﬁg. 21). In this case, we plot the maximum X max (and, optionally, the minimum X min ) that the variable X takes when running around the limit cycle. The diﬀ erence X max X min gives then the amplitude of the oscillations. Sometimes it is the period of the oscillations which is given as a function of the control parameter. The Hopf bifurcation is characterised by the passage of the real part of the complex conjugated eigenvalues from negative to positive values.

−

Figure 21: Hopf bifurcation. It could happen that the branch of the Hopf bifurcation is displayed in the domain where the steady state is stable. In this case this branch is unstable. Such a bifurcation is said sub-critical (ﬁg. 22, left), while in the other case, the bifurcation is said super-critical (ﬁg. 22, right).

Figure 22: Sub-critical (left) and super-critical (right) Hopf bifurcation. When a Hopf bifurcation is sub-critical, it often reaches a limit point where it turns back towards the value of the parameter for which the steady state is unstable. In a small region, delimited by the limit point and by the Hopf bifurcation point, a domain where the stable steady state and a stable limit cycle, separated by an unstable limit cycle is present. This situation is said hard excitation (ﬁg. 23). This term stands for the fact that a perturbation suﬃciently hard has to be given to the system to make it switch from the stable steady state to the stable limit cycle (or the opposite).

19

Figure 23: Hard excitation. Finally, note that chaos can be obtained after a cascade of period doubling (Feigenbaum sequence) (ﬁg. 24). The succession of limit cycles are sometimes refered to as limit cycle, period-2 limit cycle, period-4 limit cycle, period-8 limit cycle,...(ﬁg. 25).

Figure 24: Period doubling cascade leading to chaos.

Figure 25: Period doubling cascade leading to chaos, trajectory in the phase space.

20

6

Applications

6.1

One-variable system: synthesis and degradation of one compound

Consider one compound, X, synthetised with a rate constant ks and degraded at rate kd : ks

kd

−→ X −→ Evolution equation: The evolution equation of the concentration X is:

dX = F (X ) = k s dt

− k X d

(30)

(31)

Note that this is a linear system. Steady state:

dX =0 dt

⇒ X = k /k s

s

d

Linear stability analysis: The evolution of the perturbation x is given by:

dx = dt

dF dx

x = X S

−k x

d

(32)

The solution of this diﬀ erential equation is: x = e

−kd t

(33)

This means that the perturbation will be damped, in an exponential way, with respect to time. The steady state is therefore stable. Analytical solution: Note that this system is relatively simple and the solution of equation

(30) can be found analytically2 : −kd t

X = X 0e

+

ks 1 kd

−kd t

− e

(34)

where X 0 = X (0) We can check that lim x(t) =

t→∞

2

Recall that:

1

p+qX

dX = 1q ln( p + qX )

21

ks = X S kd

(35)

6.2

One-variable system: logistic synthesis and linear degradation

Consider now one compound, X, synthetised with a logistic synthesis rate and linearly degraded: dX X = F (X ) = k s X 1 (36) kd X dt N

− −

Steady states:

X s1 = 0 ks X s2 = N

−k

d

(37)

(exists if ks > kd )

ks

(38)

Linear stability analysis:

k − N X − k X k F = k − 2 X − k N F (X ) = k − k F (X ) = k − k s

F = k s X

2

d

s

′

s

d

′

s1

s

d

s2

d

s

′

(39)

(40)

(41) (42)

The following table summarizes the stability conditions for each steady state:

X s1 X s2

ks < kd ks > kd stable unstable stable

Graphical analysis:

Figure 26: Graphical analysis and bifurcation diagram (transcritical bifurcation). Analytical solution: Note that Eq. (36) can be rewritten as a standard logistic equation

and can be solved analytically.

22

6.3

Two-variable system: two mutually activated compounds (bistability)

Consider a system involving two compounds which activates each other according to a Hill-type kinetics:

Evolution equations:

dX Y n = v 1 n dt θ + Y n dY X n = v 2 n dt θ + X n

− k X 1

(43)

− k Y 2

To simplify, we assume that v 1 = v 2 = 1 and k1 = k 2 = 1. The equations become:

Steady state:

dX Y n = n dt θ + Y n dY X n = n dt θ + X n

− X

(44)

− Y

Let’s θ = 1/2. We can check that X S = Y S = 1/2 is a steady state. We also see that X S = Y S = 0 is also a steady state. Note that those two states may not be the only solutions... Representation in the phase plane:

We can represent the nullclines in the phase space. The nullclines are deﬁned by: dX =0 dt dY =0 dt

⇒ ⇒

Y n = X θn + Y n X n = Y θn + X n

(45)

(46)

Linear stability analysis:

The Jacobian matrix for the steady state (1/2, 1/2) is:

J =

˙ ∂ X ∂ X

˙ ∂ X ∂ Y

˙ ∂ Y ˙ ∂ Y ∂ X ∂ Y

23

(47)

Figure 27: Phase space and nullclines corresponding to the system (44) for n = 1 (left panel) and n = 4 (right panel). Recall that: X n ∂ n θ + X n ∂ X

(X n ) (θn + X n ) (X n ) (θn + X n ) nθn X n 1 = = (θn + X n)2 (θn + X n )2 ′

If θ = 1/2:

X n ∂ n θ + X n ∂ X

X n ∂ n θ + X n ∂ X

n 21 X n

−

=

1 2n

=

S

n 2

+ X

n 21

1 2n−1 2 1 + 21n 2n n

=

−

We can see that the trace T is always negative and that T 2 The sign of ∆ depends on n: if n < 2, we have

(48)

1

n

n 2

The trace T and the determinant ∆ of the Jacobian matrix are: T = and T 2 4∆ = 4 4(1 n2 /4) = n 2

−

−

−

At the steady state, X S = 1/2:

′

−

> 0

(49)

(50)

2

−2 and ∆ = 1 − n /4

− 4∆ is always positive.

⇒ the steady state (1/2, 1/2) is thus a stable node. • if n > 2, we have ∆ < 0 ⇒ the steady state (1/2, 1/2) is thus an (unstable) saddle •

∆

point.

24

Bifurcation diagram

In the previous analysis, we assumed v1 = v2 . We can now sketch the behaviour of the system when v2 is varied. Fig. 28 shows how the nullclines are aﬀ ected when v2 is decreased. In particular we see that bistability is lost when v1 is below a critical value, around v2 = 0.8. Fig. 29 shows the steady state of X as a function of v2 (obtained by numerical simulation). When v2 > 0.8, the system displays bistability: two stable nodes are separated by a (unstable) saddle point. When v2 < 0.8, the single (stable) steady state is the trivial state (0,0). The bifurcation point, at which the upper stable state and the middle saddle point coalesce and disappear is called “Saddle-Node” (SN) bifurcation. Through numerical simulation it is also possible to follow the SN bifurcation point in the (v1 , v2) plane (ﬁg. 30). Such stability diagram shows the region of bistability as a function of these two parameters. 1

v =1

v =0.85

1

2

v =0.7

1

2

2

0.8

0.8

0.8

Y0.6

Y0.6

Y0.6

0.4

0.4

0.4

0.2

0.2

0.2

0 0

0.5

0

1

0

0.5

X

0

1

0

0.5

X

1

X

Figure 28: Nullclines for 3 diﬀ erent values of v2 (v1 = 1, n = 4). 2

1.8

1.6

stable node

1.4

1.2 S

X

1

Saddle!Node 0.8

bifurcation

0.6

unstable saddle

0.4

0.2

stable node 0

0

0.2

0.4

0.6

0.8

1

v

1.2

1.4

1.6

1.8

2

2

Figure 29: Bifurcation diagram as a function of v2 (v1 = 1, n = 4). 2

1.8

1.6

Bistability

1.4

1.2 2 1

v

0.8

SN bifurcation

0.6

Single steady state

0.4

(X =0, Y =0) S

0.2

0

0

0.2

0.4

S

0.6

0.8

1

v1

1.2

1.4

1.6

1.8

2

Figure 30: Stability diagram as a function of v1 and v2 (n = 4). 25

6.4

Two-variable system: Brusselator (limit-cycle oscillations)

Consider the following system of 4 chemical reactions:

A

k1

−→ X →Y+C B+X− → 3X 2X + Y − →D X− k2

k3

k4

Evolution equations

The concentrations of A and B are supposed to be constant and noted a and b, respectively. The evolution equations of the concentrations X and Y are:

dX = k 1a k2 bX + k3 X 2Y dt dY = k 2 bX k3 X 2Y dt

− k X

−

4

(51)

−

To simplify, we will consider that k1 = k 2 = k 3 = k 4 = 1. The system then reduces to:

Steady state:

dX = a bX + X 2Y dt dY = bX X 2Y dt

− X

−

(52)

−

The steady state of the system is: X S = a

(53)

Y S = b/a

Linear stability analysis:

The Jacobian matrix is:

J =

˙ ∂ X ∂ X

˙ ∂ X ∂ Y

˙ ∂ Y ∂ X

˙ ∂ Y ∂ Y

− =

b + 2XY

− 1

b

− 2XY

X 2 2

−X

(54)

At the steady state, X = X S = a and Y = Y S = b/a, and thus: J =

b

2

−1 a −b −a

2

26

(55)

The trace T and the determinant ∆ = a 2 .

∆ of

the Jacobian matrix are: T = b

−1−a

2

and

The characteristic equation is: λ2

− T λ + ∆ = 0

(56)

We have to study the sign of T 2 T 2

− 4∆

− 4∆ = (b − 1 − a ) − 4a = (b − 1 − a − 2a)(b − 1 − a = (b − (a + 1) )(b − (a − 1) ) 2 2

2

2

2

2

+ 2a)

2

(57)

The determinant ∆ is always positive. The trace T is positive if b > a2 + 1 and negative otherwise. T 2 4∆ is negative if (a 1)2 < b < (a + 1) 2 and positive otherwise.

−

−

The following table summarizes the diﬀ erent possible behaviors as a function of the parameters a and b: b T ∆ 2

T 4∆ type of steady state

−

(a + + stable node

− 1) + 0

2

+ stable focus

a2 + 1 0 + -

+ + unstable focus

(a + 1) 2 + + 0

+ + + unstable node

When parameter b increases, the steady state turns from a stable node to a stable focus, then, it losts its stability and the steady state turns from an unstable focus to an unstable node. When the steady state is unstable, the system then evolves towards a limit cycle. These behaviours are shown in ﬁgure 31 (results obtained by numerical integration of the diﬀ erential equations): •

For a = 2 and b = 0.5: the system evolves towards a steady state (stable node)

•

For a = 2 and b = 4: the system evolves towards a steady state (stable focus)

•

For a = 2 and b = 6: the system leaves its steady state (unstable focus) to reach a limit cycle (sustained oscillations)

•

For a = 2 and b = 12: the system leaves its steady state (unstable node) to reach a limit cycle (sustained oscillations)

The stability diagram showing the stable and unstable regions as a function of a and b is given in ﬁgure 32. The solid curves, satisfying b = a 2 + 1 delimits the stability region. The dotted curve corresponding to b = (a + 1) 2 and b = (a 1)2 separate the node from the focus in the stable and unstable regions respectively.

−

Figure 33 is a bifurcation diagram: it shows how the steady state of X changes as a function of the parameter b (for a ﬁxed to 2). Figure 34 shows how the period varies in the oscillatory domain. These diagrams have been obtained numerically. 27

0.4 2.5

X Y

a=2, b=0.5

2

0.3

Y1.5 , X

Y

0.2

1

0.5 0

0.1 0

5

10

15

0

20

1

2

X

Time 2.5 X Y

a=2, b=4

3

2

Y2 , X

Y 1.5

1 0

1 0

5

10

15

20

0

2

4

X

Time 8

a=2, b=6

8

X Y

6

6

Y , X4

Y4

2

2

0

0 0

10

20

30

40

50

0

5

10

X

Time 25

a=2, b=12

25

X Y

20

20 15

Y15 , X

Y

10

10

5

5 0

0 0

10

20

30

40

50

0

20

40

X

Time

Figure 31: Diﬀ erent kinds of behaviors obtained for the Brusselator model with diﬀ erent parameter (a and b) values. Left panels: time evolution. Right panels: phase space. Note that the period at the bifurcation can be calculated. We know that the frequency is given by the imaginary part of the eigen values. At the bifurcation point, T = 0 and the eigen values are λ 1 = λ 2 = ∆ = a. The period is therefore equal to 2π /a. For a = 2 and b = 5, the period is thus equal to π 3.14.

√

≈

28

12

10

unstable node

unstable focus

stable focus

stable node

8

b r e t e 6 m a r a p 4

2

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

parameter a

Figure 32: Stability diagram for the Brusselator. 18

16

maximum of X (oscillations)

14

n i m12

X r o

10 x a m X 8

r o

Hopf bifurcation

S 6

X

4

stable steady state

unstable steady state

2

minimum of X 0 0

2

4

6

8

10

12

Parameter b

Figure 33: Bifurcation diagram for the Brusselator as a function of parameter b (with a=2). The steady state looses its stability and limit cycle oscillations emerge at b = 5 (Hopf bifurcation). 14

12

10

d 8 o i r e P 6

4

2

0 0

2

4

6

8

10

12

Parameter b

Figure 34: Period for the Brusselator as a function of parameter b (with a=2). At the bifurcation point (b = 5), the period is = π 3.14.

≈ 29

Nullclines and direction ﬁeld:

The nullclines, deﬁned by: dX =0 dt dY =0 dt

2

⇒ Y = (1 + b)/X − a/X

(X-nullcline)

⇒ Y = b/X

(Y-nullcline)

(58)

are shown in ﬁgure 35. They delimit regions in the phase space where the vector ﬁeld has a particular direction (ﬁgure 36). 9

X nullcline Y nullcline

8 7 6 5

Y 4

stable steady state (focus)

3 2 1 0

0

0.5

1

1.5

2

2.5

3

3.5

4

X

Figure 35: Brusselator: nullclines (for a=2 and b = 6).

9 8 7 6 5

Y 4 3 2 1 0

0

0.5

1

1.5

2

2.5

3

3.5

4

X

Figure 36: Brusselator: direction ﬁeld (for a=2 and b = 6).

30

7 7.1

Appendix Detailed study of the perturbation evolution in a 2-variable system

The evolution of the perturbation around the steady state in the case of a 2-variable system is given by equation (27):

x(t) = x˜1 eλ+ t + x˜2 eλ− t y(t) = y˜1eλ+ t + y˜2eλ− t

(59)

Note that λ + and λ are related to the trace and the determinant of the Jacobian matrix (see Fig. 9). −

As explained in the text, two cases should be distinguished: (a) The eigen values λ+ and λ are real, or (b) they are complex conjugated. We describe the evolution of the perturbation for these both cases in a bit more details. −

(a) λ+ and λ are real −

In order to sketch the trajectories of x and y in the plane (x, y), it is convenient to calculate the ratio dy/dx: dy y˜1 λ+ eλ+ t + y˜2λ eλ− t = (60) dx x˜1 λ+ eλ+ t + x˜2 λ eλ− t −

−

we can rearrange eq (60) as dy y˜1λ+ + y˜2 λ e(λ− = dx x ˜1 λ+ + x˜2 λ e(λ− −

−

−λ+

)t

−λ+

)t

(61)

If we assume that λ < λ + < 0, then the limit −

lim

t→∞

dy y˜1 = dx x ˜1

(62)

This is the equation of a straight line crossing the origin (0,0) (see line d 1 on ﬁg. 37) that describes the asymptotic approach of the evolution of the perturbation. We can also calculate the limit lim

t→−∞

dy y˜2 = dx x˜2

(63)

This is the equation of a straight line crossing the origin (see line d2 on ﬁg. 37) that describes the slope of the trajectories of the evolution of the perturbation ”initially”. > λ+ > 0, the trajectories will leave the steady state, initially being dy y˜1 asymptotic to the line d1 := = dx x˜1 When λ and λ + have opposite signs, the trajectory approaches the point (0,0) along the line d1 but then goes away following the line d 1 (see ﬁg. 38). Similarly, if λ

−

−

31

Figure 37: Evolution of the perturbation in the case λ+ < 0 and λ < 0 (stable node). −

Figure 38: Evolution of the perturbation in the case λ+ > 0 and λ < 0 (saddle). −

(b) λ+ and λ are complex conjugates −

When the eigen values λ + and λ are complex conjugates, let’s say λ + = p ± iq , equations 59 can be rearranged as −

−

x(t) = Ae pt cos qt + Be pt sin qt y(t) = A e pt cos qt + B e pt sin qt ′

′

(64)

which can then be rewritten as

x(t) = Ke pt cos(qt + φ) y(t) = K e pt cos(qt + φ ) ′

′

(65)

Indeed, recalling that cos(a + b) = cos a cos b

− sin a sin b

(66)

we can show the equivalence between equations (64) and (65): x = Ke pt (cos qt cos φ sin qt sin φ) = K cos φ.e pt . cos qt K sin φ.e pt . sin qt

− −

(67) (68)

Deﬁning A = K cos φ and B = K sin φ, we get A B 2 2 A + B = K 2 tan φ =

⇒ ⇒ 32

A B A2 + B 2

φ = arctan

K =

√

(69) (70)

Similarly, for y , we can show that: C D A 2 + B 2

φ′ = arctan

√ K = ′

′

(71)

′

Equation (65) is the equation of a spiral in the plane (x, y). If p < 0, the spiral converges to the origin (0,0) and the trajectory corresponds to damped oscillations of x and y with the time (ﬁg. 40). If p > 0, the spiral diverge from the origin (0,0) and the trajectory corresponds to ampliﬁed oscillations of x and y (not shown). If λ+ and λ are purely imaginary ( p = 0), then the oscillations are not damped, neither ampliﬁed. Their amplitude remains constant and depends on the perturbation applied to the system. In this particular case, Eq. (64) becomes −

which can be written as:

x(t) = A cos(qt) + B sin(qt) y(t) = A cos(qt) + B sin(qt) ′

′

x(t) = K cos(qt + φ) y(t) = K cos(qt + φ ) ′

′

(72)

(73)

Equation (73) is the equation of a close curve in the pane (x, y) and corresponds to self-sustained oscillations of x and y. Note that its amplitude and period, respectively governed by K and φ are depending on A, B, A and B , i.e. on the initial conditions of the perturbation x(0) and y(0). ′

′

Figure 39: Evolution of the perturbation when λ+ and λ are complex conjugates (λ± = p ± iq ) with real part p > 0 (stable focus). −

33

Figure 40: Evolution of the perturbation when λ+ and λ are purely imaginary (center, marginal stability). −

7.2

Examples of trajectories around the steady state for a 3variable system

A 3-variable system has three eigen values. Again, the type of behavior can be characterized as a function of the position of the these eigen values in the Re/Im plane. Five non-degenerated cases can be distinguished: (1) the three eigen values are real and negative (stable steady state), (2) the three eigen values are real, Two of them are negative (unstable steady state), (3-4) two eigen values are complex conjugates with a negative real part and the third one real is negative (stable steady state) (two cases can be distinguished depending on the relative value of the real part of the complex eigen values and of the real one, and (5) two eigen values are complex conjugates with a negative real part and the third one real is positive (unstable steady state). The cases (3) and (5) are illustraed on ﬁg. 41.

Figure 41: Evolution of the perturbation for a 3-variable system (a) when two eigen values are complex conjugates with a negative real part and the third one is real and negative (stable steady state) (left panel) and (b) when two eigen values are complex conjugates with a negative real part and the third one is real and positive (unstable steady state) (right panel).

34

7.3

Stability of the steady state for a N-variable system: RouthHurwitz criteria

Let’s consider the case of a N variable system: dX 1 = f 1 (X 1 , X 2 , ...X N ) dt dX 2 = f 2 (X 1 , X 2 , ...X N ) dt ... dX N = f N (X 1 , X 2 , ...X N ) dt

(74)

This can be rewritten is a more compact form: dX i = f i (X 1, X 2, ...X N ) dt

(75)

with i = 1, ...N or in the vectorial form:

dX = f (X) dt

(76)

By deﬁnition, the steady state X S must satisfy:

dX dt

= f (XS) = 0

(77)

X =X S

Linearization around the steady state leads us to consider the jacobian:

J = i.e.

J =

This a N x N matrix.

∂ f 1 ∂ X1 ∂ f 2 ∂ X1

... ∂ f N ∂ X1

∂ F ∂ X

∂ f 1 ∂ X 2 ∂ f 2 ∂ X 2 ∂ f N ∂ X 2

(78) X =X S

... ... ...

∂ f 1 ∂ X N ∂ f 2 ∂ X N

∂ fN ∂ X N

(79)

X =X S

The eigen values are obtained by solving the following equation: det(J

− λI) = 0

(80)

The characteristic equation is a polynom of degree N : λN + a1 λN −1 + a2 λN −2 + ... + aN = 0

35

(81)

If N = 2, we obtain the quadratic equation discussed in details for the case of a 2-variable system (remember that in this case, a1 = T race(J) and a 2 = Det(J)). When N > 2, it is diﬃcult (usually impossible) to ﬁnd all the roots of equations (81). We can nevertheless obtain informations about their signs (and hence about the stability of the steady state). Suppose λ1 , λ2,...λN all (known) eigenvalues of the linearized system:

−

dX = JX dt

(82)

The solution of this equation can be written: X i = X iS + x˜1 eλ1 t + x˜2 eλ2 t + ... + x˜N eλ

N

t

(83)

If one eigen value λ has a real part > 0, the peturbation will ultimately increase and the steady state is thus unstable. How to determine if all the eigen value have a negative real part? This can be done by checking some conditions, known as the Routh-Hurwitz criteria. We can deﬁne N matrices as follows:

H j =

H1 =

1 a2 a4

0 a1 a3

a1 a3 a5 ... a2 j

1

−

a2 j

a1

2

−

a2 j

, H2 =

... ... ...

0 1 a2 3

−

a2 j

a1 1 a3 a2

4

−

0 0 0

... a j

where the (l, m) element in the matrix H j is a2l 1 0

−m

, H3 =

a1 1 0 a3 a2 a1 a5 a4 a3

, ..., HN =

,...

a1 1 0 ... 0 a3 a2 a1 ... 0 ... 0 0 0 ... aN

(84)

for 0 < 2l m < k for 2l = m for 2l < m or 2l > k + m

−

Then, all eigen values have negative real parts (steady steady stable) if and only if the determinants of all Hurwitz marices are positive: det(H j ) > 0

∀ j

(85)

Robert May (1973) summarized the stability criteria in the cases N = 2, N = 3, and N = 4 as follows: N N = 2 N = 3 N = 4

Conditions for stability a1 > 0, a 2 > 0 a1 > 0, a3 > 0, a1a2 > a3 a1 > 0, a1a2 > a3, a4 > 0, a1 a2 a3 > a23 + a21 a4

An application of the Routh-Hurwitz criteria to a 3-variable system is given in EdelsteinKeshet, pp.234-235. 36

7.4

Cusp and other co-dimension 2 bifurcations

The cusp bifurcation (also called cusp catastrophe) is a bifurcation, observed in 2-parameter space, at which two branches of saddle-node bifurcations vanish. The cusp bifurcation is associated to the presence of bistability and to hysteresis phenomenon. Because the cusp bifurcation can only be seen in a 2-parameter plane, it is called a codimension 2 bifurcation. Another example of co-dimension 2 bifurcation is the degenerate Hopf bifurcation at which a super-critical Hopf bifurcation becomes sub-critical. In a way these co-dimension 2 bifurcations show how a bifurcation changes when a second parameter is changed. Further examples of co-dimension 2 bifurcations are described in [Guckenheimer and P. Holmes (1983) Nonlinear Oscillations, Dynamical systems and Bifurcations of Vector Fields. Springer] and brieﬂy reviewed in the appendix of the paper by Tyson et al, Biophys J (2006).

Figure 42: Cusp bifurcation

37

7.5

Poincar´ e-Bendixson theorem

It is diﬃcult (usually impossible) to proof the existence of a limit cycle for any N -variable non-linear dynamical system. In the simple case of a 2-variable system, the Poincar´ eBendixson theorem, based on the topology of the phase space, provides however a way to determine if a limit cycle is present. The Bendixson criteria can sometimes also be applied to rule out the existence of a limit cycle.

Poincar´ e-Bendixson theorem If, for t t0 , a trajectory is bounded and does not approach any singular point (steady state), then the trajectory is either a closed periodic orbit or approaches a closed periodic orbit, i.e. a limit cycle, for t (Fig. (43)).

≥

→∞

Figure 43: Illustration of the Poincar´e-Bendixson theorem We give below an application of this theorem. Another application to a simple model (the Schnakenberg model) can be found in Edelstein-Keshet (pp. 357-360).

Application of the Poincar´ e-Bendixson theorem The theorem of Poincar´ e-Bendixon is here illustrated on the Selkov model for glycolysis (cf Strogatz book, p. 203): dx = x = ˙ x + ay + x2 y dt dy = y = b ˙ ay x2 y dt

−

− −

(86)

The procedure is in two steps. First we need to ﬁnd a trapping region for the system. Then we need to ﬁnd conditions on parameters a and b under which the steady state is unstable. Finding a trapping region requires an analysis in the phase space. First we ﬁnd the nullclines: x x = ˙ 0 y = (87) a + x2 b y = ˙ 0 y = (88) a + x2

⇔ ⇔

These nullclines are sketched in Fig. 44. They delimit various regions of the phase space distinguished by the direction of the vectors. 38

Figure 44: Selkov model: nullclines Now consider the region bounded by the dashed line shown in Fig 45. Let’s demonstrate that this is a trapping region. To this end we need to show that all the vectors on the boundary point into the box. On the vertical and horizontal axes there is no problem: the system is such that starting from positive values of x and y, these variables can not become negative (i.e. when x = 0, x˙ 0 and when y = 0, y˙ 0). The tricky part of the construction is the diagonal line of slope -1 extending from the point (b,b/a) to the nullcline (1).

≥

≥

Figure 45: Selkov model: trapping region

39

The get the right intuition, consider x and ˙ ˙ the limit of very large x. Then y in 2 x = x ˙ y 2 y = x ˙ y

so,

(89) (90)

x˙ =1 y˙

(91)

along trajectories. Hence the vector ﬁeld is roughly parallel to the diagonal line. This is conﬁrmed by a more rigorous calculation, which consists of comparing the size of x and ˙ y: ˙

− x˙ − (−y) ˙ = −x + ay + x y + (b − ax − x y) = b − x 2

2

(92)

Said otherwise,

−y˙ > x˙ if x > b

(93)

This inequality implies that the vector ﬁeld point inward on the diagonal line, as illustrated in Fig. 45, because dy/dx is more negative than -1, and therefore the vectors are steeper than the diagonal line. Thus the region is the trapping region.

Figure 46: Selkov model: trapping region + unstable steady state Now we have to calculate the steady state and to determine the condition in which it is unstable (Fig. 46). The steady state in found by solving dx = x + ay + x2 y = 0 dt dy = b ay x2 y = 0 dt

−

− − 40

(94)

We then ﬁnd the steady state (i.e. the intersection between the two nullclines): xs = b b ys = a + b2

(95)

(96)

The Jacobian maxtrix is J =

−

1 + 2xy 2xy

−

2

a + x2 (a + x2 )

−

The determinant is ∆ = a + b > 0 and the trace is T =

−

b4 + (2a

Hence the ﬁxed point is unstable if T > 0, i.e. when b2 = 21 (1 deﬁnes a curve in the parameter space (a, b), as shown in Fig. 47.

−

(97) 2

− 1)b

+ (a + a2 )

a + b2 2a ± 1

√ − 8a).

This

Figure 47: Selkov model: stability diagram Thus, for parameter values satisfying T > 0, we have guaranteed that the system has a close orbit (a limit cycle in this case). This is conﬁrmed by numerical simulation of the system for the case a = 0.08, b = 06 (see Fig. 48).

Figure 48: Selkov model: numerical analysis 41

e-Bendixon theorem is one of the central results in nonlinear dyRemark: The Poincar´ namics. It says that the dynamical possibilities in the phase space are very limited: if a trajectory is conﬁned to a close, bounded region that contains no ﬁxed points then the trajectory must eventually approach a close orbit. Nothing more complicated is possible. This result depends crucially on the 2-dimensionality of the plane. In higher dimensions, The Poincar´e-Bendixon theorem no longer applies. Something new can happen: trajectories may wander around forever in a bounded region without setting down to a ﬁxed point or a close orbit. They converge to more complex attractors (e.g. chaos or quasi-periodic torus).

Bendixson criteria: dx dy ∂ F ∂ G Let’s call = F and = G. If + = 0 for all x and y and does not change sign dt dt ∂ x ∂ y in D, there is no close orbit (no limit cycle)

̸

The proof - which requires advanced mathematics - can be found in Edelstein-Keshet p. 379-380.

42

7.6

Characterization of complex behaviors

It exists several methods to characterize complex behaviors, in particular to diﬀ erentiate e maps , Fourier between chaos and quasi-periodicity. These methods include Poincar´ transforms, and Lyapunov exponents . We describe here the principle of the Poincar´e map. The idea is to “cut” the attractor with a plane and report the intersection between the trajectory and the plane, i.e. the points where the trajectory cross the plane (selecting only one direction). A simple limit cycle would thus give a poincar´ e section with a single point, a chaotic attractor would give a poincar´e section with a open line of points (ﬁg. 49), and a quasi-periodic attractor would give a poincar´ e section with a close line of points (ﬁg. 50). Note that in high dimension, the situation is often more complex...

Poincare map 20

18

16

14

12

Z10 8

6

4

2

0 3

4

5

6

7

8

9

10

11

12

X

Figure 49: Poincar´e map for chaos.

Poincare map 0.17

0.16

0.15

0.14

2 x 0.13

0.12

0.11

0.1 0.13

0.132 0.134 0.136

0.138

0.14

y1

Figure 50: Poincar´e map for quasi-periodicity.

43

0.142

0.144 0.146 0.148

0.15

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