Mathematical Modeling I

Transcription

1 Mathematical Modeling I Dr. Zachariah Sinkala Department of Mathematical Sciences Middle Tennessee State University Murfreesboro Tennessee 37132, USA November 5, 2011

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3 1d systems To understand more complex multi-dimensional systems of ordinary differential equations it is valuable to first study one dimensional systems. They are also valuable as a simplest case to study bifurcations.

4 General 1d systems The general 1d system can be expressed dy(t) dt = F(y(t),t) If the right hand side of the equation does not depend on time, the system is called autonomous otherwise non-autonomous. Each solution to the equation y(t) departing from some initial condition y(t 0 ) = y0 is called a trajectory. The mapping φ(y 0,t) which takes some initial condition and a time value and returns the value of solution departing from y 0 at time t is called a flow.

5 Linear systems In the general case the solutions of 1d systems might not exist for every value of t (trajectories might diverge). It is also difficult to find solutions analytically. There are clever methods of solving differential equations, but they only work for certain special cases. Nevertheless there are a couple of important special cases which can be solved analytically. These are linear equations of the form dy(t) = ay(t)+b dt It is easy to verify (via differentiation) that the solutions are: y(t) = b ( ) b a + a +y 0 e a(t t 0)

6 In particular equation dy(t) dt has a simple analytic solution = ay(t) y(t) = y 0 e a(t t 0) which can be simulated efficiently y(t +1) = αy(t) where α = e aτ is pre-computed (τ is the time step)

7 Homogeneous linear equation of the form d n y(t) dt n +a 1 d n 1 y(t) dt n a n 1 dy(t) dt The associated characteristic polynomial λ n +a 1 λ n a n 1 λ+a n = 0. +a n y(t) = 0 Now assume λ i is a root of the characteristic polynomial.

8 In such case e λ ix is a solution of the differential equation. Since the equation (and the differential operator) are linear, then functions e λ ix for all roots λ i of characteristic equation form a linear basis and any solution is a linear combination. In the general case if the root λ i has multiplicity m then for any k {1,2,...,m 1}x k e λ ix is a solution, and any linear combination of such solutions is also a solution. If the coefficients of the equation are real, then roots of the characteristic equation are complex conjugates.

9 Phase line analysis The basic factor determining the behavior of trajectories is the right hand side function F(y(t)). Lets assume for a moment that it does only dependent on y. Since y R the domain of F is R. We will call the real line a this context a phase line. The phase line contains all the possible states of the system but in general a phase space can be multidimensional. Plotting F(y) for y R gives a whole lot of information.

10 Equilibria If for some y c, F(y c ) = 0, we have dy dt = F(y c) = 0 We call such a y c an equilibrium. If F is positive (negative) for some point, the trajectory at that point will be increasing (decreasing respectively) We will assume that F is a continuous differentiable function. How can F look like near y c?

11 There are five qualitatively different cases in which F(yc ) = 0 F (y c ) > 0 F (y c ) < 0 F (y c ) = 0 and F (y c ) = 0 F (y c ) = 0 and F (y c ) > 0 F (y c ) = 0 and F (y c ) < 0 The first two we will call hyperbolic while the latter three non-hyperbolic. The last two we will also call degenerate. There is also possibility that the second derivative is not defined at all. We will skip these pathological cases for now.

12 Simple 1d systems can exhibit interesting phenomena A system can have a multiple stable equilibria stability. In particular two stable equilibria lead to bistability When a system is (multi) bistable, it can be pushed by some input stimulus and rest at one of the equilibria. Then it could be pushed again into different stable equilibrium. In a sense, the system exhibits memory. This phenomenon is called hysteresis.

13 Phase portraits Phase portraits are a convenient way to qualitatively describe properties of the dynamical system One begins the construction of a phase portrait with the plot of right hand side function F Then one finds its roots, that is the equilibrium points. By looking at the plot one can easily verify the type of equilibrium point Lastly one can determine attraction domains and convergence rates

14 Topological equivalence Two dynamical systems with essentially identical phase portraits behave in similar ways. The rate of convergence of certain trajectories may differer, nevertheless there is the same structure of equilibria and attraction domains. The keyword here is essentially identical. What does this mean? We will assume two phase portrait essentially identical when there exist continuous, invertible mapping from one to the other phase line that preserves all equilibria and attraction domains Two systems with topologically equivalent phase portraits we will call equivalent

15 Local equivalence (1) The total equivalence of the whole phase portrait is a rather strong notion (2) Many real world systems are not totally equivalent, but locally their phase portraits look alike. In such case it is useful to introduce local equivalence It is particularly useful to associate a dynamical system in the vicinity of hyperbolic equilibrium with its linear approximation. Under certain assumptions such association can be complete, that is linear systems completely determines the systems behavior sufficiently close to the equilibrium

16 Hartman-Grobman Theorem Hartman-Grobman Theorem states that near a hyperbolic equilibrium the system is equivalent to its linearization The equivalence here is somewhat stronger: not only trajectories follow the same directions, but sufficiently near the equilibrium trajectories of original and linear systems are indistinguishable. Formally the theorem states that there is a homeomorphism working from some open subset E to some open subset L (both containing the equilibrium) which for some time segment [0, T] continuously transforms each trajectory f(x e,t),x e E,t [0,T] into f l (x l,t),x l L,t [0,T]. Therefore unless the equilibrium is non-hyperbolic, it is sufficient to study linear system. We will get back to Hartman-Grobman Theorem with two dimensional systems.

17 Bifurcations Now lets assume, that the right hand side function F(y(t),β) depends on some parameter, β By changing the parameter, the shape of the function can change, and consequently the phase portrait might undergo a qualitative change Such an event, which transforms one phase portrait into topologically non-equivalent other is called a bifurcation. Bifurcations are in the focus of our interest, since as we will see, they are responsible for various dynamical behaviors A prominent example of a bifurcation in 1d systems is the saddle node bifurcation in which stable and unstable equilibrium annihilate in a continuous manner

18 Saddle node bifurcation How to check if is the system dy dt = F(y,β) is at a bifurcation with values y 1,β 1? It has to satisfy three conditions: Non hyberbolicity (at y 1 ): Non degeneracy (at y 1 ): F(y,β 1 ) y = 0 Traversality (at β 1 ): 2 F(y,β 1 ) y 2 0 F(y 1,β) β 0

19 Slow transition The behavior of trajectories may signal that the system is near a bifurcation One of such symptoms is the slow transition When the system is near the saddle node bifurcation, trajectories passing the so called attractor ruins are being slowed down

20 Quadratic Integrate Note that the simplest possible system that may undergo saddle-node bifurcation is dy(t) dt = y(t) 2 +a so that the right hand side function F is a square parabola, and parameter a controls the shift of the curve over the axis. Such a simplest equation that is able to reproduce a certain bifurcation is sometimes called a topological normal form for that bifurcation.

21 Definition In general two dimensional systems are of the form: dx(t) dt = F(x,y,t) dy(t) = G(x,y,t) dt If functions F and G depend on x,y, such systems are called autonomous. The system can also be expressed in compact vector form: d x dt = F( x,t) where x = [x 1 (t),x 2 (t)] T is a vector.

22 Definition Any pair of functions x(t) and y(t) that satisfy initial condition x(t 0 ) = x 0,y(t 0 ) = y 0 and the differential equation we will call a solution or trajectory. Trajectories can be plotted on two dimensional phase plane as implicit functions.

23 The trajectories can converge to an equilibrium, diverge to infinity. 2d case trajectories can be periodic. Any non-autonomous 1d system dx dt = F(x,t) can be dx(t) expressed as autonomous 2d system: dt = F(x,y) and dy(t) dt = 1.

24 Definition To understand a 2d system, one has to deal with a function [F,G] : R 2 R 2 Such a function takes a point from 2d plain and returns another 2d point. It is better however to think a that it returns a vector. We then have a function that attaches a vector to any point in the x,y plain. We call such a assignment a vector field.

25 Definition An important concept for the 2d phase plane analysis is the so called nullcline. Nullclines are the solutions: and F(x,y) = 0 G(x,y) = 0 Geometrically, nullcline is a locus of points in which the vector field changes direction with respect to x or y axis. In other words on the nullcline the vectors are always parallel to either x or y axis.

26 Equilibria 2d systems have equilibria whenever F(x,y) = 0 and G(x,y) = 0. It easy to see that both conditions are satisfied on the intersection of the nullclines. If F and G are continuous any equilibrium point must be on the intersection of nullclines. Equilibria can be either stable, unstable or neutral (neutrally stable), depending on the properties of the (total) derivative of function [F,G].

27 Total derivative A derivative of a function F : U R 2 R 2 at some point (x 0,y 0 ) is a linear map df(x 0,y 0 ) : R 2 R 2 such that: F(x,y) F(x 0,y 0 ) df(x 0,y 0 )(x x 0,y y 0 ) lim = 0 (x,y) (x 0,y 0 ) (x,y) (x 0,y 0 ) In other words, the function can be approximated in point (x 0,y 0 ) via a linear map df(x 0,y 0 ). There is a weaker notion of partial derivative with respect to some variable F(x,y) x. If the function F is continuous differentiable, the matrix of the total derivative coincides with matrix of partial derivatives (Jacobian matrix)

28 Linearization Taylor expansion, we have [ F(x,y) G(x,y) ] = [ F(x0,y 0 ) G(x 0,y 0 ) ] [ F x + G x F y G y ] [ x x0 y y 0 ] +... Since at the equilibrium F and G are zero we have [ ] [ ] F F [ F(x,y) x y x x0 = G(x,y) y y 0 G x G y ] + Higher order terms... If the linear part is non zero, the behavior of the system near equilibrium can be fully derived from the properties of the matrix (particularly its eigenvalues).

29 Eigenvalues The eigenvalues are the roots of characteristic polynomial where I is the identity matrix. Take λ k satisfying det(a λi) det(a λ k I) = 0 Zero determinant means that A λ k I is not invertible, that is it has non trivial kernel V k. Take any vector v k V k, we have: (A λ k I)v k = 0 Av k = λ k Iv k Av k = λ k v k

30 When the characteristic polynomial has roots λ 1...λ n and the matrix in non defective there exists a basis (eigenvector basis) in which matrix A can be expressed: λ λ A λ n Such a diagonal matrix has a particularly simple geometrical interpretation - it stretches the space in the eigenvector directions by a factor proportional to the corresponding eigenvalue.

31 In the general case, the matrix can be defective (it has less than n linearly independent eigenvectors). In this case the general representation is given by Jordan block form: J J A J k where each matrix J k has the form λ k, λ k, J k = λ k,i

32 A defective matrix induces shear transformation (transvection) along certain axes. If the eigenvalues are complex conjugate (they have to be conjugate, since for real matrices the characteristic polynomial has real coefficients), the matrix induces a rotation. In general any non singular real matrix transformation is a combination of scaling, shear and rotation Shear is something in between scaling and rotations - the vectors seem to rotate, but they never make the full π, instead they scale up to infinity... A matrix is called singular it has at least one zero eigenvalue

33 Hartman-Grobman theorem An equilibrium point is called hyperbolic if the Jacobian matrix at that point is not singular (doesnt have zero eigenvalues)

34 Hartman-Grobman theorem Theorem Let f : R n R n be a smooth map with a hyperbolic fixed point p. There exists a neighborhood U of p and a homeomorphism h : U R n such that f = h 1 df p h that is, in a neighborhood U of p, f is topologically conjugate to its linearization.

35 The behavior of a dynamical system near a hyperbolic equilibrium is fully determined by its linearization.

36 For the 2d system at a fixed point we have: where a = F x Let F(x,y) = a(x x 0 )+b(y y 0 )+Higher order terms G(x,y) = c(x x 0 )+d(y y 0 )+Higher order terms (x0,y0),b = F y G G (x0,y0),c = x (x0,y0),d = ] A = [ a b c d y (x0,y0). To determine stability, we have to find roots of the characteristic polynomial: [ ] a λ b det(a Iλ) = det = (a λ)(d λ) bc c d λ

37 We see that the characteristic polynomial: can be written as λ 2 (a+d)λ+ad bc λ 2 τλ+ where τ = tra = a+d (trace of the matrix A) and = deta = ad bc. The roots are then: λ 1 = τ + τ 2 4,λ 2 = τ τ They are real when τ 2 4 > 0, complex conjugate when τ 2 4 < 0 and real of multiplicity two when τ 2 4 = 0.

38 Having the eigenvalues we can fairly easily find eigenvectors. When there are two distinct eigenvalues λ 1 and λ 2 we solve the equations Ax 1 = λ 1 x 1 and Ax 2 = λ 2 x 2. When the eigenvalue is a double (multiple) root the case is somewhat more difficult. There are two possibilities: We solve Ax = λx. If the resulting kernel is two dimensional, were done, the vectors that span the kernel are the eigenvectors. If the kernel is one dimensional say {x 1 } (matrix A is defective), we have to find the generalized eigenvector x g which satisfies (A Iλ) 2 x g = 0. We can achieve that by solving (A Iλ)x g = x 1. The resulting normal form matrix is the so called Jordan block. For the general n-dimensional matrix things are a bit more complex, for more information search for Jordan normal form.

39 Once we have the eigenvectors v 1 and v 2, the solutions of a linear system are given by: [ ] v(x) = c w(y) 1 e λ1t v 1 +c 2 e λ2t v 2 Where c 1 and c 2 are constants dependent on the initial condition. For the real eigenvalues we have exponential convergence (divergence) along the eigenvectors. For complex conjugate eigenvalues we have complex eigenvectors. Nevertheless we can obtain the real basis of solutions by taking u = eλt +e λt 2 = e R(λ)t cos(i(λ)t) and v = eλt e λt 2 = e R(λ)t sin(i(λ)t). In that case u and v will be cosines and sines of t possibly multiplied by an exponential function (of t).

40 Proof. We have [ d c1 e λ1t v 1,1 +c 2 e λ2t ] v 2,1 dt c 1 e λ1t v 1,2 +c 2 e λ2t = v 2,2 [ a b c d ][ c1 e λ 1t v 1,1 +c 2 e λ 2t v 2,1 c 1 e λ 1t v 1,2 +c 2 e λ 2t v 2,2 ] [ c1 λ 1 e λ 1t v 1,1 +c 2 λ 2 e λ 2t v 2,1 c 1 λ 1 e λ 1t v 1,2 +c 2 λ 2 e λ 2t v 2,2 ] = [ a b c d ][ c1 e λ 1t v 1,1 +c 2 e λ 2t v 2,1 c 1 e λ 1t v 1,2 +c 2 e λ 2t v 2,2 ] but since v 1 and v 2 are the eigenvectors, we know that Av 1 = λ 1 v 1 etc. So: [ c1 λ 1 e λ1t v 1,1 +c 2 λ 2 e λ2t ] [ v 2,1 c1 λ c 1 λ 1 e λ1t v 1,2 +c 2 λ 2 e λ2t = 1 e λ1t v 1,1 +c 2 λ 2 e λ2t ] v 2,1 v 2,2 c 1 λ 1 e λ1t v 1,2 +c 2 λ 2 e λ2t v 2,2

41 What kind of equilibria we can have in a 2d system? Assuming hyperbolicity, we have the following choices: 1 Eigenvalues are real and positive - this results in a repelling, unstable node 2 Eigenvalues are real and negative - this results in a stable node 3 Eigenvalues are real, one positive the other negative - this results in a saddle 4 Eigenvalues are real and equal, but the matrix is defective - this gives either stable or unstable shear node 5 Eigenvalues are complex conjugates with non zero real parts - this gives either stable or unstable focus 6 Eigenvalues are purely imaginary - this gives neutrally stable node

42 Phase portrait analysis Equilibria are not the only invariant sets with respect to 2d system. There are also invariant cycles, called limit cycles. The cycle is invariant, when any trajectory departing from a point on the cycle, will always stay on the cycle. Limit cycles like equilibria can be attracting, repelling or neutral. Like in the 1d case, the phase portrait can be divided into attraction domains, but in this case two attracting equilibria do not have to divided by a repelling equilibirum, but also by a special trajectory called a separatrix (which can be a cycle).

43 Trajectories and separatrices A separatrix is a special kind of trajectory which divides two attraction domains. It can: Originate at the infinity End up in an unstable node or a saddle Form a closed curve (cycle) A trajectory that originates in some equilibrium point may Get back to the origin after making a cycle (homoclinic trajectory) End up in another equilibrium or a cycle (heteroclinic trajectory) Trajectories may never cross! (why?)

44 Bifurcations Similarly to 1d systems, two phase portraits are equivalent, if there exists a continuous transformation from one to another, which preserves all the equilibria, attraction domains, separatrices, cycles etc. Whenever due to a change of some parameter the system changes qualitatively its phase portrait it is said to undergo a bifurcation. If a bifurcation occurs when a single parameter changes its value, it is called a codimension one bifurcation. Some bifurcations on 2d plane can only occur when two parameters are changing together. Such bifurcations or of codimension two.

45 Bifurcation Recall that a bifurcation is any qualitative change in the phase portrait (resulting portraits are not equivalent in topological sense) as some parameters of the system change. And 2d phase portraits can be very complex... Nevertheless there are only four main ways of getting from resting to spiking (bifurcations of equilibria), and four ways of getting from spiking to resting (bifurcations of cycles).

46 Basic terminology Consider a bifurcation of some system with some parameter. Assume the parameter changes in such a direction that the number of stable elements (cycles, equilibria) increases as the parameter passes the bifurcation value. If stable elements appear, the bifurcation is said to be supercritical If unstable elements appear the bifurcation is said to be subcritical If the same number of stable and unstable elements appear, the bifurcation is transcritical

47 Possible bifurcations of equilibria When there are two real eigenvalues, and one of them is changing sign as the bifurcation parameter passes we have the saddle node bifurcation. When there are two complex conjugate eigenvalues, and their real parts change sign as as the bifurcation parameter passes we have Andronov-Hopf bifurcation, which as we will se can be subcritical and supercritical. When both eigenvalues become zero, the bifurcation degenerates, and we can no longer derive any useful information from the Jacobian matrix...

48 Saddle node bifurcation Recall that we have seen the saddle node bifurcation in 1d systems. It appears when the saddle and a stable node collide, which in turn gives birth to the semistable saddle-node. As the bifurcation parameter increases the saddle node disappears, leaving the ghost attractor that may slow down any passing nearby trajectory The necessary conditions for a system to undergo a saddle node bifurcation were: non-hyperbolicity, non-degeneracy and traversality. How to express these conditions with 2d systems?

49 Non-hyperbolicity in 2d means that the Jacobian matrix has exactly one zero eigenvalue. Determining non-degeneracy and traversality in general can be quite difficult, since it might require a reduction of the system to the center manifold (center manifold theorem) A center manifold is a manifold tangent to the eigenvector of the zero eigenvalue at the bifurcation point.

50 Saddle node on invariant circle An important from the computational point of view is a special case of a saddle node bifurcation which appears on the invariant circle. Before the bifurcation there is a saddle and a node, connected by two heteroclinic orbits. These orbits form the invariant circle (it is invariant, since any trajectory departing from a point on the circle has to stay on it). When the bifurcation parameters crosses critical value, the invariant circle changes into a limit cycle. The trajectories while spinning at the cycle fly by the ghost attractor, resulting in a slow transition...

51 Andronov-Hopf bifurcation When two complex conjugate eigenvalues become purely imaginary, we have the Andronov-Hopf bifurcation. In this case the equilibrium changes its stability, but also gives birth to either stable or unstable limit cycle. Consider the system dv dt = F(v,u,b) du dt = G(v,u,b)

52 Assume the system has an equilibrium at the origin and bifurcation parameter b = 0. The system undergoes Andronov-Hopf bifurcation if the following conditions are satisfied: Non-hyperbolicity: Jacobian matrix L = [ F v G v F u G u has a pair of purely imaginary eigenvalues ±iω C. trl = F v + G u = 0 and ω2 = detl = F G v u F G u v > 0 ]

53 Linear change of variables v = x and F u u = F vx ωy converts the system into: dx = ωy +f(x,y) dt dy = ωx +g(x,y) dt where f(x,y) = F(v,u)+ωy and g(x,y) = ( F F vf(v,u)+ u G(v,u)) /ω ωx. Now we are ready to state the next condition:

54 Non-degeneracy: a = 1 ( 3 ) f 16 x f x y g x 2 y + 3 g y ( 2 ( f 2 ) ( f 16ω x y x f y 2 2 g 2 ) g x y x g y f 2 g x 2 x f 2 ) g y 2 y 2 0

55 Transversality: Let c(b)+iω(b) be the eignvalues of the Jacobian matrix near 0, with c(0) = 0 and ω(0) = ω. The real part of the eignvalues must not be degenerate, that is c (0) 0 (they truly cross the imaginary axis, not just touch it). Since only one condition implies equality trl = 0, the codimension of the bifurcation is one. The sign of a determines the type of bifurcation: a < 0 - supercritical bifurcation, stable limit cycle emerges from stable equilibrium (which itself becomes unstable) a > 0 - subcritical bifurcation, unstable limit cycle emerges from unstable equilibrium (which itself becomes stable)

56 Subcritical Andronov-Hopf bifurcation A subcritical Andronov-Hopf bifurcation occurs when an unstable cycle shrinks into stable equilibrium causing it to lose stability.

57 Saddle homoclinic orbit The bifurcation gives birth to stable (or unstable) limit cycle. One of the heteroclinic trajectories departing from the saddle (via unstable manifold) becomes homoclinic as the bifurcation parameter changes. Therefore the orbit becomes a limit cycle, though the period of the cycle is infinite at the bifurcation point (transition through the saddle takes infinite amount of time) As the bifurcation parameter progresses, the homoclinic orbit becomes a full blown limit cycle flying somewhat neat the saddle... Depending on the saddle quantity (sum of eigenvalues at the saddle point), the bifurcation can be either subcritical or supercritical. In neural model we will only see the supercritical case.

58 Supercritical Andronov-Hopf Recall that with the supercritical Andronov-Hopf bifurcation a stable limit cycle emerges from stable equilibrium This process can also reverse itself, that is a stable cycle eventually shrinks into an equilibrium Much like with the saddle node on invariant circle, this case is not particularly interesting due to symmetry

59 Fold limit cycle bifurcation Recall that with subcritical Andronov-Hopf bifurcation an unstable limit cycle shrinks into a stable equilibrium which eventually looses stability. The state of the system can then fall onto a large stable cycle If we reverse that process however, even though the unstable cycle emerges and the equilibrium becomes stable, the state of the system can still follow the stable cycle.

60 The mysterious bifurcation that is taking place is called a fold limit cycle bifurcation. In this case a stable cycle collides with an unstable one, forming the neutrally stable fold cycle The bifurcation resembles the saddle node, but in this case instead of equilibria we have cycles... We can actually spot the unstable cycles by simulating models backwards (with time reversed).

61 Recapitulation The phase portrait in 2d can be very complex, nevertheless the number of possible bifurcations (especially those of codimension one) is very limited.

62 Bifurcations On the last lecture weve met four bifurcations of equilibria: saddle node, saddle node on invariant circle, supercritical Andronov-Hopf, subcritical Andronow-Hopf We also met two bifurcations of cycles: saddle homoclinic orbit fold cycle All of these bifurcations are of codimension 1, that is they have one control parameter. In neural excitability it is usually the input current.

63 Heteroclinic orbit bifurcation We have one more codimension one bifurcation in 2d left, namely heteroclinic orbit bifurcation In this case a heteroclinic trajectory changes its destination, hitting a saddle point This bifurcation is global, does not change the stability of any equilibria, does not create cycles.

64 We have therefore saddle node, saddle node on invariant circle, supercritical Andronov-Hopf, subcritical Andronow-Hopf supercritical saddle-homoclinic orbit subcritical saddle-homoclinic orbit heteroclinic orbit fold cycle These are all codimension one bifurcations in 2d. But how do we know that?

65 Poincare-Bendixon theorem Theorem Given an autonomous ODE on the plain dx dt = F(x) with continuous F, assume that solution x(t) stays in a bounded region for all times. Then x(t) converges as t to an equilibrium (with F(x) = 0) or to a single limit cycle.

66 Remark The theorem rules out chaos in 2d.

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68 Transversality: Let c(b)+iω(b) be the eignvalues of the Jacobian matrix near 0, with c(0) = 0 and ω(0) = ω. The real part of the eignvalues must not be degenerate, that is c (0) 0 (they truly cross the imaginary axis, not just touch it). Since only one condition implies equality trl = 0, the codimension of the bifurcation is one. The sign of a determines the type of bifurcation: a < 0 - supercritical bifurcation, stable limit cycle emerges from stable equilibrium (which itself becomes unstable) a > 0 - subcritical bifurcation, unstable limit cycle emerges from unstable equilibrium (which itself becomes stable)

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