Numerical simulations

We now present some numerical simulations of dissipative homeomorphisms.

Again, our aim is to compare the theoretical results with the reality of numerical sim-ulations: for simple homeomorphisms and reasonable orders of discretization, do we have convergence of the dynamics of the discretizations to that of the homeomorphism, as suggested by the above theorems?

We simulate homeomorphisms of the form

f(x, y) = (R◦Q◦P)(x, y),

where P and Q are two homeomorphisms of the torus that modify only one coordinate:

P(x, y) =

x, y+p(x)

and Q(x, y) =

x+q(y), y ,

so that the homeomorphism Q◦P preserves Lebesgue measure. We discretize these examples according to the uniform grids on the torus

E_N= i₁

N,· · ·,i_n N

∈Tⁿ

∀j,0≤i_j ≤N−1

.

We have tested two homeomorphisms:

– To begin with we studiedf₁= R₁◦Q◦P, with p(x) = 1

209cos(2π×187x) + 1

271sin(2π×253x)− 1

703cos(2π×775x), q(y) = 1

287cos(2π×241y) + 1

203sin(2π×197y)− 1

841sin(2π×811y) and

R₁(x, y)

x=x−0.00227 sin(2π×95(x+α)) + 0.000224 cos(2π×197(y+α))

−0.00111 sin(2π×343(x+α)) R₁(x, y)

y=y−0.00376 sin(2π×107(y+β))

−0.000231 cos(2π×211(x+β)) + 0.00107 cos(2π×331(y+β)),

with α= 0.00137 and β= 0.00159. This dissipative homeomorphism is a small C⁰ perturbation of the identity, whose derivative has many oscillations whose amplitudes are close to 1. That creates many fixed points which are attractors, sources or saddles.

– It has also seemed to us useful to simulate a homeomorphism close to the iden-tity in C⁰topology, but with a small number of attractors. Indeed, as explained heuristically by J.-M. Gambaudo and C. Tresser in [GT83], a homeomorphism like f₁ can have a large number of attractors whose basins of attraction are small. It turns out that the dissipative behaviour off₁ cannot be detected for reasonable orders discretization. We therefore defined another homeomorphism close to the

identity in C⁰topology, but with much less attractors, sayf₂= R₂◦Q◦P, with P

Remark that we have chosen to define the homeomorphisms we compute with lacu-nary trigonometric series, to “mimic” the action of Baire theorem.

4.3.1 Algorithm used for the calculus of invariant measures

The algorithm we used to conduct simulations is quite fast (in fact, it is linear in the number of points of the grid). It detects all the periodic orbits of the discretizationsf_N in the following way. It takes a first pointx₁∈E_N and iterates it until the orbit meets a point that has ever been visited by the orbit. The points belonging to the orbit of x₁ are labelled as falling into the periodic orbit number 1. The algorithm also notes the number of points that have been attracted by this orbit, and the coordinates of the points of the periodic orbit. It then takes another pointx₂∈E_Nwhich does not belong to the orbit ofx₁. There are two cases: either an iterate ofx₂is equal to an iterate ofx₁, and in this case it updates the number of points which fall into the periodic orbit number 1;

or an iterate ofx₂meets another iterate ofx₂, and in this case it creates a periodic orbit number 2, which attracts all the orbit ofx₂. This procedure is iterated until there is no more points of E_Nthat have not been visited. Remark that this algorithm computes the image of a point at most twice.

This algorithm allows to compute quantities like the cardinality of the recurrent set Ω(f_N), the number of periodic orbits off_N, the maximal size of a periodic orbit off_N, etc. It also allows to represent the invariant measureµ^f_X^N off_N. Recall that this measure is defined as the limit in the Cesàro sense of the push forward of the uniform measures on E_Nby the discretizationsf_N. It is supported by the unionΩ(f_N) of the periodic orbits off_N; the measure of each of these periodic orbits is proportional to the size of its basin of attraction.

We present images of sizes 128×128 pixels representing in logarithmic scale the density of the measuresµ^f_X^N: each pixel is coloured according to the measure carried by the set of points of E_N it covers. Blue corresponds to a pixel with very small measure and red to a pixel with very high measure. Scales on the right of each image corre-sponds to the measure of one pixel on the log 10 scale: if green correcorre-sponds to−3, then a green pixel will have measure 10⁻³forµ^f_X^N. For information, when Lebesgue measure is represented, all the pixels have a value about−4.2.

We also compute the distance between the measureµ^f_X^N and Lebesgue measure. The distance we have chosen is given by the formula

d(µ,ν) =

This distance spans the weak-* topology, which makes compact the set of probability measures onT². In practice, we have computed an approximation of this quantity by summing only on thek∈~0,7.

From a practical point of view, we have restricted ourselves to grids of sizes smaller than 2¹⁵×2¹⁵: the initial data become quickly very large, and the algorithm creates temporary variables that are of size of the order of five times the size of the initial data.

For example, for a grid 2¹⁵×2¹⁵, the algorithm needs between 25 and 30 Go of RAM, and takes about two days of calculus for a single order of discretization.

4.3.2 Combinatorial behaviour

We simulated some quantities related to the combinatorial behaviour of discretiza-tions of homeomorphisms, namely:

– the cardinality of the recurrent setΩ(f_N), – the number of periodic orbits off_N, – the maximal size of a periodic orbit off_N.

We calculated these quantities for discretizations of orders 128k for k from 1 to 150, and represented it graphically (see Figure4.2). For information, if N = 128×150, then q_N'3.6.10⁸.

Theoretically, the degree of recurrence, that is, the ratio between the cardinality of Ω(f_N) andq_N, should tend to 0 (see Corollary4.14); this is what we observe on simula-tions. This is not really surprising: we will even see it for discretizations of conservative homeomorphisms (see Figure5.7). In this context, it is interesting to compare the be-haviour ofΩ(f_N) in the conservative and the dissipative case. The result is a little dis-appointing: the graphic forf₁, the dissipative homeomorphism, is very similar to that off₃, the corresponding conservative homeomorphism, while in theory they should be very different. This is quite different forf₂, where the cardinality of Ω((f₂)_N) is more or less linear in N. We have no explanation to the linear shape of this function; if the mapsf_N were typical random maps, their degree of recurrence would be linear in N, with a value would close to 2.4.10⁴for N = 150×128 (here the value is about three times bigger).

The theoretical results assert that the number of periodic orbits of f_N should tend to +∞(as a generic dissipative homeomorphism has an infinite number of attractors).

We can hope that this quantity reflects the fact that the dynamics converges to that of the initial homeomorphism: among others, we can test if it is of the same order as the number of attractors of the homeomorphism. In practice, this number of periodic orbits off_N first increases rather quickly, to stabilize to around a value of 1.5.10⁴ for f₁ and 9.10³ for f₂. We could be tempted to interpret this phenomenon by the fact that after a while, the discretization has detected all the attractors of f and thus, the number of periodic of the discretizations reflects the number of attractors off. This idea may be reasonable for f₂ (as we will see in observing the invariant measures of (f₂)_N in Figure4.5), but if we compare these graphs in the dissipative case with that of the conservative case (Figure 5.7), we see that they are as alike as two peas in a pod.

Thus, this is not clear at all that this behaviour is due to the dissipative character of the homeomorphism or not.

Since the dynamics of discretizations is assumed to converge to that of the initial homeomorphism, we could expect that the length of the longest periodic orbit of dis-cretizations (f_i)_N is almost always a multiple of that of an attractive periodic orbit of

Figure 4.2: Size of the recurrent set Ω((f_i)_N) (top), number of periodic orbits of (f_i)_N (middle) and length of the largest periodic orbit of (f_i)_N (bottom) depending on N, for f₁(left) andf₂(right), on the grids E_Nwith N = 128k,k= 1,· · ·,150.

f_i. The graphic of this length for f₁ looks like the conservative case (see Figure 5.7), so we can say that the dissipative behaviour of this homeomorphism is not detected in practical by this quantity. Forf₂, the length of the longest orbit is much smaller than forf₁ (up to a factor 10), and seem to increase linearly in N. This may be imputed to the fact thatf₂is “almost conservative” around its attractive periodic points; thus it has a conservative behaviour, but at a smaller scale thanf₁.

4.3.3 Behaviour of invariant measures

We have computed the invariant measuresµ^(f_T2^i)N of dissipative homeomorphismsf₁ andf₂as defined on page53. Our aim is to test whether Theorem4.16applies in

prac-tice or if there are technical constraints such that this behaviour cannot be observed on these examples. For a presentation of the representations of the measures, see page54.

Figure 4.3: Distance between Lebesgue measure and the measureµ^f_T^N₂ depending on N forf₁(left) andf₂(right), on the grids E_N with N = 128k,k= 1,· · ·,150.

On Figure 4.3, we have represented the distance between the measure µ^(f_T2^i)N and Lebesgue measure. Theorem4.16says that for a generic dissipative homeomorphism, this quantity converges to the distance between µ^f_T2 and Lebesgue measure. Clearly, this is not what happens in practice forf₁: the distance between both measures globally increases when N increases. Locally, the behaviour of the map N 7→ dist(µ^f_T^N₂,Leb) is quite erratic: there is no sign of convergence to any measure. When we compare this with the conservative case (Figure5.8), we see that both behaviours are very similar. In other words, we do not see the dissipative nature off₁on simulations. The behaviour of the distance between the measureµ^(f_T2^2)N and Lebesgue measure is much more inter-esting. First of all, we observe that this distance is smaller than forf₁. Moreover, it is globally slightly decreasing in N, and it seem to converge to a value close to 0.2. So it suggests that the measuresµ^(f_T2^2)N converge to a given measure whose distance to Leb is close to 0.2, as predicted by the theory. In the view of these graphics, we can say that the behaviour off₂is much more close to that of a generic dissipative homeomorphism that that off₁. This can be interpreted in the view of the article of J.-M. Gambaudo and C. Tresser [GT83]: forf₁, the attractors are much too small to be observed in practice for reasonable orders of discretization.

The behaviour of invariant measures forf₁(see Figure4.4), which is a small C⁰ dis-sipative perturbation of identity, is relatively similar to that of invariant measures forf₃ i.e. the corresponding conservative case (see Figure5.9): when the order discretization is large enough, there is a strong variation of the measureµ^(f_T2^1)N. Moreover, this measure has a significant absolutely continuous component with respect to Lebesgue measure.

This is very different from what is expected from the theoretical results (in particular Theorem4.16), which say that for a generic dissipative homeomorphism, the measures µ^f_T^N₂ must converge to the measureµ^f_T2. Thus, we can say that it is impossible to detect the dissipative character off₁on these simulations. Again, as noted by J.-M. Gambaudo and C. Tresser in [GT83], the size of the attractors off₁can be very small compared to the numbers involved in the definition off₁. So even in orders discretization such as 2¹⁵, the dissipative nature of the homeomorphism is undetectable on discretizations. This

Figure 4.4: Simulations ofµ^(f1)N

T² on grids E_N, with N = 2^k,k= 7,· · ·,15 (from left to right and top to bottom).

is why the discretizations off₁are very similar to those of its conservative counterpart f₃.

Recall that what happens for the dissipative homeomorphism f₁ is rather close to what happens for the conservative homeomorphismf₃. For their part, simulations of invariant measures forf₂on grids of size 2^k×2^k(see Figure4.5) highlight that we expect from a generic dissipative homeomorphism: the measuresµ^(f_T2^2)Nseem to tend to a single measure (that is also observed on a series of simulations), which is carried by the attrac-tors of f₂. The fact thatf₂ has few attractors allows the discretizations of reasonable orders (typically 2¹¹) to find the actual attractors of the initial homeomorphism, con-trary to what we observed forf₁. We also present a zoom of the density of the measures µ^(f_T2^2)N (Figure4.6). On these simulations, we can see that the attracting regions that we observe on the simulations of Figure4.5are in fact crumbled, it particular they are not connected. This is what is predicted by the theory: the set of Lyapunov periodic points

Figure 4.5: Simulations ofµ^(f2)N

T² on grids E_N, with N = 2^k,k= 7,· · ·,15 (from left to right and top to bottom).

of a generic dissipative homeomorphism is a Cantor set. On these zoomed simulations, we also observe that the dynamics of the discretizations has not completely converged at the order N = 2¹⁵: locally, the measuresµ^(f_T2^2)N are quite different for different orders N; locally, the homeomorphism still behaves like in the conservative one.

Figure 4.6: Zoom on the density of the measures µ^(f_T2^2)N on grids E_N, with N = 2^k, k= 13,14,15 (from left to right); the zoom is made on the top right of the representation of Figure4.5, on a square of size 1/8×1/8.

Discretizations of a generic