Applied nonlinear dynamics

A brief glance at continuous-time chaos

The Lorentz system

Lecturer: Dr. A. Schaum,Chair of Automatic Control, Kiel University

Motivation

Convection plays a crucial role in fluid dynamics and meteorolgy toexplain phenomena like convection rolls and cloud streets

Convection roll over Kiel, Spring 2019

The dynamics of such phenomena are clearly nonlinear andhigh-dimensional. Typically, one would expect to on-set of such behaviordue to some kind of bifurcation. . .

System dynamics

Edward N. Lorenz (1917-2008) studied cellular convection in theathmosphere [2], ending up with the following simplified model

x = σ(y − x)y = rx − y − xzz = xy − bz

Cold air

Warm air

where σ is the so-called Prandl number (i.e. the ratio of momentumdiffusivity (kinematic viscosity) to thermal diffusivity), r the Rayleighnumber (i.e. a measure for the ratio of thermal conduction throughconvection: for large r thermal conduction is primarily throughconvection, while for low r its due to conduction), and b is related to theaspect ratio of convection rolls.

Dissipativeness

Consider the change of avolume element V (t) with surface S(t)

V (t + dt) = V (t) +

∫S

f (x) · n dt dA,

where dA denotes an infinitesimal area element, and n the outwardnormal of the surface S at x . By rearranging terms, division through dt,and taking limits, the preceding equation implies

V =

∫S

f (x) · n dA =

∫V

∇ · f dV

=

∫V

(∂

∂x[σ(y − x) +

∂

∂y[rx − y − xz ] +

∂

∂z[xy − bz ]

)dV

= −(σ + 1 + b)

∫V

dV = −(σ + 1 + b)V

So the volume occupied by trajectories converges exponentially to zero.

Equilibrium points

The equilibrium conditions read

0 = σ(y − x)0 = rx − y − xz0 = xy − bz

yielding

0 = (r − 1)x − 1

bx3 =

(r − 1)

bx(b − x2

)which corresponds to the classical normal form of the pitchforkbifurcation:

up to three equilibriax = 0, x = C− = (−

√(r − 1)b,−

√(r − 1)b, r − 1)′, and

x = C+ = (√

(r − 1)b,√

(r − 1)b, r − 1)′

Pitchfork bifurcation at r = 1.

Equilibrium points

The Jacobian for an arbitrary equilibrium point is given by

J[x∗] =

−σ σ 0r − z −1 −xy x −b

.Eigenvalues of the origin are given by

λ1,2 =1

2

(−(σ + 1)±

√(σ + 1)2 − 4(1− r)σ

), λ3 = −b

implying that x = 0 is a local attractor for r < 1.

Alternatively, using theLyapunov function [4]

W (x , y , z) =1

2

(1

σx2 + y2 + z2

)> 0

W = −x2 − y2 − bz2 + (r + 1)xy

= −x2 + (r + 1)xy − (r + 1)2

4y4 +

(r + 1)2

4y4 − y2 − bz2

Equilibrium points

The Jacobian for an arbitrary equilibrium point is given by

J[x∗] =

−σ σ 0r − z −1 −xy x −b

.Eigenvalues of the origin are given by

λ1,2 =1

2

(−(σ + 1)±

√(σ + 1)2 − 4(1− r)σ

), λ3 = −b

implying that x = 0 is a local attractor for r < 1. Alternatively, using theLyapunov function [4]

W (x , y , z) =1

2

(1

σx2 + y2 + z2

)> 0

W = −x2 − y2 − bz2 + (r + 1)xy

= −x2 + (r + 1)xy − (r + 1)2

4y4 +

(r + 1)2

4y4 − y2 − bz2

Equilibrium points

Summarizing, it holds that

W = −(x − r + 1

2y

)2

−(

1− (r + 1)2

4

)y2 − bz2

which for r < 1 is non-positive. Actually, the second term vanishes onlyfor y = 0 given that r < 1, and thus W = 0 implies x = 0, y = 0, z = 0.Hence W is negative definite. This, in turn implies that the origin is theunique global attractor for r < 1, because W is radially unbounded.For the non-zero equilibrium point C+ the Jacobian becomes

J[C+] =

−σ σ 0

1 −1 −√

(r − 1)b√(r − 1)b

√(r − 1)b −b

with characteristic equation

λ3 + (σ + b + 1)λ2 + (r + σ)bλ+ 2bσ(r − 1) = 0.

Equilibrium points

Given that no additional equilibrium points can exist, and thus, in case ofC+ (and by symmetry C−) becomming unstable the next stage ofattractor would be a limit cycle (born due to a Hopf bifurcation), we nextanalyze the possibility of two purely imaginary eigenvalues

λ1,2 = ±iω.

Substituting into the characteristic equation yields

i(b(r + σ)ω − ω3

)− ω2(b + σ + 1) + 2bσ(r − 1) = 0

so that the critical value of r for a Hopf bifurcation is found to be

rH = σσ + b + 3

σ − b − 1.

Thus, one can conclude that C+ (and C−) are local attractors for

1 < r < rH .

Equilibrium points

It can be shown (see e.g. [3, 1]) that the bifurcation is subcritical .

This implies that for r < rH the equilibrium point C+ (and C−) issurrounded by an unstable limit cycle, which collapses over it atr = rH , rendering it unstable (actually it becomes a saddle point).

For r > rH there are no more stable equilbrium points. . .

r

x

1 rH

Thus, none of the attractors (equilibrium points, limit cycles)studied so far in this course exists for r > rH .

The classical analysis ends here and the following analysis is basedon numerical analysis tools.

Equilibrium points

It can be shown (see e.g. [3, 1]) that the bifurcation is subcritical .

This implies that for r < rH the equilibrium point C+ (and C−) issurrounded by an unstable limit cycle, which collapses over it atr = rH , rendering it unstable (actually it becomes a saddle point).

For r > rH there are no more stable equilbrium points. . .

r

x

1 rH

Thus, none of the attractors (equilibrium points, limit cycles)studied so far in this course exists for r > rH .

The classical analysis ends here and the following analysis is basedon numerical analysis tools.

Equilibrium points

It can be shown (see e.g. [3, 1]) that the bifurcation is subcritical .

This implies that for r < rH the equilibrium point C+ (and C−) issurrounded by an unstable limit cycle, which collapses over it atr = rH , rendering it unstable (actually it becomes a saddle point).

For r > rH there are no more stable equilbrium points. . .

r

x

1 rH

Thus, none of the attractors (equilibrium points, limit cycles)studied so far in this course exists for r > rH .

The classical analysis ends here and the following analysis is basedon numerical analysis tools.

Equilibrium points

It can be shown (see e.g. [3, 1]) that the bifurcation is subcritical .

This implies that for r < rH the equilibrium point C+ (and C−) issurrounded by an unstable limit cycle, which collapses over it atr = rH , rendering it unstable (actually it becomes a saddle point).

For r > rH there are no more stable equilbrium points. . .

r

x

1 rH

Thus, none of the attractors (equilibrium points, limit cycles)studied so far in this course exists for r > rH .

The classical analysis ends here and the following analysis is basedon numerical analysis tools.

Lorentz attractor

For b = 83 , σ = 10 and r = 28 > 24.74 ≈ rH the time response shown in

Figure 1 is obtained, showing aperiodic long-time behavior.

−20−15

−10−5

05

1015

20

−30

−20

−10

0

10

20

30

0

10

20

30

40

50

x

y

z

0 10 20 30 40−20

−15

−10

−5

0

5

10

15

20

Time t

x

0 10 20 30 40−30

−20

−10

0

10

20

30

Time t

y

0 10 20 30 400

5

10

15

20

25

30

35

40

45

50

Time t

z

Figure: Long-time behavior of the Lorentz dynamics for b = 83, σ = 10 and

r = 28 > 24.74 ≈ rH .

Lorentz map

Next, we analyze the discrete-time map defined by the dependency of themaximum of the amplitude of the time response at time tn+1 on themaximum in the previous time step tn, i.e. (see Figure 2)

z(tn+1) = f (z(tn)).

8 10 12 14 16 18

−5

0

5

10

15

Time t

x(t)

z(tn−1

)

z(tn)

z(tn+1

)

Figure: Illustration of the meaning of the Lorenz map.

Lorentz map

Figure 3 shows this, so called Lorenz map taken from a numericalsimulation with initial condition (1, 1, 1)′, simulation time T = 104, andparameters b = 8

3 , σ = 10 and r = 28 > rH .

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

z(tn)

z(tn+1)

Figure: Lorenz map for b = 83, σ = 10 and r = 28 > 24.74 ≈ rH , initial

conditions (1, 1, 1)′ and simulation time T = 104.

The slope |f ′| of the map is always greater than 1.

Strange attractor

Accordingly, for an arbitrary p ∈ N it holds that

z(tn+p) = f [z(tn+p−1)] ≈ f ′[z(tn+p−1)]z(tn+p−1)

= f ′[z(tn+p−1)]f [z(tn+p−2)] ≈ . . .

≈(

Πp−1i=1 f

′(z(tn+i ))z(tn)

> z(tn)

given that |f ′(z)| > 1 ∀ z .

Thus, if there exists an periodic orbit it is unstable.

Hence, trajectories spiral around the equilibrium points C+ and C−,jumping from time to time from one side to the other over and overagain (as seen in Figure 1), converging to an attractor set of zerovolume.

This attractor set is called a strange attractor.

Sensitivity with respect to initial conditions

The distance between trajectories with initial conditions that areclose to each other increases exponentially

||δ(t)|| := ||x1(t)− x2(t)|| ≈ ||x1(0)− x2(0)||eλt .Consider the initial conditions x1(0) = (10, 10, 10)′ andx2(0) = (1.0001, 1.0001, 1.0001)′

0 5 10 15 20 25 30−10

−5

0

5

Time t

ln(||x1−x2||)

Figure: Time evolution of the natural logarithm of the difference betweenthe two intial conditions for b = 8

3, σ = 10 and r = 28. The straight line

indicates the exact exponential growth with factor 0.9.

Deterministic chaos

A system is chaotic if its solutions show

1 Aperiodic long-term behavior, i.e. trajectories do not settle downon periodic orbits, equilibrium points, or quasiperiodic orbits (on atorus).

2 Deterministic dynamics, i.e. there are no stochastic sources in thesystem dynamics.

3 Sensitivity with respect to initial conditions, i.e. the distancebetween initial conditions which are close to each other growsexponentially fast.

References

P. G. Drazin.

Nonlinear Systems.Cambridge University Press, Cambridge, UK, 1992.

E. N. Lorenz.

Deterministic nonperiodic flow.J. Atmos. Sci., 20:130–141, doi:http://dx.doi.org/10.1175/1520–0469(1963)020¡0130:DNF¿2.0.CO;2, 1963.

J. E. Maarsden and M. McCracken.

The Hopf bifurcation and its applications.Springer, New York, 1976.

S. H. Strogatz.

Nonlinear Dynamics and Chaos, with applications to physics, biology, chemistry, andengineering.Perseus, Massachusetts,, 1994.