124
5 Analysis of Simple Nonlinear Dynamical Systems
The carried out numerical experiments showed that using the different sampling
frequency, the power spectrum and wavelet spectrum were not changed. This was also
validated by results obtained by Benettin, neural networks and Rosenstein methods
which yielded the results very close to original ones. Kantz method gave underestimated results for different frequency selection, correlating with the results obtained
for the standard sample size.
5.7 Lorenz Attractor [9]
The input hydrodynamic system is governed by the following ODEs:
⎧
⎪ ⎨
⎪ ⎩
˙
x = σ(y − x),
˙
y = x(r − z) − y,
˙
z = x y − bz,
(5.6)
where r stands for the normalized Rayleigh number (non-dimensional number defining fluid behaviour under gradient):
r =
gβT L
3
νχ
.
(5.7)
In the above equation, the following notation is used: g—gravity of Earth; L—
characteristic dimension of the fluid space; T —temperature difference between
fluid walls; ν—kinematic fluid viscosity, χ—thermal conductivity of the fluid; β—
coefficient of heat fluid extension; σ—Prandtl number (takes into account heat source
property) governed by the following equation
σ =
ν
α
=
ηC p
ℵ
,
(5.8)
where ν = η/ρ—kinematic viscosity, η—dynamic viscosity, ρ—density, α =
ℵ
ρC p
—
temperature transfer coefficient, ℵ—heat transfer coefficient, C p —specific heat
capacity under constant pressure; and ρ—information about the geometry of the
convective cell.
Moreover, for fixed: σ = 10.0, r = 28.0, b = 8/3. The original results follow:
LEs: 0.9056, 0, -14.5723; the Kaplan-York dimension: 2.06215.
The power spectrum of the attractor uniformly decreases when approaching a finite
frequency, and there is a lack of frequencies with a strongly dominating amplitude.
The latter observation is also verified by Gauss wavelet spectrum. The bifurcation
diagrams, similar to those for Rössler system, exhibit a complex structure, but the
correlation to the LLEs change is conserved. The richest/lowest dynamics of LLE is
obtained for changing parameter r/σ. Based on the reported graphs of LEs, one can
5 Analysis of Simple Nonlinear Dynamical Systems
The carried out numerical experiments showed that using the different sampling
frequency, the power spectrum and wavelet spectrum were not changed. This was also
validated by results obtained by Benettin, neural networks and Rosenstein methods
which yielded the results very close to original ones. Kantz method gave underestimated results for different frequency selection, correlating with the results obtained
for the standard sample size.
5.7 Lorenz Attractor [9]
The input hydrodynamic system is governed by the following ODEs:
⎧
⎪ ⎨
⎪ ⎩
˙
x = σ(y − x),
˙
y = x(r − z) − y,
˙
z = x y − bz,
(5.6)
where r stands for the normalized Rayleigh number (non-dimensional number defining fluid behaviour under gradient):
r =
gβT L
3
νχ
.
(5.7)
In the above equation, the following notation is used: g—gravity of Earth; L—
characteristic dimension of the fluid space; T —temperature difference between
fluid walls; ν—kinematic fluid viscosity, χ—thermal conductivity of the fluid; β—
coefficient of heat fluid extension; σ—Prandtl number (takes into account heat source
property) governed by the following equation
σ =
ν
α
=
ηC p
ℵ
,
(5.8)
where ν = η/ρ—kinematic viscosity, η—dynamic viscosity, ρ—density, α =
ℵ
ρC p
—
temperature transfer coefficient, ℵ—heat transfer coefficient, C p —specific heat
capacity under constant pressure; and ρ—information about the geometry of the
convective cell.
Moreover, for fixed: σ = 10.0, r = 28.0, b = 8/3. The original results follow:
LEs: 0.9056, 0, -14.5723; the Kaplan-York dimension: 2.06215.
The power spectrum of the attractor uniformly decreases when approaching a finite
frequency, and there is a lack of frequencies with a strongly dominating amplitude.
The latter observation is also verified by Gauss wavelet spectrum. The bifurcation
diagrams, similar to those for Rössler system, exhibit a complex structure, but the
correlation to the LLEs change is conserved. The richest/lowest dynamics of LLE is
obtained for changing parameter r/σ. Based on the reported graphs of LEs, one can