__Slow Manifold Gallery / Flow Manifold Gallery __

**Singularly Perturbed Systems
2D & 3D**

**MF 30** **Van der Pol model**

*Flow
Curvature Method*, i.e., the *1 ^{st} flow curvature
manifold* directly provides the

*1 ^{st}
flow curvature manifold*

*singular**
approximation*

[1926]
Van der Pol, B., “On
’Relaxation-Oscillations’,” Phil. Mag., 7, Vol. 2.

**MF 48 FitzHugh-Nagumo
model**

The FitzHugh-Nagumo model (FitzHugh, 1961 ; Nagumo *et al.*, 1962)
is a simplified version of the Hodgkin-Huxley model which models in a detailed
manner

activation and deactivation
dynamics of a spiking neuron. In the original papers of FitzHugh
this model was called “Bonhoeffer-Van der Pol oscillator”, since it

contains the van der
Pol oscillator as a special case for a = b = 0.

*x** *denotes the
membrane potential, *y *is a recovery variable and *b *is the
magnitude of stimulus current

*Flow
Curvature Method*, i.e., the *1 ^{st} flow
curvature manifold* directly provides the

*1 ^{st}
flow curvature manifold*

*singular**
approximation*

[1961]
FitzHugh, R., “Impulses and physiological states in
theoretical models of nerve membranes,” 1182-Biophys. J., 1, 445-466.

[1962] Nagumo J. S., Arimoto S. & Oshizawa S., “An active pulse transmission line simulating
nerve axon,” Proc. Inst. Radio Engineers. 50, 2061-2070.

**MF49** **cubic Chua’s model**

The cubic Chua’s
circuit (Chua *et al.*, 1986) is an electronic circuit comprising an
inductance *L*1, an active resistor *R*, two capacitors *C*1 and
*C*2, and a nonlinear

resistor. Chua’s circuit
can be accurately modeled by means of a system of
three coupled first-order ordinary differential equations in the variables *x
*(*t*), *y *(*t*) and

*z** *(*t*), which
give the voltages in the capacitors *C*1 and *C*2, and the intensity
of the electrical current in the inductance *L*1, respectively.

The function *k*(*x*)
describes the electrical response of the nonlinear resistor, i.e., its
characteristics which is a “cubic” function is defined by:

where the real
parameters *c*1 and *c*2 are determined by the particular values of
the circuit components.

*Flow
Curvature Method*, i.e., the *2 ^{nd} flow
curvature manifold* directly provides the

[1986] Chua, L. O., Komuro,
M. & Matsumoto, T., “The double scroll family,” IEEE Trans. on Circuits and
Systems, 33, Vol.11, 1072-1097.

**Non-singularly Perturbed
Systems 2D & 3D**

**MF 50** **Brusselator**** model**

Studying an hypothetical chemical reaction in which it is assumed
that the reactions are all irreversible a *et al.*,
1967) proposed the following model in which *x *and *y *denotes
concentrations

According to Grasman (1987) the chemical oscillation turns into *relaxation oscillations*. Thus, the Brusselator may be considered as slow-fast dynamical
systems although it has no small multiplicative parameter in its velocity
vector field and so no *singular approximation*.

Although the Brusselator model is not *singularly
perturbed *it can be considered as a *slow fast dynamical system *since
it may be shown (numerically) that its functional Jacobian
matrix exhibits a *fast *eigenvalue.

*Flow
Curvature Method*, i.e., the *1 ^{st} flow
curvature manifold* directly provides the

[1967]
Prigogine, I. & Lefever, R. “On Symmetry-Breaking
Instabilities in Dissipative Systems,” Journal of Chemical Physics, Vol. 46,
3542-3550.

[1987]
Grasman, J., *Asymptotic Methods for Relaxation
Oscillations and Applications*, vol. 63,

**MF 50** **Lorenz model**

In the beginning of
the sixties a young meteorologist of the M.I.T., Edward N. Lorenz, was working
on weather prediction. He elaborated a model derived from

the Navier-Stokes equations with the Boussinesq
approximation, which described the atmospheric convection. Although his model
lost the correspondence to the actual atmosphere in the process of
approximation, chaos appeared from the equation describing the dynamics of the
nature. Let’s consider the Lorenz model (Lorenz, 1963):

Although the Lorenz model is not *singularly perturbed *it can be
considered as a *slow fast dynamical system *

since it may be shown (numerically)
that its functional Jacobian matrix exhibits a *fast
*eigenvalue.

*Flow
Curvature Method*, i.e., the *2 ^{nd} flow
curvature manifold* directly provides the

[1963] Lorenz, E. N., “Deterministic non-periodic flows,” J. Atmos. Sc, 20,
130-141.

**MF 51** **Pikovskii-Rabinovich-Trakhtengerts**** model**

The Pikovskii-Rabinovich-Trakhtengerts model (PRT) (Pikovskii *et al.*, 1978) has been elaborated in order
to study interactions between “whistler waves” which propagate parallel to the
magnetic field and lower hybrid waves in a plasma. Such interactions are among
the important phenomena taking place in the ionosphere. This phenomenon can be modeled by means of a system of three coupled first-order
ordinary differential equations in the variables *x*(*t*), *y*(*t*)
and *z*(*t*), which give the the normal
amplitude of the wave, the normal amplitude of the ion acoustic wave and the
normal amplitude of the synchronous third wave produced, respectively. The
amplitudes are assumed to be constant in space. The evolution equations of the
(PRT) model may be written in dimensionless form:

where the amplitudes
have been nondimensionalized; *h *is proportional
to the amplitude of the electric field of the “whistler” and *v*1 and *v*2
are the damping decrements of the excited hybrid and acoustic waves normalized
to the damping of the decay-induced third wave.

It has been
recently established (Llibre *et al.*, 2008)
that this dynamical system exhibits a four wings butterfly chaotic attractor
for a particular set of parameters.

Although the (PRT) model is not *singularly perturbed *it can be
considered as a *slow fast dynamical system *

since it may be shown (numerically)
that its functional Jacobian matrix exhibits a *fast
*eigenvalue.

*Flow
Curvature Method*, i.e., the *2 ^{nd} flow
curvature manifold* directly provides the

[1978]
Pikovskii, A. S., Rabinovich,
M. I. & Takhtengerts, V. Y., “Onset of stochasticity in decay confinement of parametric
instability,” Soc. Phys. J. E. T. P., t. 47, 715-719.

[2008]
Llibre, J., Messias, M.
& Silva, P. R., “On the global dynamics of the Rabinovich
system,” J. Phys. A: Math. Theor. 41, 275210.

**Singularly Perturbed Systems
4D & 5D**

**MF 55** **fourth-order cubic Chua’s model**

The fourth-order
cubic Chua’s circuit (Thamilmaran *et al.*,
2004, Liu *et al.*, 2007) may be described starting from the same set of
differential equations as previously but while replacing the piecewise linear
function by a smooth cubic nonlinear.

The function *k *(*x*1)
describing the electrical response of the nonlinear resistor is an
odd-symmetric function similar to the piecewise linear nonlinearity *k *(*x*1)
for which the parameters *c*1 = 0*.*3937 and *c*2 = *-*0*.*7235
are determined while using least-square method (Tsuneda,
2005) and which characteristics is defined by:

*Flow
Curvature Method*, i.e., the *3 ^{rd} flow
curvature manifold* directly provides the

[2004]
Thamilmaran, K., Lakshmanan
M. & Venkatesan A. “Hyperchaos
in. a modified canonical Chua’s circuit,” *Int. J. Bifurcation and Chaos*,
vol. 14, 221-243.

[2007]
Liu X., Wang, J. & Huang, L., “Attractors of Fourth-Order Chua’s Circuit
and Chaos Control,” *Int. J. Bifurcation and Chaos *8, Vol. 17, 2705
-2722.

**MF 56** **fifth-order cubic Chua’s model**

The fifth-order
cubic Chua’s circuit (Hao *et al.*, 2005) may be
described starting from the same set of differential equations as previously but
while replacing the

piecewise linear function by a smooth
cubic nonlinear.

The function *k *(*x*1)
describing the electrical response of the nonlinear resistor is an
odd-symmetric function similar to the piecewise linear nonlinearity *k *(*x*1)
for which the parameters *c*1 = 0*.*1068 and *c*2 = *-*0*.*3056
are determined while using least-square method (Tsuneda,
2005) and which characteristics is defined by:

*Flow
Curvature Method*, i.e., the *4 ^{th} flow
curvature manifold* directly provides the

[2005] Hao, L., Liu, J. & Wang, R., “Analysis of a Fifth-Order
Hyperchaotic Circuit,” in *IEEE Int. Workshop VLSI
Design & Video Tech.*, 268-271.

[2005]
Tsuneda, A., “A gallery of attractors from smooth
Chua’s equation,” *Int. J. Bifurcation and Chaos*, vol. 15, 1- 49.

**Non-singularly Perturbed
Systems 4D & 5D**

**MF 57** **Homopolar**** dynamo model**

Homopolar dynamo model are used
for the understanding of spontaneous magnetic field generation in magnetohydrodynamics. In the seventies H. K. Mofatt (1978) proposed a model of coupled Faraday-disk homopolar dynamos. By introducing fluxes variables instead
of the currents this model has been transformed (Hide *et al.*, 1999) into
the following dimensionless one:

Although the homopolar dynamo model is not *singularly perturbed *it
can be considered as a *slow fast dynamical system *since it may be shown
(numerically) that its functional Jacobian matrix
exhibits a *fast *eigenvalue.

*Flow Curvature Method*, i.e., the *3 ^{rd} flow curvature manifold*
directly provides the

[1978]
Mofatt, H. K., *Magnetic field generation in electrically
conducting fluids*, Monograph in CUP series on Mechanics and Applied
Mathematics. (Translated from Russian).

[1999] Hide, R.
& Moroz,

**MF 58** **Magnetoconvection**** model**

A fifth-order
system for magnetoconvection (Knobloch
*et al.*, 1981) is designed to describe nonlinear coupling between
Rayleigh-Bernard convection and an external magnetic field. This type of system
was first presented by Veronis (Veronis,
1966) in studying a rotating fluid. The fifth-order system of magnetoconvection is a straightforward extension of the
Lorenz model for the Boussinesq convection
interacting with the magnetic field. The fifth-order autonomous system of magnetoconvection is given as follows:

where *x*1(*t*)
represents the first-order velocity perturbation, while *x*2 (*t*), *x*3
(*t*), *x*4 (*t*) and *x*5 (*t*) are measures of the
first- and the second-order perturbations to

the temperature and to
the magnetic flux function, respectively. With the five real parameters where is the magnetic *Prandtl** *number (the ratio of the

magnetic to the thermal
diffusivity), * * is the *Prandtl**
*number, *r *= 14*.*47 is a normalized *Rayleigh *number, *q
*= 5 is a normalized *Chandrasekhar *number, and

* *
is a
geometrical parameter

Although the magnetoconvection model is not *singularly perturbed *it
can be considered as a *slow fast dynamical system *since it may be shown
(numerically) that its functional Jacobian matrix
exhibits a *fast *eigenvalue.

*Flow Curvature Method*, i.e., the *4 ^{th} flow curvature manifold*
directly provides the

[1966]
Veronis, G., “Motions at subcritical values of the
Rayleigh number in a rotating fluid,” *J. Fluid Mech.*, Vol. 24, 545–554.

[1981]
Knobloch, E. & Proctor, M., “Nonlinear periodic
convection in double-diffusive systems,” *J. Fluid Mech. *108, 291-316.

**Non-Autonomous Dynamical
Systems**

**MF 60** **Forced Van der Pol model**

In order to
highlight the efficiency of the *Flow
Curvature Method*, i.e., that it may extended to non-autonomous dynamical
systems let’s consider the forced Van der Pol model (Guckenheimer et al., 2002)
which may be written as:

A variable changes may transform this *non*-*automous**
*system into an *autonomous *one which reads:

Although this
transformation increases the dimension of the system the

*Flow Curvature* *Method *enables
to directly compute the *slow manifold analytical equation* associated
with system.

*Flow Curvature Method*, i.e., the *3 ^{rd} flow curvature manifold*
directly provides the

[2002] Guckenheimer, J., Hoffman, K. & Weckesser,
W., “The forced van der Pol
equation I: the slow flow and its bifurcations,” SIAM J. App. Dyn. Sys., 2, 1-35.