**
2.
Using Normal Form Theory to Identify the
Cyclic Patterns of Population Dynamics **
**Full text (PDF)**

Tahmineh Azizi

**
Pages:
16-****29**

**
Abstract:
** In
this paper, we study a three-parameter family of discrete time
population models which describes a host-parasitoid interaction, and
generalizes a model that originally was proposed by Nicholson and
Bailey. In contrast to the last case, where there was a density
dependent factor exp(r(1-H(n)=k-P(n))) in the equation of host
population, the general case of this family consists of
exp((r(1-P(n))^(2 a-1)-H(n)/k)) displays a somewhat almost similar range
of behaviors for a in Z+. In particular, this general model also
undergoes a Neimark Sacker bifurcation that produces an attracting
invariant closed curve in some areas of the parameter space. We perform
the local stability analysis to discover the local dynamics of
equilibrium points. Using the normal form theorem, we will present the
topological normal form for Neimark-Sacker bifurcation. From the
viewpoint of biology, the invariant closed curve corresponds to the
periodic or quasi-periodic

oscillations between host and parasitoid populations.

**
3. Constructing some discrete 4-D
hyperchaotic systems****
**
**Full text (PDF)**

N. K. K. Dukuza

**
Pages:
30-4****0**

**
Abstract:
** Modeling real life phenomena
often leads to complex nonlinear dynamics such as bifurcation and chaos.
The study of such problems has attracted interest of many scientists
over the past decades. In this paper, we present a method for
constructing some discrete four dimensional (4-D) hyperchaotic systems.
A nonclassical procedure for discretising autonomous 4-D continuous
hyperchaotic systems is applied; a parameter is introduced in this
process. By adjusting this parameter, until we obtain exactly two
equal-positive Lyapunov exponents, a new discrete 4-D hyperchaotic
system is realised. We prove that these discrete systems are
bounded-input bounded-output (BIBO) stable. Our illustrative results
show that the constructed discrete systems and their continuous
counterparts have similar phase portraits.

**
4. Synchronized population models in
biology****
**
**Full text (PDF)**

Tahmineh Azizi

**
Pages:
41-53**

**
Abstract:
** Chaotic behaviors and
synchronized cycles of population models has been investigated by many
researchers in di erent areas. In this paper, we use a recently
developed method to synchronize discrete-time dynamical system. Using
this coupling approach, we are able to nd a threshold to completely
synchronize a dynamical system and we will apply this result on a
quadratic population model. This model reveals di erent types of
dynamics depending on parameter values from stable equilibrium to
periodic behavior and chaos. This rigorous method helpes us to
synchronize successfully the chaotic attractors of the original system
and its coupled one. Finally, we will use different numerical tools such
as the mean phase difference and the mean amplitude difference, time
series and bifurcation diagram for different threshold and parameter
values to test the analytic results.

**
5. On some universal dynamics of a 2-D
Hénon-like mapping with an unknown bounded function****
**
**Full text (PDF)**

Zeraoulia Elhadj

**
Pages:
54-63**

**
Abstract:
** This paper investigates the
dynamics of a 2-D H enon-like mapping with an unknown bounded function.
The values of parameters and the range of initial conditions for which
the dynamics of this equation is bounded or unbounded are rigorously
derived. The results given here are universal and do not depend on the
expression of the nonlinearity in the considered map.