Limit cycles in a quadratic discrete iteration

Applied Mathematics and Computation, 2006

We show that the p-periodic logistic equation x n+1 = l n mod p x n (1 À x n ) has cycles (periodic solutions) of minimal periods 1, p, 2p, 3p, . . . Then we extend Singer's theorem to periodic difference equations, and use it to show the p-periodic logistic equation has at most p stable cycles. Also, we present computational methods investigating the stable cycles in case p = 2 and 3.

Applied Mathematics Letters, 2003

In this paper, we consider a discrete logistic equation x(n + 1) : x(n)exp Jr(n)(1x(n) K(n) ] J ' where {r(n)} and {K(n)} are positive w-periodic sequences. Sufficient conditions are obtained for the existence of a positive and globally asymptotically stable w-periodic solution. Counterexamples are given to illustrate that the conclusions in [1] are incorrect. (~

International Journal of Computer Applications, 2013

In this paper, we study basic dynamical facts for logistic growth models in population dynamics and its dynamical behavior. Different logistic growth curves have been developed and more general biological logistic growth curve are studied. We also discuss the concept of bifurcation in the context of logistic growth models.

Acta Physica Polonica A, 2015

Motivated by a possibility to optimize modelling of the population evolution we postulate a generalization of the well-know logistic map. Generalized difference equation reads: x n+1 = rx p n (1 − x q n), (1) x ∈ [0, 1], (p, q) > 0, n = 0, 1, 2, ..., where the two new parameters p and q may assume any positive values. The standard logistic map thus corresponds to the case p = q = 1. For such a generalized equation we illustrate the character of the transition from regularity to chaos as a function of r for the whole spectrum of p and q parameters. As an example we consider the case for p = 1 and q = 2 both in the periodic and chaotic regime. We focus on the character of the corresponding bifurcation sequence and on the quantitative nature of the resulting attractor as well as its universal attribute (Feigenbaum constant).

Electronic Journal of Qualitative Theory of Differential Equations, 2015

On infinitesimally short time intervals various processes contributing to population change tend to operate independently so that we can simply add their contributions (Metz and Diekmann, The dynamics of physiologically structured populations, 1986, p. 3). This is one of the cornerstones for differential equations modeling in general. Complicated models for processes interacting in a complex manner may be built up, and not only in population dynamics. The principle holds as long as the various contributions are taken into account exactly. In this paper we discuss commonly used approximations that may lead to non-removable dependency terms potentially affecting the long run qualitative behavior of the involved equations. We prove that these terms do not produce such effects in the simplest and most interesting biological case, but the general case is left open. Our main result is a rather complete analysis of an important limiting case. Once complete knowledge of the qualitative properties of simple models is obtained, it greatly facilitates further studies of more complex models. A consequence of our analysis is that standard methods can be applied. However, the application of those methods is far from straightforward and require non-trivial estimates in order to make them valid for all values of the parameters. We focus on making these proofs as elementary as possible.

Applied Mathematics Letters, 1996

2009

In this paper, using the global bifurcation theory, we complete the qualitative analysis of a quartic dynamical system which models the dynamics of the populations of predators and their prey that use the group defense strategy in a given ecological system. In particular, we prove that such a system can have at most two limit cycles.

International Journal of Theoretical Physics, 1997

The relation between chaotic behavior and complexity for one-dimensional maps is discussed. The one-dimensional maps are mapped into a binary string via symbolic dynamics in order to evaluate the complexity. We apply the complexity measure of Lempel and Ziv to these binary strings. To characterize the chaotic behavior, we calculate the Liapunov exponent. We show that the exact normalized complexity for the logistic map~ [0, l] ~ [0, l],f(x) = 4x(l-x) is given by 1. Since the discovery of chaotic attractors, chaos has become an important concept in nearly all branches of the natural sciences. The difference and differential equations which are believed to govern our natural world and may exhibit chaotic attractors are widely discussed in literature (see, for example, Steeb 1992a,b, 1996). The simplest systems showing chaotic behavior are one-dimensional maps J2 1-~/, where I is an interval. A definition for chaos is as follows: Let X be a set. The mapping g: X-~ X is said to be chaotic on X if (1) g has sensitive dependence on initial conditions, (2) g is topological transitive, and (3) periodic points are dense in X. Here we use the Liapunov exponent to characterize chaos. The exponent measures the sensitive dependence on initial conditions. Many different definitions of complexity have been proposed in the literature. Among them are: algorithmic complexity (Kolmogorov-Chaitin)

On infinitesimally short time interval various processes contributing to population change tend to operate independently so that we can simply add their contributions (Metz and Diekmann (1986)). This is one of the cornerstones for differential equations modeling in general. Complicated models for processes interacting in a complex manner may be built up, and not only in population dynamics. The principle holds as long as the various contributions are taken into account exactly. In this paper we discuss commonly used approximations that may lead to dependency terms affecting the long run qualitative behavior of the involved equations. We prove that these terms do not produce such effects in the simplest and most interesting biological case, but the general case is left open.

International Journal of Mathematics and Mathematical Sciences

In this paper, we employ the logistic map and the cubic map to locate the relaxation and the convergence to the periodic fixed point of a system, specifically, the period—1 fixed point. The study has shown that the period—1 fixed point of a logistic map as a recurrence has its convergence at a transcritical bifurcation having its power-law fit with exponent β = − 1 when α = 1 and μ = 0 . The cubic map shows its convergence to the fixed point at a pitchfork bifurcation decaying at a power law with exponent β = − 1 / 2 α = 1 and μ = 0 . However, the system shows their relaxation time at the same power law with exponents and z = − 1 .