Volterra Mapping Research Articles

Integrable maps in 4D and modified Volterra lattices

In recent work, we presented the construction of a family of difference equations associated with the Stieltjes continued fraction expansion of a certain function on a hyperelliptic curve of genus $g$. As well as proving that each such discrete system is an integrable map in the Liouville sense, we also showed it to be an algebraic completely integrable system. In the discrete setting, the latter means that the generic level set of the invariants is an affine part of an abelian variety, in this case the Jacobian of the hyperelliptic curve, and each iteration of the map corresponds to a translation by a fixed vector on the Jacobian. In addition, we demonstrated that, by combining the discrete integrable dynamics with the flow of one of the commuting Hamiltonian vector fields, these maps provide genus $g$ algebro-geometric solutions of the infinite Volterra lattice, which justified naming them Volterra maps, denoted ${\cal V}_g$. The original motivation behind our work was the fact that, in the particular case $g=2$, we could recover an example of an integrable symplectic map in four dimensions found by Gubbiotti, Joshi, Tran and Viallet, who classified birational maps in 4D admitting two invariants (first integrals) with a particular degree structure, by considering recurrences of fourth order with a certain symmetry. Hence, in this particular case, the map ${\cal V}_2$ yields genus two solutions of the Volterra lattice. The purpose of this note is to point out how two of the other 4D integrable maps obtained in the classification of Gubbiotti et al. correspond to genus two solutions of two different forms of the modified Volterra lattice, being related via a Miura-type transformation to the $g=2$ Volterra map ${\cal V}_2$. We dedicate this work to a dear friend and colleague, Decio Levi.

Secure access microcontroller system based on fingerprint template with hyperchaotic encryption

Authentication systems based on biometric and microcontroller systems are used around the world to control the access of users in restricted sites such as in offices, banks, or hospitals. Moderns authentication systems use cloud connectivity using the Internet. Nevertheless, biometric data are considered confidential, which is susceptible to hacker attacks with the aim of fraud, extortion or identity theft. In this work, we propose a low-cost secure control access microcontroller system that uses the Secure Hash Algorithm 1 (SHA-1) and hyperchaotic encryption of fingerprint template to increase the security against cybernetic attacks. The main findings can be briefly described as (1) chaotic encryption of the fingerprint template provides strong security by using the hyperchaotic 2D Lotka–Volterra map with just one round of diffusion–confusion; (2) the fingerprint template injected to SHA-1 increases the robustness against differential attack by combining a unique 160-bit hash key with the personal key; (3) the authentication system implemented in a low-cost embedded system based on 32-bit microcontroller provides high security at low cost; (4) several security analysis of data generated in the embedded authentication system validates the effectiveness of proposed scheme. Based on the high security level achieved in proposed authentication scheme, the system provides double padlock in secure access biometric systems providing robustness against malicious attacks, avoid identity theft and the unauthorized access in highly restricted areas.

Interior periodic points of a Lotka–Volterra map

Given the plane triangle and the transformation , we prove the existence of interior periodic points of periods . One of the periodic orbits of period 6 is given explicitly. We also prove that for any lower periodic saddle point, there is an interior periodic point with the same itinerary (with respect to the natural decomposition of Δ given by the vertical middle line).

Transitivity of a Lotka-Volterra map

The dynamics of the transformation $F: (x,y)\rightarrow (x(4-x-y),xy)$ defined on the plane triangle $\Delta$ of vertices $(0,0)$, $(0,4)$ and $(4,0)$ plays an important role in the behaviour of the Lotka--Volterra map. In 1993, A. N. SharkovskiĬ (Proc. Oberwolfach 20/1993) stated some problems on it, in particular a question about the trasitivity of $F$ was posed. The main aim of this paper is to prove that for every non--empty open set $\mathcal{U} \subset \Delta$ there is an integer $n_{0}$ such that for each $n>n_{0}$ it is $F^{n}(\mathcal{U}) \supseteq \Delta \setminus P_{\varepsilon}$, where $P_{\varepsilon} = \{ (x,y) \in D : y<\beta, \mbox{ $where$ F(t,\varepsilon)=(\alpha,\beta) \mbox{ and } t \in[0,2] \}$ and $\varepsilon \rightarrow 0$ for $n \rightarrow \infty$. Consequently, we show that the map $F$ is transitive, it is not topologically exact and it is almost topologically exact. Additionally, we prove that the union of all preimages of the point $(1,2)$ is a dense subset of $\Delta$.

Dynamics of a Lotka–Volterra map

Dynamics of a Lotka–Volterra map

On the exponentially self-regulating population model

A new type of discrete dynamical systems model for populations, called an exponentially self-regulating (ESR) map, is introduced and analyzed in considerable detail for the case of two competing species. The ESR model exhibits many dynamical features consistent with the observed interactions of populations and subsumes some of the most successful discrete biological models that have been studied in the literature. For example, the well-known Tribolium model is an ESR map. It is shown that in addition to logistic dynamics – ranging from the very simple to manifestly chaotic one-dimensional regimes – the ESR model exhibits, for some parameter values, its own brands of bifurcation and chaos that are essentially two-dimensional in nature. In particular, it is proved that ESR systems have twisted horseshoe with bending tail dynamics associated to an essentially global strange attractor for certain parameter ranges. The existence of a global strange attractor makes the ESR map more plausible as a model for actual populations than several other extant models, including the Lotka–Volterra map.

Operator Constructions and Volterra Mappings in Differential Games

It is proved that B. N. Pshenichnyi's e-strategies and strategies based on Volterra mappings are equivalent. It is shown that Volterra mapping sought for is a boundary for some directednesses of prelimit mappings.

Bifurcations and Transitions to Chaos in the Three-Dimensional Lotka–Volterra Map

The three-dimensional cyclically invariant Lotka–Volterra map is studied, pointing out, by analytic and numeric techniques, the bifurcations of fixed points that have no counterpart in the corresponding system of ordinary differential equations. In particular, the Hopf-bifurcation of the three-population equilibrium point, and the related dynamics in a bounded invariant domain, are analysed. The analytic description of stable periodic orbits in a weak-resonant case is given, and the transitions to chaotic attractors via sequences of Hopf-bifurcations and period-doublings are discussed.