arXiv subjects

V. V. M. S. Chandramouli

Publications and source records attributed to V. V. M. S. Chandramouli.

5 recordsLinked to original sources

arXiv2026

On the horizon problem over the product of self-similar fractals

In this article, we first investigate the box dimension results for the graphs of continuous functions defined on the product of two self-similar fractals ({\mathfrak{F}}s), where the fractal {\mathfrak{F}} satisfies the open set condition. Following this, we manifest the existence of a prevalent surface, defined by a continuous function over the product of two {\mathfrak{F}}s, that satisfies the horizon property (associated with the box dimension), i.e., the box dimension of the prevalent surface is greater than that of its horizon up to a constant η, where η is the box dimension of the self-similar fractal {\mathfrak{F}}.

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arXiv2026

Dynamical analysis of r-Chialvo neuron map with cosine memristive

In this work, we construct a novel two-dimensional discrete neuron map by incorporating a cosine-based memristor into the reduced Chialvo neuron map to examine the dynamical analysis of electromagnetic modulation. The nonlinear current-voltage characteristics of the memristor enrich the neuron map's behavior, leading to diverse firing regimes, stability behaviors, and chaotic attractors. This study begins to establish the equilibrium points using both analytical and numerical methods. Additionally, we determine the conditions on parameters under which the proposed map exhibits a Neimark-Sacker bifurcation. Further, the numerical study reveals the antimonotonicity structure through the forward and backward bifurcation diagrams. The model exhibits a wide range of codimension-one and codimension-two bifurcation patterns, including Neimark-Sacker, period-doubling, saddle-node, generalized period-doubling, cusp-point, fold-flip, and various resonance structures (1:1, 1:2, 1:3, and 1:4). We also observe that the coexistence of multistable attractors including a stable limit cycle, a period-five attractor, and a chaotic attractor, along with their respective basins of attraction. Furthermore, we extend this analysis to the network of neurons under the ring-star configuration and discuss several spatiotemporal patterns. This network investigation reveals complex collective patterns, including imperfect synchronization, clustered patterns, and multi-chimera state phenomena, which have not been previously observed in existing Chialvo-based studies. These results highlight the potential of the discrete memristor-based neuron map for advancing theoretical neurodynamics and offer a robust framework for investigating low-dimensional yet dynamically rich neuron systems.

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arXiv2020

Renormalization of symmetric bimodal maps with low smoothness

This paper deals with the renormalization of symmetric bimodal maps with low smoothness. We prove the existence of the renormalization fixed point in the space $C^{1+Lip}$ symmetric bimodal maps. Moreover, we show that the topological entropy of the renormalization operator defined on the space of $C^{ 1+Lip}$ symmetric bimodal maps is infinite. Further we prove the existence of a continuum of fixed points of renormalization. Consequently, this proves the non-rigidity of the renormalization of symmetric bimodal maps.

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arXiv2020

Period tripling and quintupling renormalizations below $C^2$ space

In this paper, we explore the period tripling and period quintupling renormalizations below $C^2$ class of unimodal maps. We show that for a given proper scaling data there exists a renormalization fixed point on the space of piece-wise affine maps which are infinitely renormalizable. Furthermore, we show that this renormalization fixed point is extended to a $C^{1+Lip}$ unimodal map, considering the period tripling and period quintupling combinatorics. Moreover, we show that there exists a continuum of fixed points of renormalizations by considering a small variation on the scaling data. Finally, this leads to the fact that the tripling and quintupling renormalizations acting on the space of $C^{1+Lip}$ unimodal maps have unbounded topological entropy.

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arXiv2007

Chaotic Period Doubling

The period doubling renormalization operator was introduced by M. Feigenbaum and by P. Coullet and C. Tresser in the nineteen-seventieth to study the asymptotic small scale geometry of the attractor of one-dimensional systems which are at the transition from simple to chaotic dynamics. This geometry turns out to not depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point which is also hyperbolic among generic smooth enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that in the space of $C^{2+α}$ unimodal maps, for $α$ close to one, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main results states that in the space of $C^2$ unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to get a priori bounds. In this smoother class, called $C^{2+|\cdot|}$ the failure of hyperbolicity is tamer than in $C^2$. Things get much worse with just a bit less of smoothness than $C^2$ as then even the uniqueness is lost and other asymptotic behavior become possible. We show that the period doubling renormalization operator acting on the space of $C^{1+Lip}$ unimodal maps has infinite topological entropy.

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