arXiv subjects

Sachin Bhalekar

Publications and source records attributed to Sachin Bhalekar.

34 records · Page 2Linked to original sources

arXiv2022

Study of Low-dimensional Nonlinear Fractional Difference Equations of Complex Order

We study the fractional maps of complex order, $α_0e^{i r π/2}$ for $0<α_0<1$ and $0\le r<1$ in 1 and 2 dimensions. In two dimensions, we study H{é}non and Lozi map and in $1d$, we study logistic, tent, Gauss, circle, and Bernoulli maps. The generalization in $2d$ can be done in two different ways which are not equivalent for fractional-order and lead to different bifurcation diagrams. We observed that the smooth maps such as logistic, Gauss, and H{é}non maps do not show chaos while discontinuous maps such as Lozi, Bernoulli, and circle maps show chaos. The tent map is continuous but not differentiable and it shows chaos as well. In $2d$, we find that the complex fractional-order maps that show chaos also show multistability. Thus, it can be inferred that the smooth maps of complex fractional-order tend to show more regular behavior than the discontinuous or non-differentiable maps.

nlin.CD↗

arXiv2022

Can a Fractional Order Delay Differential Equation be Chaotic Whose Integer-Order Counterpart is Stable?

For the fractional order systems \[D^αx(t)=f(x),\quad 0<α\leq 1,\] one can have a critical value of $α$ viz $α_*$ such that the system is stable for $0<α<α_*$ and unstable for $α_*<α\leq 1$. In general, if such system is stable for some $α_0\in(0,1)$ then it remains stable for all $α<α_0.$ In this paper, we show that there are some delay differential equations \[D^αx(t)=f(x(t),x(t-τ))\] of the fractional order which behave in an exactly opposite way. These systems are unstable for higher values of fractional order and stable for the lower values. The striking observation is the example which is chaotic for $α=0.27$ but stable for $α=1$. This cannot be observed in the fractional differential equations (FDEs) without delay. We provide the complete bifurcation scenarios in the scalar FDEs.

math.DS↗

arXiv2022

Stability and Bifurcation Analysis of a Fractional Order Delay Differential Equation Involving Cubic Nonlinearity

Fractional derivative and delay are important tools in modeling memory properties in the natural system. This work deals with the stability analysis of a fractional order delay differential equation \begin{equation*} D^αx(t)=δx(t-τ)-εx(t-τ)^3-px(t)^2+q x(t). \end{equation*} We provide linearization of this system in a neighbourhood of equilibrium points and propose linearized stability conditions. To discuss the stability of equilibrium points, we propose various conditions on the parameters $δ$, $ε$, $p$, $q$ and $τ$. Even though there are five parameters involved in the system, we are able to provide the stable region sketch in the $qδ-$plane for any positive $ε$ and $p$. This provides the complete analysis of stability of the system. Further, we investigate chaos in the proposed model. This system exhibits chaos for a wide range of delay parameter.

math.DS↗

arXiv2022

Stability analysis of fixed point of fractional-order coupled map lattices

We study the stability of synchronized fixed-point state for linear fractional-order coupled map lattice(CML). We observe that the eigenvalues of the connectivity matrix determine the stability as for integer-order CML. These eigenvalues can be determined exactly in certain cases. We find exact bounds in one-dimensional lattice with translationally invariant coupling using the theory of circulant matrices. This can be extended to any finite dimension. Similar analysis can be carried out for the synchronized fixed point of nonlinear coupled fractional maps where eigenvalues of the Jacobian matrix play the same role. The analysis is generic and demonstrates that the eigenvalues of connectivity matrix play a pivotal role in stability analysis of synchronized fixed point even in coupled fractional maps.

math.DS↗

arXiv2021

Stability and Dynamics of Complex Order Fractional Difference Equations

We extend the definition of $n$-dimensional difference equations to complex order $α\in \mathbb{C} $. We investigate the stability of linear systems defined by an $n$-dimensional matrix $A$ and derive conditions for the stability of equilibrium points for linear systems. For the one-dimensional case where $A =λ\in \mathbb {C}$, we find that the stability region, if any is enclosed by a boundary curve and we obtain a parametric equation for the same. Furthermore, we find that there is no stable region if this parametric curve is self-intersecting. Even for $ λ\in \mathbb{R} $, the solutions can be complex and dynamics in one-dimension is richer than the case for $ α\in \mathbb{R} $. These results can be extended to $n$-dimensions. For nonlinear systems, we observe that the stability of the linearized system determines the stability of the equilibrium point.

math.DS↗

arXiv2020

Modified DJ method: Application to Boussinesq equation

In this paper we present a modification of DJ Method [J. Math. Anal. Appl. 316 (2006), 753-763] to solve the nonlinear equations more efficiently. It is observed that the modified DJ method is faster and hence it has accelerated convergence rate as compared to the old one. We use this new method to find the analytical solutions of Boussinesq equation. The reported results are compared with the exact solutions. Further, we compare the absolute error in our solution with those in other iterative methods. It is observed that the presented method is simple and generates more accurate solutions as compared with other methods.

math.AP↗

arXiv2020

A new fractional order chaotic dynamical system and its synchronization using optimal control

In this work, we introduce a new three-dimensional chaotic differential dynamical system. We find equilibrium points of this system and provide the stability conditions for various fractional orders. Numerical simulations will be used to investigate the chaos in the proposed system. A simple linear control will be used to control the chaotic oscillations. Further, we propose an optimal control which is based on the fractional order of the system and use it to synchronize new chaotic system.

nlin.CD↗

arXiv2019

Nonexistence of invariant manifolds in fractional order dynamical systems

Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in planar polynomial systems. We provide the conditions for the invariance of linear subspaces in fractional order systems. Further, we provide an important result showing the nonexistence of invariant manifolds (other than linear subspaces) in fractional order systems.

math.DS↗

arXiv2019

Hybrid functions approach to solve a class of Fredholm and Volterra integro-differential equations

In this paper, we use a numerical method that involves hybrid and block-pulse functions to approximate solutions of systems of a class of Fredholm and Volterra integro-differential equations. The key point is to derive a new approximation for the derivatives of the solutions and then reduce the integro-differential equation to a system of algebraic equations that can be solved using classical methods. Some numerical examples are dedicated for showing efficiency and validity of the method that we introduce.

math.NA↗

arXiv2019

Analysis of solution trajectories of linear fractional order systems

The behavior of solution trajectories usually changes if we replace the classical derivative in a system by a fractional one. In this article, we throw a light on the relation between two trajectories $X(t)$ and $Y(t)$ of such a system, where the initial point $Y(0)$ is at some point $X(t_1)$ of trajectory $X(t)$. In contrast with classical systems, trajectories $X$ and $Y$ do not follow the same path. Further, we provide a Frenet apparatus of both trajectories in various cases and discuss their effect.

math.DS↗

arXiv2019

A new class of special functions arising from the solution of differential equations involving multiple proportional delays

Proportional delay is a particular case of time dependent delay. In this article, we consider differential equations involving multiple delays. The series solution of this equation leads to a class of special functions. This class of special functions is independent from all the existing special functions obtained as a solution of differential equations. We analyze the basic properties of this class and discuss various identities and relations.

math.DS↗

arXiv2019

Can we split fractional derivative while analyzing fractional differential equations?

Fractional derivatives are generalization to classical integer-order derivatives. The rules which are true for classical derivative need not hold for the fractional derivatives, for example, we cannot simply add the fractional orders $α$ and $β$ in ${}_0^{C}\mathrm{D}_t^α{}_0^{C}\mathrm{D}_t^β$ to produce the fractional derivative ${}_0^{C}\mathrm{D}_t^{α+β}$ of order $α+β$, in general. In this article we discuss the details of such compositions and propose the conditions to split a linear fractional differential equation into the systems involving lower order derivatives. Further, we provide some examples, which show that the related results in the literature are sufficient but not necessary conditions.

math.DS↗

arXiv2018

Singular points in the solution trajectories of fractional order dynamical systems

Dynamical systems involving non-local derivative operators are of great importance in Mathematical analysis and applications. This article deals with the dynamics of fractional order systems involving Caputo derivatives. We take a review of the solutions of linear dynamical systems ${}_0^C\mathrm{D}_t^αX(t)=AX(t)$, where the coefficient matrix $A$ is in canonical form. We describe exact solutions for all the cases of canonical forms and sketch phase portraits of planar systems. We discuss the behavior of the trajectories when the eigenvalues $λ$ of $A$ are at the boundary of stable region i.e. $|arg(λ)|=\frac{απ}{2}$. Further, we discuss the existence of singular points in the trajectories of such systems in a region of $\mathbb{C}$ viz. Region II. It is conjectured that there exists singular point in the solution trajectories if and only if $λ\in$ Region II.

math.DS↗

arXiv2018

Analyzing stability of a delay differential equation involving two delays

Analysis of the systems involving delay is a popular topic among applied scientists. In the present work, we analyze the generalized equation $D^α x(t) = g\left(x(t-τ_1), x(t-τ_2)\right)$ involving two delays viz. $τ_1\geq 0$ and $τ_2\geq 0$. We use the the stability conditions to propose the critical values of delays. Using examples, we show that the chaotic oscillations are observed in the unstable region only. We also propose a numerical scheme to solve such equations.

math.CA↗

arXiv2016

Novel Special Function Obtained from a Delay Differential Equation

This paper deals with the series solution of a linear delay differential equation (DDE) y'(x) = ay(x)+ by(q x), 0<q<1 with proportional delay. We discuss the convergence of this novel series. We establish the relation between the special function given in terms of this series and its differentials. We also discuss the bounds on this function and present the relation with other special functions viz. Kummer's functions, generalized Laguerre polynomials, incomplete gamma function, beta function and regularized incomplete beta function. Further, we discuss various properties and contiguous relations for the novel special function. Finally, we generalize this series by solving fractional order DDE and a system of DDE.

math.CA↗

arXiv2016

A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations

In this paper we introduce a numerical method for solving nonlinear Volterra integro-differential equations. In the first step, we apply implicit trapezium rule to discretize the integral in given equation. Further, the Daftardar-Gejji and Jafari technique (DJM) is used to find the unknown term on the right side. We derive existence-uniqueness theorem for such equations by using Lipschitz condition. We further present the error, convergence, stability and bifurcation analysis of the proposed method. We solve various types of equations using this method and compare the error with other numerical methods. It is observed that our method is more efficient than other numerical methods.

math.NA↗