Conference on Nonlinear Systems & Dynamics IISER Pune, 15-18 December 2022 Fractional Maps of complex fractional order with poles Divya D. Joshi * November 2022 The existence of chaos in nonlinear maps of complex orders has been recently in [1]. Various maps have been studied in this paper. Smooth maps like logistic, Gauss, Dufﬁng, and H´enon maps do not show chaos. While discontinuous maps such as Bernoulli, and circle and non-differentiable map like The tent and Lozi map show chaos. These systems are not differentiable or discontin- uous at some point and we argued that it is necessary that the function is not analytic. The following work can be seen as an extension of this work. In this work, I study a function which has simple poles at some point and shows that these functions show chaos as well. In this context, I study the q-deformed logistic map and ﬁnd that they show chaos for certain deformations. The logistic map is deﬁned as : xn+1 = λxn(1 −xn), where λ is the parameter that lies within the interval [0,4]. As per [1], the logistic map being a smooth map deﬁned by an analytic function, chaos is absent for complex or- der fractional logistic map. I introduce q-deformation in the complex order fractional logistic map. Earlier, q-deformation in nonlinear maps have been studied by Jaganathan and Sinha in [2] where the q-deformation is given as : xdef = x 1−ϵ(x−1), where ϵ is the deformation parameter. The fractional order logistic system with q-deformation can be formulated as : x(t) = x(0) + 1 Γ(α) t X j=1 Γ(t −j + α) Γ(t −j + 1) ×[λxdef(j −1)(1 −xdef(j −1)) −x(j −1)]. (1) Here, α is the complex fractional order, and x(0) is the initial condition. For the purpose of simplicity I set α = α0eirπ/2 with 0 < α0 < 1 and 0 ≤r < 1. In our previous work, we observed that maps that were discontinuous or non-differentiable showed chaos. Here I have a q-deformed fractional logistic map of complex or- der, which has second-order poles which is non-analytic. The existence of chaos, in this case, strengthens our hy- pothesis. Figure(1(a) and 1(b)) shows bifurcation diagram for ϵ = 0 and ϵ = −1.5 respectively. No chaos is seen in the case of ϵ = 0. Chaos exists in the case of ϵ = −1.5. Also the system does not show chaos for the values of ϵ ≥−1. *Divya D. Joshi are with the Department of Physics, Rash- trasant Tukadoji Maharaj Nagpur University, email: divya- joshidj27@gmail.com. 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 R e ( x ( t ) ) l (a) 0 1 2 3 4 - 3 . 0 - 2 . 5 - 2 . 0 - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 R e ( x ( t ) ) l (b) Figure 1: Bifurcation diagram for α0 = 0.7, r = 0.01, x(0) = 0.1 for (a) ϵ = 0 and (b) ϵ = −1.5 A multistable system has coexisting attractors for the same parameters with different initial conditions. Figure(2) shows the existence of multistability for a q- deformed fractional logistic map of complex order. The result suggests a relation between the analyticity of the complex fractional ordered functions and the existence of multistability. 2 . 5 3 . 0 3 . 5 4 . 0 - 3 - 2 - 1 0 1 2 3 x ( 0 ) = 1 8 + 1 9 i x ( 0 ) = 0 . 1 R e ( x ( t ) ) l Figure 2: Multistability observed for α0 = 0.7, r = 0.01, ϵ = −1.5 with different initial conditions. While concluding, I note the following observations made in this work. Existence of chaos for the q-deformed logistic map of complex fractional order. In this case, the system does not show chaos for the values of ϵ ≥−1. The system shows multistability. References [1] Divya D Joshi, Prashant M Gade, and Sachin Bhalekar. Study of low-dimensional nonlinear fractional difference equations of com- plex order. Chaos: An Interdisciplinary Journal of Nonlinear Sci- ence, 32(11):113101, 2022. [2] Ramaswamy Jaganathan and Sudeshna Sinha. A q-deformed non- linear map. Physics Letters A, 338(3-5):277–287, 2005.