Publications

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How do you compute Common-BOA (C-BOA) for Nonlinear Dynamical Systems with Multiple Equilibria?

Paper Title: 2-Contraction and Poincaré-Index Theory-Based Framework for Stability Analysis of Nonlinear Dynamical Systems with Multiple Equilibria

Abstract: This work leverages the 2-contraction theory, an extension of classical contraction theory, to develop a systematic stability framework for systems with multiple equilibria. Coupled with powerful geometric tools such as Poincaré index theory, the 2-contraction theory enables the stability analysis of planar nonlinear systems instead of undertaking it locally around equilibrium points. Using index theory and 2-contraction, we characterize the nature of equilibrium points and delineate regions in R2 where periodic solutions, closed orbits, or stable solutions may exist. A key focus of this work is the identification of regions in the state space where periodic solutions may exist and the regions that guarantee the non-existence of such solutions. Additionally, we address a crucial problem: determining the basin of attraction (BOA) for stable equilibrium points. For systems with multiple equilibria, identifying candidate BOAs is highly nontrivial. We propose a novel methodology adopting the 2-contraction theory to approximate a common BOA for a class of nonlinear systems with multiple equilibria. Theoretical findings are substantiated through benchmark examples and numerical simulations, demonstrating the efficacy of the proposed approach. Furthermore, we extend our framework to analyze networked systems, showcasing its efficacy in an opinion dynamics problem.

Authors: Riddhi Mohan Bora, Bhabani Shankar Dey, Indra Narayan Kar

Journal: International Journal of Control Automation and Systems

Publication Details: https://doi.org/10.1007/s12555-026-00148-w

C-BOA for a planar system with two stable and one saddle equilibrium points

C-BOA for a planar Opinion Dynamics model of interaction between 2-agents

How do you initialize in Iterative Algorithms associated with a non-convex optimization problem?

Paper Title: A 2-Contraction Framework for Initialization Analysis in Non-Convex Optimization

Abstract: Non-convex optimization problems often exhibit multiple local minima, maxima, and saddle points, making gradient-based methods sensitive to initialization. This paper applies the 2-contraction theory to the gradient-flow dynamics induced by a continuously differentiable non-convex objective function. Using Hessian spectral properties, we characterize the 2-contraction region and remove the saddle region to obtain a candidate region for stable equilibria. A forward-invariant sublevel-set condition is then imposed to construct a certified initialization set. The resulting method provides a-priori initialization guidance for standard gradient descent without modifying its update rule. While focusing on low-dimensional, continuous-time problems, this work also addresses scalability issues and discrete-time implementation for completeness.

Authors: Riddhi Mohan Bora, Bhabani Shankar Dey, Indra Narayan Kar

Conference: IFAC World Congress - 23rd WC 2026

Location: Busan, Republic of Korea

Publication Details: Accepted for Publication

Landscape of a two-variable non-convex objective function and its corresponding phase space in respective gradient flow dynamics

The certified initialization region for a benchmark non-convex optimization problem

Output Regulation problem using I/O Feedback Linearization with MULTI-STABLE ZERO DYNAMICS?

Paper Title: Input-Output Feedback Linearization: Case Study on 2-Contractive Zero Dynamics

Abstract: Zero dynamics are essential to the design and analysis of input-output feedback linearization control, particularly in nonlinear systems. This study revisits an input-output feedback linearization problem through an illustrative example where the zero dynamics admit multiple equilibrium points. The effect of such zero dynamics on system performance and control input has also been investigated. Conventional tools such as 1-contraction and Lyapunov theory are not directly applicable to the analysis of zero dynamics with multiple equilibria. Therefore, the theory of 2-contraction is employed to perform the analysis effectively. The impact of such zero dynamics on the external control input, tasked with regulating the system's output, is also analyzed. Extensive numerical simulations are conducted to validate and exemplify the theoretical results.

Authors: Riddhi Mohan Bora, Indra Narayan Kar

Conference: 19th IEEE International Conference on Control & Automation (ICCA)

Location: Tallinn, Estonia

Publication Details: https://doi.org/10.1109/ICCA65672.2025.11129836

Control profile when the zero dynamics has multi-stability is not 2-contractive

Control profile when the zero dynamics has multi-stability is 2-contractive

How to synchronize two Non-Identical Chaotic systems, both having different dimensions?

Paper Title: Reduced Order Synchronization of Two Non-Identical Special Classes of Strict Feedback Systems Via Back-Stepping Control Technique

Abstract: This work offers a systematic technique to achieve ROS (Reduced Order Synchronization) between two different-order general classes of chaotic systems in a master-slave configuration. In this study, the dynamics of the master and slave systems are assumed to follow a special class of strict-feedback form, namely the Generalized Triangular Feedback form. The main objective is to design a suitable scalar controller using a Lyapunov theory-based back-stepping approach such that the mth-order slave system gets synchronized with the nth order master system. Due to the difference in the order of the systems (m < n and m = (n-1)), it is only possible to achieve synchronization between m states of the slave systems with (n-1) states of the master system, respectively. We can not conclude on the stability of the nth state of the master system, as there is no counterpart (state) available in the slave system to be synchronized with. Adding the (m+1)th state dynamics along with a nonlinear feedback controller (U1) to the slave system ensures that the nth state of the master system is synchronized with the (m+1)th state dynamics of the slave system. With the suggested technique proposed in this article, complete state-to-state synchronization can be achieved with only two controllers. The analytical results are successfully validated through numerical simulations presented at the end.

Authors: Riddhi Mohan Bora, Bharat Bhushan Sharma

Journal: Journal of Computational and Nonlinear Dynamics (ASME)

Publication Details: 10.1115/1.4063721

Some glimpses of synchronized states of the Hyper-Jerk and the Genesio-Tesi systems

Some glimpses of synchronized states and error dynamics of the Hyper-Jerk and Genesio-Tesi systems

How to synchronize two Chaotic systems in the presence of Time-Delays?

Paper Title: Investigation on Synchronization of Two Identical Classes of Chaotic Systems Using Back-Stepping Control Technique in the Presence of Time Delays

Abstract: Time delay analysis is very crucial when it comes to the synchronization of chaotic systems. All practical dynamical systems are described by differential equations; the presence of time delay in dynamics becomes a pivotal part of the system analysis. So, in this article, an attempt has been made to study the synchronization problem of two identical, same-ordered time-delayed chaotic systems, in strict-feedback form as described in (2) and (3) in master andslave configuration. Unlike other conventional nonlinear control techniques, the back-stepping control design strategy is used to serve the purpose of synchronization. The main advantage of using the back-stepping control technique is that it utilizes only a single scalar controller to attain synchronization among different chaotic systems. A generalized control function u ∈ R is designed for the same. For verification of the theoretical results, the 3D time-delayed Genesio-Tesi chaotic system is considered both a master and slave system. Simulation results are provided to support the proposition.

Authors: Riddhi Mohan Bora, Bharat Bhushan Sharma

Conference: 9th 2023 International Conference on Control, Decision and Information Technologies (CoDIT 2023)

Location: Rome, Italy

Publication Details: https://doi.org/10.1109/CoDIT58514.2023.10284061

Phase portrait of Time-Delayed Chaotic System

Glimpses of un-synchronized error dynamics

Is there an adaptive way to synchronize two different order Chaotic systems?

Paper Title: LMI-Based Adaptive Robust Control Scheme for Reduced-Order Synchronization (ROS) for a Class of Chaotic Systems

Abstract: In this paper, we tried to achieve synchronization between two chaotic systems of different orders in a master-slave configuration. We mainly focused on the reduced-order synchronization case. Achieving synchronization of two non-identical non-linear systems is often a very challenging problem. This paper proposes an adaptive law while utilizing the linear matrix inequality (LMI) framework so as to identify the conditions to attain the synchronization goal. All natural systems involve uncertainty and are exposed to external disturbances in real practice. Here, a detailed analysis to tackle such dynamical systems that contain such effects is performed. Using Lyapunov stability theory, the asymptotic convergent behavior of the error dynamics between master and slave systems of the proposed class under the control effect is proved. The efficacy of the proposed control scheme is tested on different chaotic systems, provided as examples. Finally, detailed simulation results are provided at the end of the article to illustrate the effectiveness of the proposed work.

Authors: Riddhi Mohan Bora, Bharat Bhushan Sharma

Conference: 7th International Conference on Advances in Control and Optimization of Dynamical Systems (ACODS, 22)

Location: Assam, India

Publication Details: https://doi.org/10.1016/j.ifacol.2022.04.042

Synchronized error dynamics under the effect of Noise

Glimpses of synchronized error dynamics under the effect of Adaptive Control Law

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