Dynamical System Modeling and Stability Investigation�DSMSI-2025

May 08-10, 2025, Kyiv, Ukraine

Synchronization of Two Coupled Nonlinear Dynamical

Systems

Roman Voliansky, “Igor Sikorsky Kyiv Polytechnic Institute”,

Nina Volianska, Taras Shevchenko National University of Kyiv,

Iurii Shramko,Technical University “Metinvest Polytechnic” LLC,

Andri Pranolo, Universitas Ahmad Dahlan

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Problems to solve

• To perform system modeling in continuous and discrete-time domains,
• To find the system perturbed motion equations.
• To solve the inverse dynamic problem and design closed-loop control system.
• To show the use of our approach by constructing controllers for Duffing pendulum with driven exciter.

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Dynamical System Modeling and Stability Investigation, DSMSI-2025

Model of the Generalized Coupled Dynamical System in Continuous and�Discrete Time Domains�

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Coupled continuous-time dynamical system

Coupled discrete-time dynamical system

Discrete-time derivative approximation

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Perturbed Motion of the Generalized Coupled Interval System�

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Difference between subsystem trajectories and its derivatives

Generalized system state variable

System perturbed motion equations

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Interval Perturbed Motion Model

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Domain where nonlinearity is localized

System interval equation

System interval matrix equation

System state variables

Continuous -time perturbed motion

Discrete -time perturbed motion

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Synchronization Controller Design for the Coupled Exactly-defined Subsystems

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Desired closed-system motion

Desired open-system motion

Single-channel control signal

Dual-channel control signal

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Synchronization Controller Design for the Coupled Interval Subsystems

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Dual-channel interval control

Controller output

Controller output

Interval control signal

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Duffing Pendulum Model

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Duffing pendulum equation

Equation of Duffing pendulum with integrated exciter

Oscillations in Duffing pendulum

Attractor of Duffing pendulum

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Duffing Pendulum with Driven Exciter

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Equation of Duffing pendulum with driven exciter

Oscillations in Duffing pendulum

Attractor of Duffing pendulum

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Duffing Pendulum’s Interval Model

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Piecewise linear interval approximation

of pendulum’s nonlinearity.

Duffing pendulum interval equations

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Simulation Results for Duffing Pendulum’s Interval Model

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Duffing pendulum interval matrix equations

Interval Duffing pendulum position

Interval driven exciter output

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Duffing Pendulum’s Perturbed Motion

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Interval perturbed motion equation

Interval Duffing pendulum

perturbed position

Interval Duffing pendulum

perturbed speed

Interval Duffing pendulum

perturbed acceleration

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Master-Slave Subsystem Synchronization

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Control signal

Control boudaries

Interval Duffing pendulum

and exciter position

Interval Duffing pendulum

exciter controller output

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Dual-channel Synchronization

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Lower and upper boundaries of virtual control

Control signals for each channels

Interval Duffing pendulum

and exciter position

Interval Duffing pendulum

dual-channel controller output

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Conclusion

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• Performed studies prove the possibility of using interval methods to study and design closed-loop control systems to synchronize various parts of coupled systems. Such a synchronization twice reduce coupled system order and according to Poincare theorem about chaotic motions makes system motions regular. �Designed in such a way control systems can be considered as discontinuous control systems with variable minimal and maximal values of produced control signals. �One can define the control signal boundaries by taking into account the system perturbed motion equation and solving on its basis the inverse dynamic problem. �Such a solution has some general patterns which allow to simplify the
• control signal determination.
• We see the future development of our work in applying our approach to high-order chaotic systems for designing based on the second-order sliding mode approach continuous and discrete control systems to perform its synchronization and setup secured data transmission channels in environment with interconnected transmitter and receiver

Dynamical System Modeling and Stability Investigation, DSMSI-2025

Thank you for your attention

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