| dc.description.abstract | Nonsmooth dynamical systems are widely used in many engineering applications. Because of its nonsmooth property, it is very common to observe coexisting attractors in this type of dynamical systems. These coexisting attractors are extremely sensitive to noise due to their fractal basin boundaries. For some certain requirements and application scenarios, some of them are not desireable, which should be avoided. Obviously, achieving the switching among these attractors could offer the dynamical systems more flexibilities. Hence, studying the control of coexisting attractors in nonsmooth dynamical systems is vital. In this thesis, the study focuses on developing new control strategies and computational methods for evaluating the controlling process, and in particular, the near-grazing dynamics of the nonsmooth dynamical systems with and without delay. To be more specific, the key contents of this thesis are summarised as follows:
• Due to the infinite-dimensional nature of dynamical systems with delay, analytical studies of such models are difficult and can provide in general only limited results, in particular when some kind of nonsmooth phenomenon is involved, such as impacts, switches, impulses, etc. Also, there exists so far no dedicated software package to carry out numerical continuation for such type of models. In order to overcome this problem, an approximation scheme for nonsmooth dynamical systems with delay was proposed so that a numerical bifurcation analysis can be allowed via continuation (path-following) methods, using existing numerical packages, such as COCO.
• Lyapunov exponent is a widely used tool for studying dynamical systems. When calculating Lyapunov exponents for piecewise-smooth systems with time-delayed arguments one faces a lack of continuity in the variational problem. In this thesis, the part of our works studies how to build a variational equation for the efficient construction of Jacobians along trajectories of a delay nonsmooth system. Trajectories of a piecewise-smooth system may encounter the so-called grazing event where the trajectory approaches a discontinuity surface in the state space in a non-transversal manner. For this event a grazing point estimation algorithm was developed to ensure the accuracy of trajectories for the nonlinear and the variational equations. Through adopting this algorithm, the eigenvalues of the Jacobian matrix computed by the algorithm converge with an order consistent with the order of the numerical integration method, therefore guaranteeing the reliability of the proposed numerical method.
• For the nonsmooth dynamical systems, the coexisting attractors are widely existing. But this property increases the complexity of the system’s dynamics. For example, the dynamical system with this property can have many different motions under some different initial conditions. It is easy for the system to present some undesired attractors, which should be avoided. In order to suppress the complex dynamics, a delay feedback control was considered for the nonsmooth dynamical systems to achieve the switch from the undesired attractors to the desired one. The efficiency and dynamical property of this control were demonstrated numerically for nonsmooth dynamical systems.
• In order to control coexisting attractors a control strategy for switching stable coexisting attractors of a class of non-autonomous dynamical systems was developed. The central idea is to introduce a continuous path for the system’s trajectory to transition from its original undesired stable attractor to a desired one by varying one of the system parameters according to the information of the desired attractor. The behavior of the control strategy was demonstrated numerically for both nonsmooth and smooth dynamical systems. It was shown that the proposed control concept can be implemented through either using an external control input or by varying a system parameter. | en_GB |
