Research On The Fast And Slow Effects As Well As Their Mechanism Under Co-dimension Three Triple Zero Bifurcations

In natural science and engineering,many nonlinear systems are composed of multiple subsystems that operate on different time and space scales.The evolution process from the behavior of individual subsystems to the behavior of the entire system often involves fast and slow coupling bursting oscillations.Scholars have researched these oscillations and found that they exhibit complex bifurcation characteristics under high codimensional bifurcation.This can result in the simultaneous existence of different bifurcation characteristics in the same burster.While most of the work has focused on analyzing the codimension-1 bifurcation mechanism,the corresponding silent state and excited state forms are also relatively single.Hence,much is still to be explored about the generation mechanism and forms of bursting oscillation.This paper explores the bursting oscillations of fast and slow systems under the condition of a triple zero bifurcation singularity of codimension three and their generation mechanism.Using the canonical type of second-order truncation as an example,we derive the universal unfolding form of the canonical type by perturbing on constant and linear terms,due to its Jacobian matrix singularity.Employing the fast and slow analysis method,we study the codimension-1bifurcation in the fast subsystem and calculate corresponding bifurcation conditions.We then discuss two types of codimension-2 bifurcation in the plane of unfolding parameters that may lead to complex bursting oscillations.We find that a class of bursting oscillations with quasiperiodic characteristics induced by codimension-2 fold-Hopf bifurcation at a single equilibrium point evolves into the Torus-Torus type with increased excitation amplitude.By overlapping the equilibrium branches and transformed phase portrait(TPP),we determine the bifurcation structures of several bursters.Finally,we discuss a class of chaotic bursting phenomena induced by the period-doubling cascade.Next,we analyzed the simplest normal form(SNF)with third-order truncation and external excitation.By using the nonlinear transformation coefficient that wasn’t used in the process of calculating the canonical form,we can eliminate the cubic term coefficient and obtain the SNF truncated to the cubic term.The external excitation is seen as a slow-changing unfolding parameter that perturbs the constant term.We will give the dynamic behavior of all possible bursting oscillation regions in the unfolding parameter plane.We’ll also discuss the complex dynamic behavior of perturbation parameters passing through different dynamic regions in the plane of unfolding parameters.This includes high codimensional bifurcation and global bifurcation of the system at the equilibrium point,which will ultimately lead to the form of bursting oscillation of the system from simple to complex.Lastly,we will analyze the generation mechanism of bursting oscillation in four cases by overlapping the TPP and the equilibrium branch.At last,an analysis was conducted on a parametric perturbation model of a double pendulum system with a codimension three triple zero bifurcation singularity.Using the fast and slow analysis method,the model was considered a generalized autonomous system,taking into account the gap between the excitation frequency and the system’s natural frequency.The periodic excitation acted as a slow-varying parameter that perturbed the linear term.As the parameter trajectories crossed through different dynamic regions in the two-parameter plane,complex dynamic behaviors occurred,including high co-dimensional bifurcation and bursting oscillation patterns ranging from simple to complex.The induction mechanism of four types of bursters was analyzed by overlapping TPP and equilibrium branches.Furthermore,two evolutionary processes resulting from the breaking of symmetry structures leading to chaotic bursting were discussed.Studying related work has both scientific and practical benefits.It helps us gain a deeper understanding of new bursting phenomena and strengthens the application of basic disciplines like mechanics and mathematics in engineering.Additionally,it promotes cross-integration between different fields.