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The universal unfolding of a normal form can be employed to reveal the general behaviors of a specific local bifurcation, while the computation of the normal form for high codimensional bifurcation still remains unsolved. This paper focuses on a vector field with codimension-3 triple Hopf bifurcation. Besides 1:1 internal resonance for two frequencies in semi-simple form, two cases are considered, corresponding to internal resonance and noninternal resonance between the first two frequencies and the third frequency, respectively. Based on a combination of center manifold and normal theory, all the coefficients in the normal form and the nonlinear transformation are derived explicitly in terms of the coefficients of the original vector field. Upon the recursive procedure established, a user friendly computer program can be easily developed using a symbolic computation language Maple to compute the coefficients up to an arbitrary order for a specific vector field with triple Hopf bifurcation. Furthermore, universal unfolding of the normal form is obtained, which can be used to display the topological structure in the neighborhood of bifurcation point. It is pointed out that different choices of the remaining terms in the nonlinear transformation may lead to different expressions of the normal form and the unfolding, which are qualitatively equivalent to each other. [ABSTRACT FROM AUTHOR]

This paper aims to report complex bursting patterns in a parametrically and externally excited Jerk circuit system. Typically, the common compound bursting with two clusters in each period can be observed in this system. By the fast-slow analysis, we find that the active states related to this type of bursting are created by supercritical Hopf bifurcations. Interestingly, with the increase of parametric excitation amplitude, the common bursting pattern evolves into the ones that display complex structures. We show that the parametric excitation amplitude plays a vital role in influencing the dynamical behaviors, which is apparent from the two-parameters analysis. Then, a rich variety of dynamical phenomena including period-doubling and inverse period-doubling cascades in orbits of period 2, 4, and 8 can be observed with the increase of parametric excitation amplitude, thereby leading to complex bursting patterns with multiple-periodically active states. Besides, as a result of the period-doubling cascades, chaotic attractors are eventually discovered, which can be corroborated by the maximal Lyapunov exponent. In particular, periodic windows with period-doubling and inverse period-doubling cascades also emerge in the chaotic region when the parametric excitation amplitude increases to higher values. These give rise to the complex bursting patterns with chaotically active states. Based on this, two types of bursting patterns with complex structures, i.e., bursting patterns with multiple-periodically active states and bursting patterns with chaotically active states are revealed. Our results enrich the routes to complex bursting and deepen the understanding of bursting patterns with complex structures. [ABSTRACT FROM AUTHOR]

For a vector field with Hopf bifurcation at the origin, when periodic excitation is introduced in which the frequency of the excitation is far less than the natural frequency, bursting oscillations, characterized by the alternations between the large-amplitude and the small-amplitude oscillations, can often be observed. The paper tries to answer the common question of how many types of bursting attractors may appear in such a vector field. Based on the normal form of Hopf bifurcation up to the third order, by regarding the whole exciting term as a slow-varying bifurcation parameter, all possible equilibrium branches of the generalized autonomous system are derived, the bifurcation sets of which divide the parameter space into regions associated with different types of dynamical behaviors. It is found that, there exist totally four possible combinations of the regions that the trajectory of the full system may visit as the slow-varying parameter varies. Accordingly, four qualitatively different behaviors, i.e. periodic symmetric oscillations, periodic symmetric mixed-mode oscillations, fold/Hopf/Hopf/fold and fold-Hopf/fold-Hopf bursting attractors, are presented, the mechanism of which is obtained by overlapping the transformed phase portrait and the equilibrium branches in the generalized autonomous system. Furthermore, the dynamics of the generalized autonomous system in different regions in the parameter space may influence the structures of the bursting oscillations when the trajectory enters the corresponding regions, while the bifurcations at the boundaries of the regions may lead to changes between the quiescent states and spiking states in the bursting attractors. [ABSTRACT FROM AUTHOR]

Purpose: This paper primarily investigates five novel compound bursting patterns in the parametrically amplified Mathieu-Duffing nonlinear energy harvesters (PAMDNEH). The five new compound bursting patterns, i.e. compound "subHopf/LPC-delayed subHopf/LPC-subHopf/LPC" bursting, compound "subHopf/LPC-subHopf/LPC" bursting, compound "subHopf/LPC-subHopf/LPC" bursting via "subHopf/subHopf" hysteresis loop, compound "subHopf/supHopf-supHopf/LPC" bursting via "subHopf/subHopf" hysteresis loop and "subHopf/LPC" bursting via "subHopf/subHopf" hysteresis loop, are explored. In addition, two normal bursting patterns, i.e. "subHopf/subHopf" bursting and "fold/fold" bursting, are also investigated. Method: By using the two-bifurcation analysis method, the compound "subHopf/LPC-delayed subHopf/LPC-subHopf/LPC" bursting and its dynamical evolutions is investigated. The correctness of the results is verified by the numerical simulation. Results: We find that the parameter plane can be divided into six different areas according to the types and numbers of the Hopf bifurcation points, and different bifurcation structures and different patterns of the clusters will be created, which lead to different bursting oscillations, including compound bursting types and normal bursting patterns. Because bursting oscillations is created when the variables switch between/among different attractors, the bursting oscillation amplitude is big enough to surmount the potential obstacles of the energy harvesters. Therefore, bursting oscillations may have a very important application in energy harvesting. [ABSTRACT FROM AUTHOR]

In this paper, we present a numeric bifurcation analysis of the normal form of degenerate Hopf bifurcation truncated up to seventh order with an equilibrium point located at the origin. By applying the genericity nondegenerate conditions and normal form theory, we study the bifurcation analysis of the codimension-3 Takens–Hopf bifurcation for the difficult case, where a rich bifurcation scenario is displayed. The third Lyapunov coefficient is used to distinguish the different cases of a codimension-3 Takens–Hopf bifurcation point, which can be efficiently computed with the aid of a software program based on the symbolic package Maple, presented in Appendix A. The normal form analysis results can be used to depict the complete bifurcation diagrams and phase portraits. In order to investigate the mechanism of the transitions between equilibrium and limit cycles, the methods of two scales in frequency domain are employed to study the evolutions. [ABSTRACT FROM AUTHOR]

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations. [ABSTRACT FROM AUTHOR]

In this paper, mixed-mode oscillations and bifurcation mechanism for a Filippov-type system including two time-scales in the frequency domain are demonstrated. According to classic Chua's system, we investigate a non-smooth dynamical system including two time-scales. As there exists an order gap between the exciting frequency and the natural one, the whole external excitation term can be considered as a slow-changing parameter, which results in two smooth subsystems divided by the non-smooth boundary. In addition, the critical condition about fold bifurcation (FB) is studied, and by applying the Hopf bifurcation (HB) theorem, specific formulas for determining the existence of HBs are presented. By introducing an auxiliary parameter via differential inclusions theory, the non-smooth bifurcations on the boundary are discussed. Then, the equilibrium branches and the bifurcations are derived, and two typical cases associated with different bifurcations are considered. In light of the superposition between the bifurcation curve and the transformed phase portrait, the dynamical behaviours of the mixed-mode oscillations as well as sliding movement along the non-smooth boundary are obtained, which reveal the corresponding dynamical mechanism. [ABSTRACT FROM AUTHOR]

This paper proposes a novel three-dimensional chaotic system with multiple coexisting attractors, where different values of a constant control parameter may drive the chaotic behaviors to evolve from single-scroll to double-scroll attractors. When the controlling term is replaced by a periodic harmonic excitation where the exciting frequency is far less than the natural frequency, chaotic movement may disappear, while periodic bursting oscillations will take place. Based on the fact that during a period defined by the natural frequency, the exciting term keeps almost a constant, the whole exciting term can be regarded as a slow-varying parameter resulting in a generalized autonomous system, its equilibrium branches as well as the related bifurcations occurring with the variation of the slow-varying parameter are derived. With the increase of the exciting amplitude, asymmetric and symmetric bursting attractors can be observed, for which the mechanism can be analyzed by the overlap of the equilibrium branches and the transformed phase portraits. With different values of the exciting amplitude corresponding to the change region of the slow-varying parameter, different bifurcations such as fold and Hopf bifurcations may involve the bursting structures, leading to different types of bursting oscillations. Furthermore, the phase space can be divided into two regions by a line boundary because of the symmetry of the vector field. When the trajectory from one region returning to the region arrives at the boundary, two asymmetric bursting attractors located in different regions coexist, which are symmetric to each other. However, when the trajectory passes across the boundary, an enlarged symmetric bursting attractor can be observed, whose trajectory connects the two original asymmetric attractors. Furthermore, it is found that when the trajectory runs along a stable equilibrium branch to the bifurcation point, it may move almost strictly along an unstable equilibrium branch of the fast subsystem because of the delay influence of the bifurcation. [ABSTRACT FROM AUTHOR]

The mechanism of bursting oscillations in high dimensional systems with the coupling of two scales is an open problem in nonlinear dynamics, since there may coexist stable attractors in the fast subsystem with different bifurcations. Here we consider the bursting oscillations in the normal form up to the third order of a vector field with triple Hopf bifurcation at origin. When slow-varying external excitation is introduced, with the increase of the exciting amplitude, Hopf bifurcations in the three sub-planes occur in turn, which leads to three stable limit cycles. The dynamics may evolve from a 2-D torus bursting oscillations with two-mode and three-mode, respectively, the mechanism of which can be presented by overlapping the transformed phase portrait and the equilibrium states as well as the bifurcations of the fast subsystem. Synchronization and non-synchronization between the state variables in different sub-planes can be observed, which is explained by the bifurcations. An interesting phenomenon is found that the three-mode bursting oscillations still behave in quasi-periodic type instead of high dimensional torus. The projections of the trajectory in the sub-planes alternate between quiescence and repetitive spiking state with the oscillating frequency approximated at the frequency of the corresponding limit cycle. • Triple Hopf bifurcation may result in stable limit cycle in sub-planes. • Behavior evolves from single-mode to three-mode bursting attractor. • Synchronization and non-synchronization between the state variables appear. • Non-resonance leads to quasi-periodic bursting oscillations. • Delay effect may cause disappearance of influence of associated bifurcations. [ABSTRACT FROM AUTHOR]

• Bursting oscillations of the vector field with a Double Hopf bifurcation are observed. • Bifurcation mechanism of bursting oscillations is presented. • Different forms of bursting oscillations with both single mode and mixed modes are obtained. Up to now, most of the work related to the slow-fast dynamical systems are based on the low dimensional systems with only codimension-one bifurcations at the transitions between the quiescent states and the spiking states, while there usually only exists one slow variable in the system. Since the coupling effect of two scales on the behaviors of high dimensional systems with two or more slow variables as well as high co-dimensional bifurcations lacks exploration, in this paper, we investigate the slow-fast effect on a 6-dimension system, which can be regarded as the coupling of two subsystems. The fast subsystem is expressed as the normal form of the vector field with a double Hopf bifurcation at the origin, while the slow-varying parameter in the fast subsystem is controlled by the slow subsystem. For the fast subsystem, the trajectory is projected onto two independent sub-planes for better observation, and the movement of bursting oscillators can be synthesized with information from each dimension. In the sub-plane, the mechanism of bursting oscillations is derived by employing the overlap of the transformed phase portrait and the equilibrium branches as well as the bifurcations. It is found that, Hopf bifurcation sets may cause the fast subsystem to transform between the single mode and the mixed-mode or between different single modes, leading to bursting oscillations in certain sub-planes or the whole four dimensions of the fast subsystem. Furthermore, interesting phenomena may be caught, such as the trajectory jumping in the process of oscillation as well as transforming between different oscillation modes of the fast subsystem by the way of jump as a result of the fold bifurcation set. [ABSTRACT FROM AUTHOR]

Slow-fast analysis has been extensively used in the past to study the occurrence of bursting oscillations. Most bursting oscillations studies are performed on the low dimensional autonomous systems where only codimension-1 bifurcations take place at the transitions of quiescent and spiking states. However, in high dimensional slow-fast dynamical systems, there exist higher co-dimensional bifurcations, which may lead to much more complicated mechanisms for bursting oscillations. To reveal and understand the complicated phenomena, the present paper investigates the normal form of a four-dimensional cubic order system with parametric excitation as a slow parameter. Parametric frequency is chosen far less than the natural frequencies to make sure the coupling of fast and slow scales. In the absence of parametric excitation, the system has a double Hopf bifurcation at the origin. Several equilibrium branches and mechanisms of bursting oscillations are obtained for different amplitudes of the parametric excitation. The trajectory of the system is projected onto two independent sub-planes, while the movement of bursting oscillators is synthesized by the properties in each sub-plane. In the sub-plane, the mechanism of bursting oscillations is derived by employing the overlap of the transformed phase portrait and the equilibrium branches as well as the bifurcations. The codimension-2 double Hopf bifurcation may result in the occurrence of the bursting oscillations in four dimensions. When the excitation amplitude is relatively small, the trajectory only follows the equilibrium branches of trivial and single-mode solutions. With the increase of the excitation amplitude, the trajectory switches between the equilibrium branches of single-mode and mixed-mode. Furthermore, a few interesting phenomena are presented. • Bursting oscillations of the vector field with a Double Hopf bifurcation are observed. • Bifurcation mechanism of bursting oscillations is presented. • Different forms of bursting oscillations with both single mode and mixed modes are obtained. [ABSTRACT FROM AUTHOR]

The main purpose of this manuscript is to introduce and compare two methods for discussing the nature of Hopf bifurcation, namely, a research method combining the normal form theory and the center manifold theorem and the other method of multiple scales. Taking a delayed Van der Pol oscillator as an example, the local stability as well as the occurrence of Hopf bifurcation are explored via choosing time delay as the bifurcation parameter. On the basis of two different methods respectively, the characters of Hopf bifurcation which includes the direction of Hopf bifurcation as well as the stability of bifurcating periodic solutions are analyzed. Finally, numerical simulations supporting the theoretical findings are given, and it is observed that the results of two methods are consistent. • Propose a delayed Van der Pol oscillator. • Discuss local stability and occurrence of Hopf bifurcation. • Use Center-manifold reduction and multiple scales to study Hopf bifurcation. • Numerical simulations are in accord with theoretical analysis. • The results of two methods are consistent. [ABSTRACT FROM AUTHOR]