4.1. Generation Mechanism of MTSOs
Since the discontinuity boundary
is defined by
, we redraw the phase portrait in
plane and the time history of
x of the MTSOs when
, respectively, given in
Figure 3a,b. Then we may find that every spiking oscillation contacts the discontinuity boundary
(denoted as the blue horizontal line). Meanwhile,
Figure 3a also reflects that the mixed tonic spiking oscillation with
may be not strictly periodic.
In order to exhibit more details about the oscillation structures, we choose
,
,
and
as cross sections, respectively, as shown in
Figure 3a. Poincaré maps are plotted in
Figure 4, where one may find that the response of the whole system (
3) then is a quasi-periodic oscillation when
, since all the Poincaré maps on the four cross sections are nearly closed curves.
According to the Poincaré maps in
Figure 4, it is not difficult to predict that limit cycles, i.e., the so-called spike attractors exist in the Filippov fast subsystem (
4). Furthermore, the relatively large closed curves intercepted from LAOs, labeled as LAO in
Figure 4a–c, indicate that the corresponding spiking attractor behaves in crossing motions on
. While the relatively small closed curves intercepted from SAOs, labeled as SAO in
Figure 4b–d, indicate that the corresponding spiking attractor behaves in sliding motions on
.
The relatively small closed curves intercepted from SAOs appears in the inner of the relatively large closed curves intercepted from LAOs, seen in
Figure 4b,c. It indicates that the spiking attractor behaving in crossing motions coexists with the spiking attractor behaving in sliding motions. Moreover, the relatively small closed curves intercepted from SAOs behaves in different nonsmooth structures, which means that the corresponding spiking attractor may undergo sliding bifurcation.
Considering the symmetry, as shown by the green circles in
Figure 3a, two spike attractors corresponding to the two SAOs exist in (
4), respectively, labeled as
(
) and
(
). Taking
as the example,
Figure 5 gives the corresponding geometry of sliding bifurcation, where
crosses through the visible cusp point
on
when
to undergo a multi-sliding bifurcation, seen in
Figure 5b, leading to the single sliding segment in
Figure 5a turns into two sliding segments in
Figure 5c.
We now discuss the generation mechanism for the quasi-periodic MTSOs when
by using the classical slow–fast decomposition method. Via analyzing the conventional bifurcations of spike attractors in
Figure 6a, we can give the complete details about bifurcations of spike attractors in
Table 2.
As shown by the red real arrows in
Figure 6a, the relatively large amplitude spiking attractor, denoted as
, will collide with the two unstable cycles
, respectively, leading to the transitions from
to
caused by the fold bifurcations of limit cycles
. Similarly, the two relatively small amplitude spike attractors
may also collide with unstable cycles
, respectively, leading to the transitions from
to
caused by the fold bifurcations of limit cycles
.
Considering the symmetry, in order to exhibit the generation mechanism of the quasi-periodic MTSOs, we choose to superimpose one segment of trajectory with
(
) onto the one-parameter bifurcation diagram of spike attractors, seen in
Figure 6b. During the slow variable
w changes from
to
, it can be found that the evolution process of the trajectory can be explained well by using the bifurcation structures of spike attractors. For example, the multi-sliding bifurcation
leads the SAOs behaving in single-sliding oscillations to turn into a SAO behaving in double-sliding oscillations. Meanwhile, the two fold bifurcations of limit cycles
and
lead the transition from SAOs to LAOs and the transition from LAOs to SAOs, respectively.
It should be pointed out that the two multi-sliding bifurcations lead to the change of sliding modes in SAOs rather than the transitions between attractors. We will ignore these two multi-sliding bifurcations because the main focus in our work is on the transition mechanism.
Since the whole system (
3) is a slow–fast dynamical system when
, one may be interested in knowing whether or not the quasi-periodic MTSO response in
Figure 3 should be a slow–fast dynamics behavior. The limit cycles in the fast subsystem (
4) are presented in
Figure 6a via giving the extreme points in the
plane. Without loss of generality, we can also map those limit cycles as fixed points in the
plane via discrete dynamics theory. As shown in
Figure 6c, where
,
and
denote the fixed points corresponding to
,
and
, respectively, while
and
denote saddle-node bifurcations corresponding to
and
.
Similarly, we now represent the quasi-periodic MTSOs in
Figure 3 with the maximum points of spikes, as shown by the cyan points in
Figure 6c. Based on the classical slow–fast decomposition method, we can find that the quasi-periodic MTSOs is a relaxation oscillation induced by the four saddle-node bifurcations. Accordingly, we may say that the quasi-periodic MTSOs response in
Figure 3 is a typical slow–fast dynamics behavior characterized by a relaxation oscillation on the torus.
4.2. Generation Mechanism of MBOs and TBOs
Based on the above analyses for the quasi-periodic MTSOs, the response of the whole system (
3) can be well predicted by attractors and the bifurcation structures in (
4). Meanwhile, according to the spike attractors and bifurcations in
Figure 6a, it seems that the transitions between spike attractors should be the only alternative for the whole system trajectory. However, in the MBOs when
, redrawn in
Figure 7, the appearances of QSs connecting the neighboring two SAOs do not agree with the prediction based on the two fold bifurcations of limit cycles
.
Figure 7 shows that the MBOs when
is a symmetrical period
response of system (
3), where
denotes the period of excitation
. For the convenience of analysis, we divide the whole period behavior into six equal time segments in time history according to the monotonous zones of slow variable
w, labeled as
, respectively, seen in
Figure 7b. It can be found that the QSs located on
and
represent the unique sticking motions which occur on discontinuity boundary
. Since the sticking motion is an unique nonsmooth dynamics behavior that the trajectory in Filippov systems converges to a stable PE and behaves in a certain static state, we now count PE as well as BEBs into the bifurcation diagram in
Figure 6a, as shown in
Figure 8a.
By employing (
9) and (
15) as well as the results, it is easy to check that one stable focus type pseudo-equilibrium branch
, existing in the sliding region
, will disappear at the two boundary equilibrium points
via nonsmooth fold bifurcations (respectively labeled as
). Under the separation of the four bifurcations
and
, seen in
Figure 8a, three special rest spike bistability structures, composed by stable focus type pseudo-equilibrium points (rest attractors) and nonsmooth stable cycles (spike attractors), can be observed in the fast subsystem. One is composed of
coexisting with the relatively large amplitude crossing cycle
, while the other two are composed of
and the small amplitude sliding cycles
, more details are given in
Table 3.
By then, the symmetrical period
MBOs in
Figure 7 can be divided into two different transition routes. Taking the trajectories in decreasing intervals of the excitation as the example, as shown in
Figure 8b, the segment with
(black dash–dotted curve) and the segment with
(black solid curve), respectively, converge to the special rest attractor
and the crossing spike attractor
. Then, the segment when
and the segment when
converge to the relatively small amplitude spiking attractor
, respectively, caused by the bifurcation
and the bifurcation
.
According to the bifurcation structures presented in the fast subsystem (seen in
Figure 8a), even when starting from the stable PE, the trajectories of the whole system may also transition to
via the two nonsmooth fold bifurcations
, and repeat the transition route similar with the first case with
. Clearly, the transition route in the black dash–dotted curve is non-bifurcation-induced transition. Therefore, the key to understanding the generation mechanism of MBOs in
Figure 7 is to reveal the dynamics mechanism underlying the appearance of the two coexisting distinct transition routes shown in the green circle in
Figure 8b, especially the fundamental reason that results in the transition from
to
.
One may note that the two coexisting attractors in the rest spike bistability II (III) may be very close to each other in the geometric position. For instance, see the rest spike bistability II when
in
Figure 9a. The clockwise stable cycle
, starting from the smooth region
(
), behaves in the sliding motion
(
) when it enters into
at the point
(
), until it crosses through the sliding boundary
(
) at the point
(
) to enter into the opposite side of
. We can find that the coexisting stable
is very close to the sliding segment
. We now focus on the division of attractor basins of the two coexisting attractors in the rest spike bistability II, which is important to explain the transition route from
to
.
Since the stable
is very close to the sliding segment
of spike attractor
, the sliding trajectories (denoted by
) between
and the sliding segment
will converge to one of the two coexisting attractors ultimately, According which the attractor basins of the two coexisting attractors in rest spike bistability II can be analyzed. Meanwhile considering that the sliding vector field (
9) is linear, all the sliding trajectories
can be derived analytically according to that
is a stable focus, expressed by
where the two coefficients
can be calculated by submitting the corresponding initial conditions of a sliding trajectory
into (
18).
It is not difficult to verify that the two cusp points on sliding boundaries
both are visible, computed as
via (
17). Taking
as the initial conditions of (
18), one special sliding orbit, labeled as
, can be computed with negative
. Certainly, the sliding trajectories between
and
will slide to the stable
in the forward time. However, what is in doubt is whether all the sliding trajectories between
and
will converge to the spike attractor
, that can be answered via the following numerical method.
One may note that all the sliding trajectories between
and
inescapably cross through the visible fold points between
and
. Thus, the attractor basin structure can be easily obtained by which of the rest spike bistabilities II the trajectories from these visible fold points eventually converge to. Via numerical verification, the point
divides the sliding boundary
between
and
into two segments. As shown in
Figure 9b,where trajectories starting from visible points between
and
converge to
to behave in sticking motions after a minimal oscillation in
, while trajectories starting from visible points between
and
oscillate to spike attractor
. Similarly, another special sliding orbit, denoted as
, can be obtained numerically by taking
as the initial condition of (
18). It can be regarded as an attractor basin boundary of the rest spike bistability II in sliding region.
Meanwhile, the classical slow–fast decomposition method reflects a fundamental fact in the family of slow–fast dynamical systems, referring to the fact that the dynamics behaviors of the fast subsystem are modulated continuously by the slow variable. According to that, the attractor basin structures of rest spike bistability II have changed after the whole system completes one spiking oscillation. Without loss of generality, taking the trajectory oscillating along with the
in
Figure 9a as the example. Assume that the whole system trajectory starts at a point in the sliding region, where the corresponding slow variable is
sin
, the trajectory completes a spike after one period of
(denoted as
). The slow variable has changed to be
sin
, i.e., there exists a slight variation
. For more precision, here we name it the deviation of the slow variable (DSV for short).
Note that the excitation frequency has been assumed as
, then we have
, indicating that the DSV
will increase along with the increase of
, as shown in the sketch map in
Figure 10a. As we have mentioned that
may be very close to the sliding segment
of
, seen in
Figure 9a, the increase in DSV caused by the increase of
, even though it is relatively small, can lead to unexpected influence on the attractor basins of rest spike bistability II. To illustrate this point,
Figure 10b gives the attractor basin structures of rest spike bistability II with
, i.e.,
. The special sliding orbit
(colored in red) divides the sliding region
into two different sub-regions by the aid of sliding boundary
, respectively, denoted by
and
. The sliding trajectories then converge to
when they are located in
while the sliding trajectories converge to
(represented by the two sliding segments
) when they are located in
.
Interestingly, when we increase the DSV to a appropriate value, for instance,
, another unique sliding orbit besides
, denoted by
, can also be obtained numerically. Therefore, the attractor basin structures of rest spike bistability II can be redivided by using
as well as the two sliding boundaries
, as shown in
Figure 10c. Especially, we can find that the sliding segment
then is located in
while the sliding segment
still is located in
. Sticking motions can be observed when the whole system trajectories oscillating along with
move to the sliding segment
just in time. While spiking oscillations will persist when the whole system trajectories oscillating along with
move to the sliding segment
just in time.
Accordingly, one may find that a threshold
exists in the whole system (
3) under the parameter conditions in
Table 1, which can be approximately computed as
via tracking the critical value of
of the appearance of the dash-dotted transition route in
Figure 8b. Therefore, we have the following results for the transitions from rest spike bistability II to rest spike bistability I. When
, only the transition from
to
can be observed, which is explained well by the fold bifurcation of limit cycles
. However, when
, the special transition from
to
can also be observed besides the above. That can be explained by the disappearance of
induced by the nonnegligible DSV rather than a common bifurcation. Accordingly, we name it the transition induced by the DSV effect. Meanwhile, considering the symmetry, we have the same results about transitions from the rest spike bistability III to the rest spike bistability I.
For convenience, we can summarize the transition mechanism in a monotonous zone of slow variable w into the following two types when .
The regular transition route, meaning that the trajectory of the whole system will transition to from , and then to via the fold bifurcations of limit cycles, leading to the birth of mixed tonic spiking oscillation pattern;
The irregular transition route, meaning that the trajectory of the whole system will transition to from via DSV effect, and then to via the nonsmooth fold bifurcations, leading to the birth of bursting oscillation pattern.
By then, the generation mechanism underlying the symmetrical period
MBOs in
Figure 7 can be understood well based on the above analyses. More precisely, assuming that the trajectory starts at the maximum of
w in
, it, respectively, passes through the rest spike bistability II and III three times when it oscillates along with spike attractor
, where the transition induced by DSV effect, respectively, only takes place once. It results in a symmetrically periodic transition structure composed of two irregular transitions followed by for regular transitions in a complete oscillation period. Then, the period
MBOs, consisting of two QSs characterized by sticking motions and two SPs characterized by mixed spiking oscillations, can be observed in the whole system.
Based on the analysis of the generation mechanism of the period
MBOs when
, it is not difficult to understand the responses of whole system (
3) when
and
. We now focus on the generation mechanisms of the two responses in
Figure 11.
When
, seen in
Figure 11a, we may find that the whole system (
3) behaves in an asymmetrical period–
MBOs, and the trajectory, respectively, passes through the rest spike bistability II and III once when it oscillates along with spike attractor
. The transition induced by the DSV effect only occurs when the trajectory moves to the rest spike bistability I from the rest spike bistability III, however, it does not take effect when the trajectory moves to the rest spike bistability I from the rest spike bistability II. It results in the appearance of the asymmetrical period–
MBOs consisting of one QS characterized by a sticking motion and one SP characterized by mixed spiking oscillations.
When
, seen in
Figure 11b, the whole system response is a symmetrical period–
TBOs. Similarly, with the whole system response with
, the trajectory in this case also passes through the rest spike bistability II and III once when it oscillates along with spike attractor
, respectively. However, we can find that the transition induced by the DSV effect may take effect at the two transition locations, leading to the trajectory transition from
to
. That implies the appearance of the symmetrical period–
TBOs consisting of two QSs characterized by sticking motions and two SPs characterized by spiking oscillations without LAO.
4.3. Coexistence of Whole System Responses
According to the generation mechanism of whole system responses involving transitions induced by the DSV effect, one may find that any response of the whole system is a certain combination form of regular and irregular transition routes. Moreover, such a combination may be random in theory since period–3 movement has been obtained when , implying the appearance of an arbitrary period of whole system response, even chaos.
More interestingly, the period–
MBOs in
Figure 11a is asymmetrical, i.e., there must be another asymmetrical period–
MBOs coexisting with the one in
Figure 11a based on the symmetry. It indicates that the coexistence of attractors can also be observed in the whole system. We will give further discussions in this section.
Figure 12a gives the coexistence response of the whole system when
, and the corresponding initial conditions from top to bottom are
,
and
, respectively. Numerical results show that there also exists a symmetrical period–
MTSOs (seen in
Figure 12(a1)) besides the two coexisting asymmetrical period–
MBOs (seen in
Figure 12(a2,a3)), indicating that tristability then exists in the whole system (
3). Noting that the asymmetrical period–
MBOs in
Figure 12(a3) is symmetrical with the one in
Figure 11a, the corresponding generation mechanism can be obtained according to the symmetry. Meanwhile, for the symmetrical period–
MTSOs in
Figure 12(a1), the generation mechanism is attributed to the fact that only two regular transition routes appear in a complete period motion.
In order to show more details about the coexistence of responses of the whole system (
3), we turn to exhibit the responses of the whole system via plotting the Poincaré map. The whole system trajectory will oscillate with
when it is in the regular transition route. In contrast, the whole system trajectory may move along with
in the irregular transition route. Accordingly,
is chosen as the Poincaré cross section, which can reflect the symmetry of the whole system response and can distinguish well between the two transition routes. As shown in
Figure 12b, we give the Poincaré map of the whole system when
, in which the period–
MTSOs (colored in orange) can persistently exist in the whole frequency range.
Meanwhile, the period–
, period–
, period–
, period–
and period–
MBOs can discontinuously appear in turn along with the decrease of
. In those periodic MBOs, further numerical simulations show that the irregular transition route will appear at most once, which is absent here for brevity. Here the periodic MBOs colored in red denote the irregular transition route that only occurs nearby the rest spike bistability II. While the ones colored in green denote the irregular transition route that only occurs nearby the rest spike bistability III. Moreover, the black ones denote the irregular transition route occurs both in the rest spike bistability II and in III. Furthermore, the structures of MBOs may also alternate between asymmetry and symmetry in this progress. It leads the coexistence of responses of the whole system (
3) to alternate between tristability and bistability, more details see
Table 4.
Then, we can find that the whole system (
3) exhibits various dynamics behaviors characterized by one period–
MTSOs coexisting with one (or two) periodic MBOs when the irregular transition route is involved. It indicates that the response only involving regular transition route still exists in the whole system when
. We now turn to show another interesting coexistence of the whole system responses.
When
, tristability structure can also be observed in the whole system. As shown in
Figure 13a, one symmetrical period–
TBOs, which has been presented in
Figure 11b, can coexist with two asymmetrical period–
MTSOs, the corresponding initial conditions from top to bottom are
,
and
, respectively. Similarly, we also give the Poincaré map when
in
Figure 13b, where the symmetrical period–
TBOs (colored in orange) can persistently exist in the whole frequency range. Meanwhile, both the two asymmetrical period–
MTSOs presented in
Figure 13(a2,a3) perform the obvious period-doubling cascades along with the decrease of
, colored in light gray and gray, respectively.
Different from the former case in
Figure 12b, one may find that tristability structure always exists in the whole system (
3) when
. Moreover, the tristability can be composed not only by a period–
TBOs coexisting with two periodic MTSOs but also by a period–
TBOs coexisting two chaotic MTSOs.