In this section, we discuss the origin which represents a local singular Bogdanov–Takens and a global singular SNIC bifurcation
point; see Fig. 5. Let us first formulate general conditions under which the origin in the -diagram of the local normal form is relevant for system (3) under the condition that Assumptions 1–4 are satisfied. We look at the lower fold
point when it violates Assumption 5, i.e. we are interested in parameter values where
with defined in (18). This condition is typically violated in 1-parameter families. For SNIC bifurcations
to appear with a saddle-node near the fold point , we will hence impose the following condition.
Assumption 6
For fixed , the fold point is a singular contact point that undergoes a singular Bogdanov–Takens bifurcation
with respect to the parameters at . More precisely, we impose (on top of Assumptions 1–4):
(27)
(28)
Besides the possible singular points near occurring in this bifurcation, there are no other singular points on .
Conditions (27) imply that the fold point is a local codimension-2 singular point. Conditions (28) imply that a complete unfolding of the singularity is obtained upon varying . In fact, the conditions in (28) could be replaced by the slightly more general condition,
where , but we prefer to keep (28) in order to be able to identify I as the Hopf breaking parameter among the two parameters.
Remark 13
It can be seen that conditions (27) and (28) imply the following conditions on the normal form (15):
which are verified on the canonical model (1).
Under these conditions it is well known that in ?-dependent rescaled coordinates, a regular Bogdanov–Takens bifurcation takes place;
see 16]. As a consequence, the presence of small-amplitude homoclinics is clear in some parameter
subset. Furthermore, as (singular) Andronov–Hopf bifurcations form part of the bifurcation
diagram, canard-type orbits are present. Indeed, the double singularity in the slow
dynamics at may unfold in a way that the fold point becomes a canard point and an extra saddle-singularity
in the slow dynamics on the middle branch may appear. In that way, an incomplete canard
explosion can be observed that terminates in a canard-type saddle homoclinic (‘jump-back’,
without ‘head’). In fact, these are phenomena that appear locally near the Bogdanov–Takens
fold point.
Besides the small-amplitude phenomena near the Bogdanov–Takens point, we consider
orbits that are close to the singular saddle-node homoclinic loop ? shown in Fig. 4. We expect the existence of large-amplitude saddle-node homoclinics (SNICs) and as
in the previous section, we also expect large-amplitude saddle homoclinics as well
as relaxation oscillations.
In order to get a hold on the parameters close to , we rescale the parameters and introduce
(29)
for some large . By doing this we in fact assume that and . After the parameter rescaling (29), we study the system
(30)
The singularity at has been described in 16] as a slow–fast Bogdanov–Takens point.
In that paper, it is shown that a BT bifurcation takes place near the origin. More
importantly, it is shown that the phase portraits associated with the BT bifurcation
are the only phase portraits seen in a small neighbourhood of the origin. The paper
does not deal with any interactions with global return mechanisms, i.e. the interaction
of the (local) singular BT and the (global) singular SNIC have not been studied. We
will therefore repeat part of the local analysis, with the focus on the interaction
with the global return mechanism.
6.1 Blow-up of the Singular Fold
Near the fold, we study the system using blow-up14], 15]. We write
where denotes the half-sphere with (also known as Poincaré or blow-up sphere). The weights are chosen in a way that the higher order (big-oh) terms in (30) have also higher order in the rescaled equation.
As is usual in geometric desingularisation, we study the flow on the half-sphere in
different (coordinate) charts. Two charts are important: the chart (or the phase-directional rescaling chart), and the chart (or the family rescaling chart). The chart is used to extend the orbits along the slow manifold (which are directed towards
the fold) to a neighbourhood that is at distance from the origin. While this transition is the most technical and least obvious for
the reader who is not accustomed to the blow-up method, it fortunately is that part
where the study of system (30) agrees with the results in studies of a classical regular jump point. Hence, we
do not present detailed computations in the chart , but focus on presenting important facts and refer to the literature 14], 15] for a detailed analysis.
The phase-directional rescaling chart. Here, we explain the dynamics near the equator of the blow-up sphere . When presenting a picture in blown-up -space, where the origin is replaced by (or blown-up to) a sphere, we can position the point of view from the top of the ?-axis; looking down on the -plane we see the spherical surface with as the interior of a circle , and the equator as the circle with the outer slow–fast dynamics around it; see Fig. 9.
Fig. 9. Blow-up of the singular fold. ‘Birds-eye view’ of the upper blown-up sphere in -space. Normal hyperbolicity of the manifolds and is gained at the equator, allowing one to extend them onto the blown-up sphere near
singularities , respectively, . The two additional singularities represent connection to the fast fibre at
Calculations in reveal two hyperbolic saddle singularities, and , and two semi-hyperbolic singularities and along the equator. Combining information from the slow–fast dynamics near with information obtained in chart allows one to reconstruct the dynamics near the circle shown in Fig. 9.
Combining the global return mechanism, which defines a map , with the information from this chart allows one to show the smoothness and exponentially
contractiveness of the -family of maps
(31)
Given the uniqueness of the centre separatrix issued from , one can prove that the image of any small section under this map limits to this centre separatrix (intersected with ) as . To characterise the global dynamics, it is therefore important to know the dynamics
of this centre separatrix. In particular, a connection of to will distinguish whether or not singular points are met, or whether a regular ‘jump
point-like’ connection is possible. This study will be done in the family directional
rescaling chart.
The family rescaling chart. Once the orbits have passed the chart , we can assume that and . In the blow-up coordinates, this means that is bounded away from the equator of the sphere (since then it means ). It is well known that a study of that part of the sphere can be established by
looking at an ?-dependent rescaling
(32)
for some large . Applying this rescaling to (30), we can divide out a common factor ?, thus transforming the system into a regular perturbation family
(33)
6.2 Local and Global Codimension-2 Bifurcations
System (33) describes the flow in the interior of the sphere as shown in, e.g., Fig. 9, and it can be analysed by means of classic bifurcation analysis. Together with the
information obtained from the global return mechanism, we are able to describe all
observed local and global bifurcations in parameter space.
Bogdanov–Takens bifurcation.
Lemma 3
For, system (33) undergoes a subcritical Andronov–Hopf bifurcation when, , and a saddle–ode bifurcation of singularities when. Both bifurcation curves meet in a Bogdanov–Takens bifurcation point at. Both bifurcations persist for.
Proof
There are two singular points on , located at . The singularity at , denoted , is always a saddle. The singularity at , denoted is of focus/node type for and undergoes a change in the sign of the trace along , which indicates an Andronov–Hopf bifurcation. Along this parameter line, is weakly unstable; a basic calculation shows that the first Lyapunov coefficient
is positive. Hence the Andronov–Hopf bifurcation is subcritical.
The two singular points and collide along indicating a saddle-node bifurcation at .??
Remark 14
Let us mention, without proof, that the homoclinic saddle-loop bifurcation curve ( in Fig. 10) of the Bogdanov–Takens point (BT) at lies between the Andronov–Hopf curve (AH) and the parameter line () and tends towards this parameter line as it approaches infinity; see Fig. 10. This can be seen by studying (33) for near infinity 22].
Fig. 10. Bifurcations found in (33) for in the -parameter plane. Codimension-2: Bogdanov–Takens (BT), resonant homoclinic (Resonant) and saddle-node homoclinic (); codimension-1: saddle-node (SN), Andronov–Hopf (AH), saddle homoclinic ( and ), saddle-node of limit cycles (SNPO)
Saddle-node homoclinic bifurcation. The following proposition states some properties of system (33) for . As mentioned before, we focus on the interaction of local dynamics with the global
return mechanism (31). In particular, we want to understand the dynamics of the centre separatrix of .
Proposition 2
Along the saddle–ode bifurcation line, we have the following behaviour of (33) for (see Fig. 11):
1.
When, the separatrix coming fromconnects to a centre–stable separatrix of the saddle–ode singularity. The unique unstable centre separatrix ofconnects to.
2.
When, the separatrix coming fromconnects to the hyperbolic attracting separatrix of the saddle–ode singularity. The unique unstable centre separatrix ofconnects to.
3.
When, the separatrix coming fromconnects directly toalong a regular orbit. This is the jump scenario. In particular, the BT point () is not connected to the separatrix.
4.
The attracting separatrix of the saddle–ode pointand the separatrix coming frombreak regularly with respect to the parameter C.
Fig. 11. Behaviour of (33) on the Poincaré disc for , along the SN-curve . For , the point connects to the SN point . At , a BT point occurs, not connected to , however
Proof
The proof uses basics from planar theory of vector fields (e.g. invariant curves,
isoclines, positive invariant sets). It requires some computations, but as it concerns
basic properties we have left the details for the reader.
For , we define . Notice that the saddle-node singularity is a point on the parabola and that which is positive except at the SN point . Using information from infinity (i.e. from chart ), we see that the separatrix from enters the region which is positively invariant. Hence the ?-limit set of the separatrix has to be the vertex of the parabola. Finally, the hyperbolic
separatrix of the saddle-node singularity is tangent to which implies it lies outside the positive invariant set . This proves part (1).
For , the singularity lies on the invariant parabola from Lemma 3, which then coincides with the separatrix
coming from . It is not hard to verify that the hyperbolic eigenspace of the saddle-node singularity
coincides with the tangent space of the parabola. This proves part (2).
For , we define . One can verify that , so that is a positive invariant set. It is a symbolic computation to verify that the separatrix
coming from enters this invariant set, and hence cannot leave.
On the other hand, V computed at the saddle-node point yields . We conclude that the separatrix from cannot reach . Since is the only other option for a ?-limit, it shows part (3).
As for the regular breaking of the connection in part (4): we compute the stable separatrix
of the saddle-node and compare it with the separatrix coming from . For the comparison we choose an arbitrary section crossing and parameterise it by the levels of V. It is not hard to see that the separatrix coming from intersects any such section at V-values that are . On the other hand, a variational computation of the stable separatrix of reveals that it is given by . Since , it explains the transversality. This finishes the proof of the theorem.??
Clearly, the saddle-node curve persists within a manifold . The regular breaking property formulated in the proposition ensures that the results
persist for .
Theorem 3
There exists a parameter surfacealong which a saddle–ode singularityexists. On this surface, there exists a curvealong which a saddle–ode homoclinic () connection appears containing the hyperbolic separatrix of the saddle–ode. Foron this parameter surface, there is a SNIC connection containing a centre separatrix of the saddle–ode. For, there is no SNIC connection.
Proof
Restrict to the saddle-node surface. Let be the unstable separatrix of the saddle-node that connects to . It smoothly intersects in a point the section . The global return mechanism (31) takes this point to a point on , where lies on the centre separatrix. On the other hand, let be the hyperbolic stable separatrix of the saddle-node that intersects at a point . From Proposition 2, we know that , and, parameterizing the section by a regular coordinate ?, we also know that at . Hence, we can apply the implicit function theorem to prove the presence of a curve
along which both points coincide and a saddle-node homoclinic connection appears.
The rest of the statements follow easily from the properties at the singular limit.??
Resonant homoclinic bifurcation.
Proposition 3
Along, we have the following behaviour of (33) for; see Fig. 12:
1.
When, the centre separatrix ofconnects to the node.
2.
When, a SN–bifurcation takes place (see Proposition 2).
3.
When, a centre separatrix ofconnects to the hyperbolic saddle, and one of unstable separatrices of the saddle connects to. The ratio of eigenvalues is given by, and the saddle is strongly resonant at.
4.
For any given, the saddle connection breaks regularly with respect to the parameter A as one moves
away from.
Fig. 12. Behaviour of (33) on the Poincaré disc for , along the curve . At , the point connects to the SN point . For the connects to the node , for , there is a sequence of two heteroclinic connections from to , via the saddle , which is resonant at
Proof
Recalling V from the proof of Proposition 2, we see that along , is invariant and hence contains the centre separatrix of . It is easy to verify that when , only the node lies on , and when only the saddle lies there. In the second case, the node is found to lie in . So, the unstable separatrix of the saddle in can only connect to . The computation of the eigenvalues is direct.
Let us finally prove the regular breaking property, with a Melnikov-like approach
that is adapted to planar dynamics; see 19]. Denoting , , then
which evaluates to along . This function has a fixed sign on the separatrix from up to the saddle at . We can now directly refer to 19] (Proposition 5.7) where the regular breaking is related to a Melnikov computation,
where the integrand is exactly (multiplied by a specific exponential that implies the convergence of the Melnikov
integral). Since this function is sign-fixed, the related Melnikov integral is nonzero;
see 23] for a generalisation of Melnikov theory to arbitrary dimensions.??
Theorem 4
Let. There exists a parameter surface, along which a large–amplitude saddle homoclinic () connection exists. On this surface, there exists a curvealong which the homoclinic changes stability (resonant): for lower values of C, the homoclinic is stable, for larger values it is unstable. From this curve emerges a surfacealong which a SNPO bifurcation takes place. The surfacesandare exponentially close.
Proof
The presence of the homoclinic surface follows from a reasoning completely analogous
to the one in the proof of Theorem 3. The change of stability is simply an eigenvalue
computation: the equation is perturbed regularly under the ?-perturbation.
The emergence of an SNPO branch from the resonant saddle homoclinic is standard (see
24]), and based on three features: (i) the ratio of eigenvalues is perturbed regularly
upon variation of a parameter (C), (ii) the separatrix connection breaks regularly upon variation of another parameter
(A), and (iii) the divergence integral along the homoclinic loop is nonzero. Properties
(i) and (ii) follow directly from the singular limit analysis in Proposition 3. Property
(iii) follows from the slow–fast nature of the global return mechanism: the divergence
integral computation is dominated by the passages along the slow branches , which are both attracting and yield a contribution of the order , for some , while the fast parts and the parts near the folds yield an contribution. While this argument does not prove that the SNPO branch is uniformly
defined up to the limit, the proof of such a result is based on combining the local
Dulac map of the saddle with a return mechanism. Since all properties are uniform and since
the global return mechanism is sufficiently smooth up to and including the singular
limit, the method for showing SNPO branches is valid uniformly in ?.??
Figure 10 summarises all observed codimension-2 bifurcations and the bifurcating codimension-1
branches.
Remark 15
There are no additional bifurcations (proof omitted).
Remark 16
The homoclinic surface defined in Theorem 4 can be extended to , up to and including its intersection with the SN-surface from Theorem 3. At the singular limit, this is seen in Fig. 10, but a proof is needed for . In such a proof, one would need to blow up the vector field once more at the saddle-node
singularity and at the parameter value , using a family blow-up, in order to uniformly separate the saddle from the node.
The technical issues involved in such a construction go beyond the scope of what we
intend to expose in this paper.
Remark 17
The bifurcation curves AH, SNPO, and shown in Fig. 8 and Fig. 10 are the same. To rigorously prove this, we would need to include the parameters in the blow-up analysis, i.e. we would have to blow up the origin . Again, the technicalities involved in such a construction go beyond the scope of
what we intend to expose in this paper.