On the Double-Zero Bifurcation of Jerk Systems

: In this paper, we construct approximate normal forms of the double-zero bifurcation for a two-parameter jerk system exhibiting a non-degenerate fold bifurcation. More precisely, using smooth invertible variable transformations and smooth invertible parameter changes, we obtain normal forms that are also jerk systems. In addition, we discuss some of their parametric portraits


Introduction
The double-zero bifurcation, also called the Bogdanov-Takens bifurcation, can occur in a continuous-time dynamical system ẋ = f (x, µ), x ∈ R n , µ ∈ R 2 , n ≥ 2 when the system has at the critical value of µ an equilibrium point with two zero eigenvalues and no other eigenvalues on the imaginary axis (see, e.g., [1,2]).
The double-zero bifurcation can be met, for instance, in mechanical, electrical, and biological systems.For example, the analysis of mathematical models of an internally constrained planar beam equipped with a lumped visco-elastic device and loaded by a follower force [3] or of a non-linear cantilever beam that is externally damped and made of a visco-elastic material [4] reveals among other solutions the existence of a double-zero bifurcation.Oscillators and electronic circuits are modeled by differential equations, and in some cases, they experience a double-zero bifurcation (see, e.g., [5][6][7]).Also, a double-zero bifurcation appears in some chemical reactions (see, e.g., [8,9]) and in fluid mechanics (see, e.g., [10,11]).
The importance of the double-zero bifurcation is highlighted by the following remark: "One of the most important features of the Takens-Bogdanov bifurcation is that it warrants the existence of global connections in its vicinity (a homoclinic orbit in the non-symmetric case and a homoclinic or a heteroclinic orbit if the system is symmetric)" [12].
This local bifurcation was first analyzed by Bogdanov [13] and Takens [14] in the case n = 2. Several normal forms of this bifurcation were reported in this case (see, e.g., [1]).In fact, such a normal form is "the simplest parameter-dependent form to which any generic two-parameter system exhibiting the bifurcation can be transformed by smooth invertible changes of coordinates and parameters and (if necessary) time reparametrizations" [2].Approximate normal forms are obtained by truncation of higher-order terms.In the ndimensional case n ≥ 3, the study of the double-zero bifurcation is carried out by reduction on a local center manifold to the planar case.It is natural to ask whether such a reduction can be avoided, i.e., whether n−dimensional normal forms can be obtained.We will give an affirmative answer for n = 3.
The paper is organized as follows: In Section 2, we recall some results regarding non-degenerate fold bifurcations.In Section 3, we derive jerk approximate normal forms for system (1), and we discuss some of their parametric portraits.

The Fold Curve
Assume there are α c , β c such that system (1) displays a non-degenerate fold bifurcation when α passes through the critical value α c and β = β c is fixed.Following [21], sufficient conditions are given below: F1.There is at the critical value (α c , β c ) an equilibrium point E(x c , 0, 0) of system (1) with a simple zero eigenvalue and no other eigenvalues on the imaginary axis, i.e., j(E c ) = 0, where E c = (x c , 0, 0, α c , β c ).We have denoted: It is known that if the fold conditions hold, then "generically, there is a bifurcation curve F in the (α, β)-plane along which the system has an equilibrium exhibiting the same bifurcation" [2].For the sake of completeness, we prove this result in our case.

by the Implicit Function
Theorem (IFT), there are the functions x = x(β), α = α(β) in a neighborhood V of β c such that x(β c ) = x c , α(β c ) = α c and which verify the equations of Γ.Hence, (x(β), α(β), β) is a parametrization of Γ in a neighborhood of (x c , α c , β c ).In addition, for all β ∈ V, E β (x(β), 0, 0) is an equilibrium point of system (1) and by continuity, the other fold conditions will be satisfied.Consequently, the construction can be repeated to extend the curve further.

The Double-Zero Bifurcation
In this section, we deduce jerk approximate normal forms for the double-zero bifurcation of a jerk system.Also, we give parametric portraits of these.
When the parameters α and β vary simultaneously to track the bifurcation curve F, another real eigenvalue can approach the imaginary axis, which leads to a double-zero bifurcation generally.
Case I.The annihilation of the term proportional to y.
It is easy to see that if µ 2 2 − 4µ 1 > 0, system (8) has two equilibria E ± (x ± , 0, 0), , which collide when µ 2  2 − 4µ 1 = 0 and then disappear for µ 2 2 − 4µ 1 < 0.Moreover, the characteristic polynomial at E ± is given by The fold curve is . Following [21], if the characteristic polynomial has the form Assume c < 0. At E − , we obtain the Hopf bifurcation curve which depends on c and d.In fact, H is half of the parabola , ∞), the negative semi-axis µ 2 = 0 for d = − 2 c , and the positive semi-axis µ 1 = 0 for d = − 2 c .Moreover, for c < 0, we get that E + is an unstable equilibrium point with a two-dimensional stable manifold; thus, it does not bifurcate.Now, let d < 0. Consider the parametric portrait given in Figure 1, where H is the above Hopf curve and F + , F − are the branches of the fold curve F separated by the double-zero point (0, 0).In region 1, there are no equilibrium points.On the curve F − , an equilibrium is born and splits into the asymptotically stable node (or focus-node) E − and the unstable saddle (or saddle-focus) E + in the region 2. Hence system (8) displays a saddle-node bifurcation when (µ 1 , µ 2 ) crosses the fold curve F − .In region 3, E − is an unstable equilibrium point with a one-dimensional stable manifold; hence, it loses stability when the curve H is crossed.Moreover, a Hopf bifurcation occurs, and a stable limit cycle is born (we assume that the first Lyapunov coefficient does not vanish).The unstable equilibria E + and E − collide when (µ 1 , µ 2 ) ∈ F + and then disappear when returning to region 1; thus, a degenerate fold bifurcation occurs.We conclude that there are no other local bifurcations in the dynamics of system (8) in the case c < 0, d < 0.
We notice that the above scenario is similar to that which takes place for Bogdanov's normal form (7) (see [2]).As is pointed out in [2], ". . .finally return to region 1, no limit cycles must remain.Therefore, there must be global bifurcations 'destroying' the cycle somewhere between H and F + ".Consequently, a global bifurcation has to occur for system (8) in this case.
In the cases for which d ∈ (0, , ∞), we obtain similar parametric portraits.The difference is that the regions 2 and 3 and the curves F − and F + change their roles.Now, assume that c > 0. We obtain that E − is an unstable equilibrium point with an one-dimensional stable manifold; thus, it does not bifurcate.Moreover, in this case system (8) does not experience a Hopf bifurcation.
(d4 Again, there are no equilibrium points in region 1.Crossing the curve F + , an equilibrium is born and separates into the unstable node (or focus-node) E + and the unstable saddle (or saddle-focus) E − in region 4. Since both equilibria are unstable, system (8) does not display a saddle-node bifurcation in the classic sense: that is, a stable node and a saddle coalesce.Anyway, a fold bifurcation occurs in the considered dynamics.In region 5, E + is an unstable equilibrium point with a two-dimensional stable manifold.Therefore, crossing the curve B, the dimension of the stable manifold of E + changes.The saddles E + and E − collide when (µ 1 , µ 2 ) crosses F − and then disappear when returning to region 1; thus, a degenerate fold bifurcation occurs.We conclude that there are no other local bifurcations in the dynamics of system (8) in this case.Similar bifurcation diagrams are obtained when d < 0.
At E − , we obtain the Hopf bifurcation curve In fact, H is half of the parabola µ , and the negative semi-axis µ 2 = 0 for d = − 2 c .Now, let d ≤ 0. We obtain the parametric portrait given in Figure 4, where H is the above Hopf curve and F + , F − are the branches of the fold curve F separated by the double-zero point (0, 0).Also, the behavior of system (10) in each region is the same as of system (8) in the case c < 0, d < 0 (see Case 1).For d ∈ (0, − 2 c ) ∪ (− 2 c , ∞), we obtain similar parametric portraits.In the case c > 0, we obtain that E − is an unstable equilibrium point with a onedimensional stable manifold; thus, it does not bifurcate.In addition, a Hopf bifurcation does not occur in the dynamics of system (10).
For d > 0, we get the parametric portrait given in Figure 5, where B is the parabola In this case, there are no equilibrium points in region 1, and an equilibrium appears when (µ 1 , µ 2 ) ∈ F − .This point splits into the unstable node (or focus-node) E + and the unstable saddle (or saddle-focus) E − in the region 6.Crossing the curve B, E + changes its number of negative eigenvalues, and in region 7, it has a two-dimensional stable manifold.
The saddles E + and E − collide when (µ 1 , µ 2 ) crosses F + and then disappear when returning to region 1; thus, a degenerate fold bifurcation occurs.We conclude that there are no other local bifurcations in the dynamics of system (10) in this case.Similar bifurcation diagrams are obtained when d ≤ 0. Remark 3. It is known that the normal forms for the double-zero bifurcation given by Bogdanov [13], Takens [14], and Guckenheimer and Holmes [1] are equivalent.In our case, the approximate normal forms (8) and (10) have similar parametric portraits.Moreover, if c < 0, d < 0, the local bifurcations are the same as those obtained for the Bogdanov normal form (see [2]) and Guckenheimer and Holmes (see [1]), respectively.It remains an open problem to establish if a jerk system and the corresponding approximate normal form are locally topologically equivalent: that is, the construction of a homeomorphism that maps orbits of the first system onto orbits of the second system.

Figure 2 .
Figure 2.The transition from an asymptotically stable orbit to a homoclinic orbit of system (8) via limit cycles for c < 0, d < 0.