1 Introduction

Bifurcations and chaos in smooth dynamical systems have been well developed in both theoretical and applied researches. However,there still exist some kinds of piecewise smooth systems in the fields of science and engineering. The studies of bifurcations and chaos in them have been concerned in recent years ^[1-14]. Piecewise smooth systems,even the simplest linear piecewise systems,may exhibit extremely complex dynamics,and are worthy of being studied. The variation of time can be either continuous or discrete. The state space of a piecewise smooth system is composed of several regions,in which the system is controlled by different smooth functions. On the boundaries of these regions,the control functions are discontinuous or non-smooth. It is better to analyze the dynamical behaviors of the piecewise smooth systems in both theoretical ways and numerical ways due to the difficulties.

The results show that there exist a kind of peculiar bifurcations when the orbit of the piecewise smooth system has the tangency with the boundary. Nusse and Yorke^[15-16] coined the term “border collision bifurcation” to describe this class of nonlinear phenomena,for which the term ”C-bifurcation” was used in some Russian literatures instead. The dynamical behaviors may change acutely at this time,e.g.,periodic orbits may transit to chaos orbits directly. This kind of bifurcations can occur only in non-smooth systems instead of smooth systems. Similar to the bifurcations in smooth systems,there are many kinds of border collision bifurcations such as non-smooth fold bifurcations and non-smooth flip bifurcations.

For the piecewise smooth systems with continuous time,di Bernardo et al.^[17] obtained the Poincaré map with the piecewise form by locally linearizing the periodic solution near the border collision bifurcation,and gave the classification method of the border collision bifurcation. Without constructing the Poincaré map,Leine and Nijmeijer^[18] obtained the generalized Jacobian matrix by using the generalized Clarke derivative,and gave some classification methods of border collision bifurcations by means of the eigenvalues of this matrix. The eigenvalues of the generalized Jacobian matrix were set-valued,and formed a one-dimensional (1D) path in the complex plane. The eigenvalue may cross the imaginary axis either continuously or through a jump under the variations of the parameters. Especially,non-smooth continuous systems can also exhibit equilibrium bifurcations,for which a 1D path of eigenvalues may cross the imaginary axis for the multiple time. Multiple crossing bifurcations are much more complex than single crossing bifurcations. They do not have a smooth counterpart. As a whole,border collision bifurcations are a new kind of bifurcations in piecewise smooth systems,and are worthy of understanding the boundary effects in non-smooth dynamics.

The Rössler attractor^[19] is a famous chaotic system with single quadratic nonlinearity. But the Rössler system is relatively complicated^[20],and researchers were prompted to search similar examples^[21],in which the quadratic nonlinearity was replaced by |x| or other elementary piecewise linear functions. Sprott^[22] found chaos in the electrical circuit ,which was piecewise smooth and simple to be constructed,analyzed,and scaled for most desired frequencies,exhibiting many new dynamic phenomena which could not occur in smooth systems. Therefore,we will investigate some new border collision bifurcations in the circuit .

This paper is organized as follows. In Section 2,we show the model and the existence of the equilibrium points. The stability of the two equilibrium points (,0,0) is analyzed in Section 3. The multiple crossing bifurcation is presented in Section 4. The border collision bifurcations are given in Section 5. Conclusions are in Section 6.

2 Model and existence of equilibrium points

We consider the following circuit proposed by Sprott^[22]:

Let x=y and y=z. Then,Eq.(1) can be transformed to the following three-dimensional (3D) piecewise smooth system:

When x ＞ 0,Eq.(2) becomes

where

It is easy to see that,when is an equilibrium point of Eq.(3).

When x ＜0,Eq.(2) becomes

where

When BC ＞0, is an equilibrium point of Eq.(4). There is a unique equilibrium point O(0,0,0) for C=0.

Through the above analysis,we can see that the two equilibrium points and collide with the border x=0 simultaneously at O(0,0,0) for C=0. In order to investigate the border collision bifurcation,we will discuss the stability of the two equilibrium points in the next section.

3 Stability of equilibrium points 3.1 Stability of equilibrium point

Proposition 1 The equilibrium point is unstable for B ＞0,C＞0,and 0 ＜A＜.

Proof Denote

Then,the discriminant of Eq.(5) is

Since Δ＜0 for 0＜A＜,we have β＞0. Hence,J_has the following three eigenvalues:

where

It is easy to see that ζ ＞0 for 0 ＜A＜ and B ＞0.

Let

Then, and w₁ ＜0 only if f(ζ) ＜0. Let

Then,Re and Re(w_2,3) ＜0 only if g(ζ)＜0. The positive root of f(x)=0 is

The positive root of f(x)=0 is

Therefore,

Therefore,x₁ ＞x₂.

The graphs of f(x) and g(x) are shown in Fig. 1.

The equilibrium point is stable only if ζ(x₂,x₁). However,

and

Therefore,ζ³ ＞x₁³,which leads to ζ＞x₁. Thus, unstable.

3.2 Stability of equilibrium point

Proposition 2 The equilibrium point is unstable for 0 ＜A＜ A≤B,and C ＞0.

Proof Denote

Then,the discriminant of Eq.(6) is

It is easy to obtain that Δ ＜0 for 0＜A Hence,a ＞0. As a result,J₊ has the following three eigenvalues:

where

Since

we have η ＞0.

The equilibrium point is stable only if η∈(x₂,x₁).

In the following,we will show that

In fact,from Eq.(7),it can be obtained that i.e.,

Let

Then, =0. We have

Moreover, for Since F′(A) ＜0 forand F(A) is monotonically decreasing on the interval [0,],F(A) ＞0 when That is to say,

From Eqs.(9) and (10),we have

Then,from Eq.(8),we have and A≤ B. Therefore,

and is unstable.

Proposition 3 The equilibrium point is stable for 0 and C ＞0.

Proof From the proof of Proposition 2,we can see that the equilibrium point is stable only if η∈(x₂,x₁). Equation (8) implies that Therefore,we only need to show that η ＜x₁,which is equivalent to η³＜x₁³. We have

Since (2A²-9)A＜0 holds for we have η³＜x₁³. Hence, is stable.

4 Multiple crossing bifurcation

Denote

The discriminant is

where Δ ＜0 when Therefore,θ ＞ The generalized Jacobian matrix of the system (2) at O(0,0,0) is defined by^{[18, 22]}

which has the following three eigenvalues:

where

When we have

When we have

In order to guarantee that q≤1,we require that A≤ B. The above results show that the eigenvalues cross the imaginary axis twice. Then,we have the following theorem.

Theorem 1 The multiple crossing of the eigenvalues of J(O) occurs at C=0 for A≤ B and

It should be noticed that bifurcations may be accompanied with the multiple crossing of the eigenvalues. Referring to Ref.^[22],an example is taken for the parameter values A=0.6 and B=1. In Fig. 2,we depict the characteristic curves of J(O),where q is a path parameter. The eigenvalues γ1,γ2,and γ3 are set-valued,and form a path in the complex plane. γ1 crosses the origin at leading to a non-smooth fold bifurcation. This fact is illustrated in Fig. 3,and can be verified theoretically by the above analysis in Section 3 and Theorem 2 below. γ{2 and γ3 are purely imaginary for =0.8,leading to a non-smooth Hopf bifurcation. The periodic orbit due to the Hopf bifurcation is presented in Fig. 4 for C=0.5. As a result,the paths of the eigenvalues of J(O) show that a multiple crossing bifurcation occurs in the sense that the eigenvalues cross the imaginary axis more than once during the jump. This bifurcation can be imagined as the composite of a non-smooth fold bifurcation and a non-smooth Hopf bifurcation.

5 Border collision bifurcations

Here,we present some theoretical results about the border collision bifurcations of Eq.(2). Through the above analysis in Section 3,we have the following Theorems 2 and 3.

Theorem 2 A non-smooth fold bifurcation can occur at O(0,0,0) for and A≤ B. The two equilibrium points and caused by the bifurcation are both unstable.

This situation is a part of the multiple crossing bifurcation and cannot occur in smooth fold bifurcations.

As an example,the bifurcation diagram is plotted in Fig. 3 for A=0.6 and B=1.

Theorem 3 Non-smooth fold bifurcation can occur at O(0,0,0) forand A ＞B. At this time,the equilibrium point is stable,while the other one is unstable.

This situation is similar to the exchange of stability in smooth fold bifurcations.

The bifurcation diagram is plotted in Fig. 5 for A=1.2 and B=1,which is not a multiple crossing bifurcation

The non-smooth Hopf bifurcation of Eq.(2) can be verified by numerical simulations,while is hard to be proceeded by theoretical analyses. However,for the special case with A=B,we have the following theorem.

Theorem 4 There exists a non-smooth Hopf bifurcation at O(0,0,0) for A=B.

Proof When A=B,there are two unstable equilibrium points and In this case,Eq.(3) becomes

becomes

x≥ 0 means that . From Eq.(12),we know

After calculations,we have the first integration of Eq.(13) as follows:

i.e.,

where C₁ is an arbitrary constant. Now,we can see that the solution of Eq.(13) corresponds to a circle on the (u,v)-plane.

From Eq.(12),we have

The solution of Eq.(14) is given by

where _C2 is an arbitrary constant,and

Now,we get the solution of Eq.(12) as follows:

which tends to the periodic solution

when t→ +∞. From Eq.(16),we have

and

Therefore,under the condition

we have

In fact,in order to guarantee that x(t)≥0 in Eq.(11),we can choose the proper value of C₁ and

such that Eq.(18) holds. Together with Eq.(15),we can conclude that the periodic solution (17) is locally asymptotically stable. The center of the above circle is

It is easy to verify that it corresponds to the point in the (x,y,z)-space. We can find that the above circle corresponds to the periodic solution of Eq.(11) in the (x,y,z)-space

Let C→ 0 in Eq.(18). Then,we can get

Since ,we have C₁=0. Therefore,the periodic solution (17) is just the origin O(0,0,0).

In a word,a non-smooth Hopf bifurcation occurs. The equilibrium point is unstable,and there exists an asymptotically stable periodic orbit around it. The periodic orbit is plotted in Fig. 6 for A=1 as an example.

Theorems 2 and 4 show that a multiple crossing bifurcation occurs at O(0,0,0) for A=B,which is the composite of a non-smooth fold bifurcation and a non-smooth Hopf bifurcation as seen in Fig. 7. This is a very interesting phenomenon. In usual smooth Hopf bifurcations,the equilibrium points locate at both sides of the bifurcation point,and there exists the exchange of stability between the equilibrium points. Nevertheless,here,the two equilibrium points are both unstable,and are located at the same side of the bifurcation point,without the exchange of stability between them.

6 Conclusions

The border collision bifurcations in a 3D piecewise smooth chaotic electrical circuit simplified from the Rössler attractor are discussed.

It is found that there are three types of such bifurcations. When A≤ B,the two equilibrium points and result from the non-smooth fold bifurcation,but they both are unstable. When A ＞B,a non-smooth fold bifurcation occurs. The equilibrium point is stable,while the other one is unstable. When A=B,a non-smooth Hopf bifurcation occurs. The equilibrium point is unstable,while the other one is also unstable. Moreover,there exists an asymptotically stable periodic orbit around it.

The multiple crossing bifurcation is also studied for C=0. The paths of the eigenvalues of the generalized Jacobian matrix and bifurcation diagrams are plotted to verify the analysis. It can be thought as the composite of a non-smooth fold bifurcation and a non-smooth Hopf bifurcation with some features different from those of usual smooth bifurcations.