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Research Article, Res Rep Math Vol: 2 Issue: 4

Sufficient condition of the Hopf bifurcation

Amine Bernoussi*

Department of Mathematics, Faculty of Science, Ibn Tofail University, Kenitra, Morocco

*Corresponding Author : Amine Bernoussi
Department of Mathematics, Faculty of Science, Ibn Tofail University, BP 133, 14000, Kenitra, Morocco
E-mail: [email protected]

Received: September 15, 2018 Accepted: December 04, 2018 Published: December 11, 2018

Citation: Bernoussi A (2018) Sufficient Condition of the Hopf Bifurcation. Res Rep Math 2:4

Abstract

In this paper we propose a new sufficient condition for the Hopf bifurcation of equilibrium is derived for nonlinear autonomous differential equations with delay in three dimension, by analysis the characteristic equation.

Keywords: Hopf bifurcation; Local stability bifurcation the hopf; Differential equations

Introduction

In this paper, we propose the following sufficient condition for the local Hopf bifurcation for a differential equation system with delay the form:

The initial condition for the above system is

image

with φ=(φ1, φ2, φ3) where φi∈C(i=1,2,3), such that φi(θ) ≥ 0(-τ≤θ≤ 0,i=1,2,3). Here C denotes the Banach space C([-τ,0]R) of continuous functions mapping the interval [-τ,0] into R, equipped with the supremum norm. The non-negative cone of C is defined as C+= C([-τ,0], R+), where R+={x∈ R|x ≥ 0}. We here assume that f: R3→ R is a locally Lipschitz continuous function on R3 Suppose that the equation (1) has a equilibrium point P*; and the characteristic equation of the equation (1) around the equilibrium point P* takes the general form:

image

With

and

image

The main results are as follows

Theorem 1 (Amine Bernoussi) Assume that the equilibrium P* of system (1) is locally asymptotically stable for τ=0:

If image Then there exists a positive τ0 such that, when τ∈[0,τ0) the steady state P* is locally asymptotically stable, and a Hopf bifurcation occurs as τ passes through τ0; where τ0 is given by

image

and ω0 is the least simple positive root of equation (5), and c, b and ͞z are defined in lemma 2 and equation (5). further we have

image

The organization of this paper is as follows. In section 2, we offer basic results for the prove of theorem 1. In section 3, we establish the local asymptotic stability and the Hopf bifurcation of the equilibrium P* and prove theorem 1. Finally a discussion and conclusion is offered in section 4.

Basic Results

We offer some basic results for the prove of theorem 1. Now we return to the study of equation (2) with τ>0. Equation (2) has a purely imaginary root iω, with ω>0:

Δ(iω, τ)=0

if and only if

image

Squaring and adding the squares together, we obtain

image

with image where A, B, C, D, E and F are given by (2).

Letting z=ω2, equation (5) becomes the following cubic equation

image

Lemma 2 [5] Define

image

(i) If c < 0, then equation (6) has at least one positive root.

(ii) If c ≥ 0 and Δ ≤ 0, then equation (6) has no positive roots.

(iii) If c ≥ 0 and Δ>0, then equation (6) has positive roots if and

only if image

From lemma 2, we have the following lemma.

Lemma 3 Assume that the equilibrium P* of system (1) is locally asymptotically stable for τ=0

(i) If one of the following:

(N1) c ≥ 0, and Δ ≤ 0,

(N2) c ≥ 0, Δ>0, and ͞z ≤ 0,

(N3) c ≥ 0, Δ>0, and h( ͞z )>0,

is true, then all roots of equation (2) have negative real parts for all τ ≥ 0.

(ii) If c<0, or c ≥ 0, Δ>0, ͞z >0, and h( ͞z )≤ 0, then all roots of equation (2) have negative real parts when

image

where Δ and ͞z >0 are defined in lemma 2

Sufficient Condition of the Hopf Bifurcation

In this section, we establish the condition sufisent of the Hopf bifurcation of the equilibrium P*.

Theorem 4 Assume that the equilibrium P* of system (1) is locally asymptotically stable for τ=0.

If image Then there exists a positive τ0 such that, when τ∈[0, τ0) the steady state P* is locally asymptotically stable, and a Hopf bifurcation occurs as τ passes through τ0, where τ0 is given by

image

and ω0 is the least simple positive root of equation (5), and c, b and z are defined in lemma 2 and equation (5). further we have

image

Proof. Consider the equation

image

the discriminant Δ1 of the equation (8) is given by

image

or by the hypothes c ≥ 0 and image therefore equation (8) admits two solutions

image

then we have

image

the hypotheses image implies thatimage Therefore

image

On the other since from the hypotheses b<0 then Δ>0 then the hypotheses (iii) of the Lemma 2 is satisfied. Hence the equation (6) has at least one positive solution, more Lemma 3 implies that all roots of equation (2) have negative real parts when τ [0, τ0).

Now we will determine τ0.

Without loss of generality, we assume that the equation (6) has three positive roots, denoted by z1, z2 and z3, respectively.

Then equation (5) has three positive roots, say

image

According to equations (3) and (4) we have

image

Then ±iωl is a pair of purely imaginary roots of equation (2) with image clearly,

image

Thus, we can define

image

Next we need to guarantee the transversality condition of the Hopf bifurcation theorem [1-4].

image

And

image

Let u(τ) and ω(τ) satisfying u(τ0)=0, and ω(τ0)=ω0. By differentiating the equations (9) and (10) with respect to τ and then set τ=τ0. Doing this, we get

image

Where

image

Solving for image we get

image

Therefore we have

image

Note that if image h(0)=c≥0.

Thus if image we have the transversality condition:

image

If image and close to τ0, then equation (2) has a root λ(τ) = u(τ)+iω(τ) satisfying u(τ)>0, which contradicts (ii) of Lemma 3. This completes the proof.

Remark 5 Our main result also valid for any differential equation of higher dimension three provided that the study of the characteristic equation of the equation reduces to study the equation (2) and more generally all the problems in Rn has equation of the form [5-7]:

image

Conclusion

In this work, we presented a mathematical analysis for a nonlinear autonomous differential equation with delays in three dimensions. The originality of this work is to have a new sufficient condition for the Hopf bifurcation of equilibria. On the other hand, the main results can be generalized to differential equation of dimension greater than three (see Remark 5).

References

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