# An Approach to Robust Control of the Hopf Bifurcation

Authors: Giacomo Innocenti; Roberto Genesio; Alberto Tesi
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

The paper illustrates a novel approach to modify the Hopf bifurcation nature via a nonlinear state feedback control, which leaves the equilibrium properties unchanged. This result is achieved by recurring to linear and nonlinear transformations, which lead the system to locally assume the ordinary differential equation representation. Third-order models are considered, since they can be seen as proper representatives of a larger class of systems. The explicit relationship between the control input and the Hopf bifurcation nature is obtained via a frequency approach, that does not need the computation of the center manifold.


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## Body

## 1. Introduction

In many fields of science, spacing from physics to biology and from engineering to economics, the learning of phenomena, characterized by complex dynamics, and their properties opened several new opportunities in the management of real processes. Consequently, new problems arose in the area of control system design and among these particular attention has been devoted to what is called the “bifurcation control”.A nonlinear dynamical system frequently presents variations of its parameters, and this can happen because of actual slow structural modifications of the considered phenomenon or due to the uncertainties affecting its modeling. When such variations happen around certain critical conditions, the system exhibits a qualitative change of behaviour defined as a bifurcation. Since the related transitions can lead to substantial effects, even destructive, for the system in study, their control has been deeply considered through more than two decades, covering theoretical aspects and various concrete applications. In this perspective, the control objective is to modify an existing bifurcation, changing its characteristics or suitably moving it or introducing a new one, in order to avoid dangerous conditions and obtain a desirable behaviour (see [1, 2] for an overview).In particular, the important case of Hopf bifurcation, leading to system oscillations, has been studied by different approaches, usually reducing the system to the essential second-order dynamics, that is, that concerning its central manifold. Time domain procedures use washout filters to stabilize dynamical feedback in [3, 4], while linear and nonlinear state feedback are considered in [5–7] and refer particularly to controller normal forms in [8–12]. Frequency domain techniques have also been studied in terms of Harmonic Balance, as reported in [13–15].This paper exploits linear and local nonlinear transformations, derived from controller normal form theory, in order to rewrite the original state equations of the system as a scalar ordinary differential equation. Then, the use of a frequency approach based on the harmonic balance method leads to analytically relate the original input to the bifurcation parameters, so giving a mathematical tool to directly design the control input once the system is given. Conditions to provide a robust control are investigated and discussed, as well. In particular, third-order systems are here considered and sufficient conditions are provided to obtain the corresponding scalar ODE representation. It is worth observing that such reduced models frequently give an efficient representation of many processes, since they can exhibit a large variety of complex dynamics (see, e.g., [16] and references within). This is the case of systems characterized by fast third-order subdynamics, where the higher order components can be seen as a slow varying external unknown input. Moreover, the scalar differential equation form is an essential point to directly derive and approximate the system dynamics around the Hopf bifurcation, since it allows one to detect the nature of such a phenomenon without recurring to the center manifold. Finally, the main idea of the paper appears in principle to be suitable for an extension to higher order systems.The paper is organized as follows. In Section2, we address the considered framework. Then, we derive in Section 3 a set of linear and nonlinear transformations, which lead the system to locally assume the ODE representation. An illustrative example is presented in Section 4 and some considerations are reported in Section 5. A mathematical appendix ends the paper.


## 2. Problem Formulation

Let us consider the affine control system(2.1)ξ̇=F(ξ;μ)+G(ξ;μ)u,
where(2.2)ξ∈R3,u∈R,μ∈Γ⊂Rn,F(ξ;μ):R3×Rn⟶R3,G(ξ;μ):R3×Rn⟶R3.
Assume that μ is a parametric vector, which may change in time according to an unknown or unmodeled slow dynamics, so that the related modifications can be considered a quasistatic process. Without loss of generality, let ξe(μ) represent the equilibrium point of the autonomous part of system (2.1), that is, the point such that(2.3)F(ξe(μ);μ)=03,∀μ∈Γ,
and assume that ξe(μ) undergoes a Hopf bifurcation at μ=μ0∈Γ.In the following, we develop an analytic method to modify the nature of such bifurcation via a locally nonlinear input, that leaves unaltered the fixed points manifold and their stability properties.Then, let us perform the change of coordinate, that brings the equilibrium points manifold (2.3) in the origin, that is, define x=ξ-ξe(μ), so that(2.4)ẋ=ξ̇=F(x+ξe(μ);μ)+G(x+ξe(μ);μ)w=F(ξe(μ);μ)+∂F∂ξ(ξe(μ);μ)x+⋯+(G(ξe(μ);μ)+∂G∂ξ(ξe(μ);μ)x+⋯)w=∂F∂ξ(ξe(μ);μ)x+⋯+(G(ξe(μ);μ)+∂G∂ξ(ξe(μ);μ)x+⋯)w.
By denoting(2.5)f̃(x;μ)=F(x+ξe(μ);μ)-F(ξe(μ);μ),Ã(μ),suchthatÃ(μ)x=f̃[1](x;μ),g̃(x;μ)=G(x+ξe(μ);μ),B̃(μ)=G(ξe(μ);μ),
we obtain the transformed system(2.6)ẋ=Ã(μ)x+f̃[2](x;μ)+⋯+(B̃(μ)+g̃[1](x;μ)+⋯)w.
Since it is wellknown that, to study the super or subcritical nature of a nondegenerate Hopf bifurcation, it is sufficient to consider the quadratic and cubic local terms [17–20], without loss of generality in the following we will focus only on the subsystem(2.7)ẋ=Ã(μ)x+f̃[2](x;μ)+f̃[3](x;μ)+(B̃(μ)+g̃[1](x;μ)+g̃[2](x;μ))w.
In the next section, we narrow our theory to linearly controllable affine systems, that is, models whose linear components satisfy the controllability condition (see, e.g., [21]).


## 3. Hopf Bifurcation in Local ODE Models

Hereafter, we analytically point out a class of third-order affine control systems, which can be locally represented via ODE models. The corresponding conditions are only sufficient and, thus, the proposed method could be generalized to a large variety of systems though this would require an increased amount of computations. However, a byproduct of our theory is the explicit relationship between the coefficients of the starting model and its local ODE representation. Hence, once such a transformation has been computed, it can be used as a look-up table for every other system in that class.Then, let us focus on linearly controllable systems. The following result holds.Theorem 3.1.
Consider system (2.7) and assume that (Ã(μ),B̃(μ)) is a controllable pair for each μ∈Γ, then, an invertible T(μ)∈ℝ3×3 and
(3.1)ν(μ)=[ν1(μ)ν2(μ)ν3(μ)]
exist such that the linear transformation
(3.2)y=T(μ)x,u=ν(μ)T(μ)x+w
modifies the system into the model
(3.3)ẏ=Ay+f[2](y;μ)+f[3](y;μ)+(B+g[1](y;μ)+g[2](y;μ))u,
where the pair (A,B) is in Brunovsky form, that is,
(3.4)A=[010001000],B=[001].It is worth underlining that transformation (3.2) can be directly computed from (A.2) via standard algorithms used to point out the proper T(μ) satisfying (A.1) (see the appendix).Corollary 3.2.
System (3.3) controlled via the input
(3.5)u=ν(μ)y+k[2](y;μ)+k[3](y;μ)
and system (2.7) with
(3.6)w=k[2](T(μ)x;μ)+k[3](T(μ)x;μ)
are locally equivalent at their equilibria ξe(μ)↔03.Hereafter, we derive a class of affine control systems, which can be locally described as ODEs, when their inputs are designed as a state feedback. This approach is motivated by the existence of simple though powerful analytic mathematical tools, developed to study Hopf bifurcations in third order nonlinear ODE systems [19, 20, 22].Theorem 3.3.
Consider the linearly controllable system (3.3), with (A,B) in Brunovsky form, then, by means of a nonlinear transformation
(3.7)y=z+φ[2](z;μ)+φ[3](z;μ),(3.8)v=u+ψ[2](z,u;μ)+ψ[3](z,u;μ),
it assumes the form
(3.9)ż1=z2+γ16(μ)z32+(δ17(μ)z1+δ18(μ)z2+δ19(μ)z3)z32,ż2=z3,ż3=v,
which is locally equivalent to (3.3) around the equilibrium in the origin.Corollary 3.4.
Consider system (3.9), where the parameters γ16(μ), δ17(μ), δ18(μ), and δ19(μ) are computed from the coefficients of the polynomial terms of the nonlinearities of model (3.3) according to Tables 1, 2, and 3. If the condition
(3.10)γ16(μ)=δ17(μ)=δ18(μ)=δ19(μ)=0,∀μ∈Γ
holds, then (3.9) can be transformed into the ODE model
(3.11)ζ⃛=v,
via the linear change of coordinates
(3.12)z1=ζ,z2=ζ̇,z3=ζ̈.Table 1
Relationships between the coefficients off[2](z), g[1](z), and φ[2](z) in (A.10)–(A.12), that is, pertaining the quadratic part of the nonlinear transformation (3.7).
α11 can be freely assigned.α12=a13+α23=a13+b21α13=b11α14=12(a15+α25-b11)=12(a15+b22-b11)α15=b12α16=12b13α21=-a21α22=-a21+2α11α23=b21α24=-a14+α12=-a14+a13+b21α25=b22α26=12b23α31=-a21α32=-a22+2α21=-a22+2(-a21)α33=-a23+α22=-a23+(-a21+2α11)α34=-a24+α22=-a24+(-a21+2α11)α35=-a25+α23+2α24=-a25+b21+2(-a14+a13+b21)α36=-a26+α25=-a26+b22γ16=α26+a16-b12=12b23+a16-b12Table 2
Relationships between the coefficients off̂[3](z) and those of f[3](z), f[2](z) and φ[2](z) as they result from condition (A.19).
i=1,2,3ĉi1=ci1+2ai1α11+ai2α21+ai3α31+2αi1(-α21-a11)+αi2(-α31-a21)-αi3a31ĉi2=ci2+2ai1α12+ai2α11+ai2α22+2ai4α21+ai3α32+ai5α31+2αi1(-α22-a12+2α11)+αi2(-α21-a11)+αi2(-α32-a22+2α21)+2αi4(-α31-a21)+αi3(-a32+2α31)-αi5a31ĉi3=ci3+2ai1α13+ai3α11+ai2α23+ai5α21+ai3α33+2ai6α31+2αi1(-α23-a13+α12)+αi3(-α21-a11)+αi2(-α33-a23+α22)+αi5(-α31-a21)+αi3(-a33+α32)-2αi6a31ĉi4=ci4+2ai1α14+ai2α12+ai2α24+2ai4α22+ai3α34+ai5α32+2αi1(-α24-a14+α12)+αi2(-α22-a12+2α11)+αi2(-α34-a24+α22)+2αi4(-α32-a22+2α21)+αi3(-a34+α32)+αi5(-a32+2α31)ĉi5=ci5+ai2α14+2ai4α24+ai5α34+αi2(-α24-a14+α12)+2αi4(-α34-a24+α22)+αi5(-a34+α32)ĉi6=ci6+ai2α15+ai3α14+2ai4α25+ai5α24+ai5α35+2ai6α34+αi2(-α25-a15+α13)+αi3(-α24-a14+α12)+2αi4(-α35-a25+α23)+αi5(-α34-a24+α22)+αi5(-a35+α33)+2αi6(-a34+α32)ĉi7=ci7+2ai1α16+ai3α13+ai2α26+ai5α23+ai3α36+2ai6α33+2αi1(-α26-a26+α15)+αi3(-α23-a13+α12)+αi2(-α36-a26+α25)+αi5(-α33-a23+α22)+αi3(-a36+α35)+2αi6(-a31+α32)ĉi8=ci8+ai2α16+ai3α15+2ai4α26+ai5α25+ai5α36+2ai6α35+αi2(-α26-a16+α15)+αi3(-α25-a15+α13+2α14)+2αi4(-α36-a26+α25)+αi5(-α25-a25+α23+2α24)+αi5(-a36+α35)+2αi6(-a35+α33+2α34)ĉi9=ci9+ai3α16+ai5α26+2ai6α36+αi3(-α26-a16+α15)+αi5(-α36-a26+α25)+2αi6(-a36+α35)ĉi0=ci0+2ai1α15+ai2α13+ai3α12+ai2α25+2ai4α23+ai5α22+ai3α35+ai5α33+2ai6α32+2αi1(-α25-a15+α13+2α14)+αi2(-α23-a13+α12)+αi3(-α22-a12+2α11)+αi2(-α35-a25+α23+2α24)+2αi4(-α33-a23+α22)+αi5(-α32-a22+2α21)+αi3(-a35+α33+2α34)+αi5(-a33+α32)+2αi6(-a32+2α31)Table 3
Relationships between the coefficients ofg[1](z) and those of ĝ[2](z) and φ[2](z) as they result from conditions (A.20).
i=1,2,3d̂i1=di1+bi1α11+bi2α21+bi3α31-2αi1b11-αi2b21-αi3b31-2αi1α13-αi2α23-αi3α33d̂i2=di2+bi1α12+bi2α22+bi3α32-2αi1b12-αi2b11-αi2b22-2αi4b21-αi3b32-αi5b31+2αi1α15+αi2α13+αi2α25+2αi4α23+αi3α35+αi5α33d̂i3=di3+bi1α13+bi2α23+bi3α33-2αi1b13-αi3b11-αi2b23-αi5b21-αi3b33-2αi6b31+4αi1α16+αi3α13+2αi2α26+αi5α23+2αi3α36+2αi6α33d̂i4=di4+bi1α14+bi2α24+bi3α34-αi2b12-2αi4b22-αi5b32-αi2α15-2αi4α25-αi5α35d̂i5=di5+bi1α15+bi2α25+bi3α35-αi2b13-αi3b12-2αi4b23-αi5b22-αi5b33-2αi6b32+2αi2α16+αi3α15+4αi4α26+αi5α25+2αi5α36+2αi6α35d̂i6=di6+bi1α16+bi2α26+bi3α36-αi3b13-αi5b23-2αi6b33-2αi3α16-2αi5α26-4αi6α36The systems satisfying condition (3.9) can be explicitly pointed out by means of the parametric relationships reported in the tables. However, they can also be represented via the functional characterization of the following statement.Theorem 3.5.
A system of the form (3.9) satisfies condition (3.10) if and only if φ[2](z),φ[3](z):ℝ3→ℝ3 and ψz[2](z),ψz[3](z),ψu[1](z),ψu[2](z):ℝ3→ℝ exist, which satisfy
(3.13)Aφ[2](z)+f[2](z)-∂φ[2]∂zAz=Bψz[2](z),(3.14)g[1](z)-∂φ[2]∂zB=Bψu[1](z),(3.15)Aφ[3](z)+f[3](z)+∂f[2]∂zφ[2](z)-∂φ[2]∂zBψz[2](z)-∂φ[3]∂zAz=Bψz[3](z),(3.16)g[2](z)+∂g[1]∂zφ[2](z)-∂φ[2]∂zBψu[1](z)-∂φ[3]∂zB=Bψu[2](z).Hereafter, a local nonlinear transformation, which maintains unaltered the linear part of the system, is introduced.Theorem 3.6.
System (3.3) controlled via the input
(3.17)u=ν(μ)y+k[2](y;μ)+k[3](y;μ)
and system (3.9), or the ODE model (3.11), controlled by
(3.18)v=ν(μ)z+v[2](z;μ)+v[3](z;μ)
are locally equivalent around their equilibria in the origin, when the following conditions, related also to the polynomial functions in (3.7) and (3.8), are satisfied:
(3.19)k[2](z;μ)=v[2](z;μ)-ν(μ)φ[2](z;μ)-ψ[2](z,ν(μ)z;μ),k[3](z;μ)=v[3](z;μ)-ν(μ)φ[3](z,ν(μ)z;μ)-∂k[2]∂zφ[2](z;μ)-∂ψ[2]∂uφ[2](z;μ)-∂ψ[2]∂uk[2](z;μ)-ψ[3](z,ν(μ)z;μ).The above Theorem3.6 allows us to design the nonlinear terms of the control input of system (3.3) and, then, of the original model (2.7), just by referring to the ODE system (3.11) driven by the input(3.20)v=ν1(μ)ζ+ν2(μ)ζ̇+ν3(μ)ζ̈+v[2](ζ,ζ̇,ζ̈;μ)+v[3](ζ,ζ̇,ζ̈;μ).
Therefore, the Hopf bifurcation nature can be modified by a suitable choice of the nonlinearities v[2](ζ,ζ̇,ζ̈;μ) and v[3](ζ,ζ̇,ζ̈;μ). Since the control system (3.11)–(3.20) is an ODE model, a generalized frequency approach, as that shown in [20, 22], can still be fruitfully employed. Moreover, by computing also the explicit relationship between the coefficients of the nonlinearities in the starting model and in the final ODE system, one can also point out the simplest input w, that provides the desired result. However, in order to illustrate this frequency approach by keeping the computational burden low, hereafter we describe how the bifurcation is affected by the only quadratic term v[2](ζ,ζ̇,ζ̈;μ), when the cubic component v[3](ζ,ζ̇,ζ̈;μ) is set to zero.Proposition 3.7.
Consider the ODE system(3.21)ζ⃛=ν3(μ)ζ̈+ν2(μ)ζ̇+ν1(μ)ζ+r1(μ)ζ2+r2(μ)ζ̇ζ+r3ζ̈ζ+r4(μ)ζ̇2+r5(μ)ζ̈ζ̇+r6(μ)ζ̈2=0,
and assume that its equilibrium point in ζ=0 undergoes a nondegenerate Hopf bifurcation at μ=μ0. Then the bifurcation is supercritical, if detM(μ0)<0, and subcritical if detM(μ0)>0, where matrix M(μ) is defined as
(3.22)M=[-ν312(-r1-(-r4+r3)ν2-r6ν22)0-2r1-r3ν20Ar2(-ν2)1/20B012(-r1+(-r3-r4)ν2-r6ν22)3ν1012(-ν2)1/2(r2+r5ν2)6(-ν2)3/200B2ν3(-ν2)1/2A-2ν2-6(-ν2)3/203ν10-ν312(-r1-(-r4+r3)ν2-r6ν22)0-2r1-r3ν20-r1-(-2r4+52r3)ν2-4r6ν22r2(-ν2)1/20(-ν2)1/2(12r2-r5ν2)012(-r1+(-r3-r4)ν2-r6ν22)3ν1012(-ν2)1/2(r2+r5ν2)6(-ν2)3/2],
where 𝒜 denotes -r1-(-2r4+(5/2)r3)ν2-4r6ν22 and ℬ denotes (-ν2)1/2((-1/2)r2+r5ν2), being the dependency on μ neglected for the sake of simplicity (for a detailed proof of the statement refer to [19]).Remark 3.8.
It is worth noticing that also the limit cycle features can be set up via the same frequency approach, as illustrated in [19, 20].Proposition3.7, along with the explicit relationships of Tables 1–4 between the coefficients of the starting systems and the final ODE model, can be fruitfully exploited to design the nonlinear components of input w in order to obtain the desired results about the Hopf bifurcation at μ=μ0.Table 4
Relationships between the coefficients off[3](z), g[2](z), and φ[3](z) in (A.10)–(A.12), that is, pertaining the cubic part of the nonlinear transformation (3.7).
β11 can be freely assigned.β12=β23+ĉ13=d̂21+ĉ13β13=d̂11β14=12(β20+ĉ10-2β13)=12(d̂22+ĉ10-2d̂11)β15=13(β26+ĉ16-β10)=13(d̂24+ĉ16-d̂12)β16=d̂14β17=12d̂13β18=12d̂15β19=13d̂16β10=d̂12β21=-ĉ11β22=-ĉ12+3β11=-ĉ12+3β11β23=d̂21β24=-ĉ14+2β12=-ĉ14+2(d̂21+ĉ13)β25=-ĉ15+β14=-ĉ15+12(d̂22+ĉ10-2d̂11)β26=d̂24β27=12d̂23β28=12d̂25β29=13d̂26β20=d̂22β31=-ĉ21β32=-ĉ22+3β21=-ĉ22-3ĉ11β33=-ĉ23+β22=-ĉ23-ĉ12+3β11β34=-ĉ24+2β22=-ĉ24+2(-ĉ12+3β11)β35=-ĉ25+β24=-ĉ25-ĉ14+2(d̂21+ĉ13)β36=-ĉ26+3β25+β10=-ĉ26+3(-ĉ15+12(d̂22+ĉ10-2d̂11))+d̂12β37=-ĉ27+β20=-ĉ27+d̂22β38=-ĉ28+β27+2β26=-ĉ28+12d̂23+2(d̂24)β39=-ĉ29+β28=-ĉ29+12d̂25β30=-ĉ20+β20+2β23+2β24=-ĉ20+d̂22+2(d̂21)+2(-ĉ14+2(d̂21+ĉ13))δ17=β27+ĉ17-β10=12d̂23+ĉ17-d̂12δ18=β28+ĉ18-2β16-β17=12d̂25+ĉ18-2d̂14-12d̂13δ19=β29+ĉ19-β18=13d̂26+ĉ19-12d̂15To this aim, it is important to underline that in general the control input designed this way depends on the bifurcation parameterμ. However, in certain situations the parameters of the model, which slowly vary in time according to an unknown law, could not be accessible to compute the proper input. In such a case, where μ cannot be observed, control w solves the problem only if it does not depend on it. This scenario can be regarded as a robust bifurcation control approach. Such a case is summarized by the following statement.Proposition 3.9 (Robust control design).
Let one consider system (2.7) along with the control input (3.6), computed according to (3.19) and related conditions. Set
(3.23)v[2](z;μ)=r1(μ)z12+r2(μ)z2z1+r3(μ)z3z1+r4(μ)z22+r5(μ)z3z2+r6(μ)z32,v[3](z;μ)=0,
and denote by
(3.24)r(μ)=[r1(μ)r2(μ)r3(μ)r4(μ)r5(μ)r6(μ)]T
the control parametric vector. Moreover, consider matrix (3.22) and define the sets
(3.25)Rsup⁡={r(μ):det⁡M(μ0)<0},Rsup⁡={r(μ):det⁡M(μ0)>0},
representing all the controls of the form (3.23), which provide, respectively, supercritical and subcritical Hopf bifurcations of the equilibrium point. Then, the corresponding bifurcation of the original system (2.7) can be turned supercritical (subcritical) without regard of the model variations, that is, it can be made supercritical (subcritical) via robust control if r(μ)∈Rsup (r(μ)∈Rsub) exist, such that the corresponding w does not depend on μ.Remark 3.10.
When only the linear part of the starting system depends on the parameterμ, any constant vector r provides a robust control in the sense of the above proposition.


## 4. Example

In order to illustrate the proposed approach to Hopf bifurcation control, let us consider a system already in the form (3.3), that is,(4.1)ẏ1=y2+y12-2y3y12+(y1-3y3y1)u,ẏ2=y3-2y2y1-2y13+(y2-3y3y2)u,ẏ3=u,
which corresponds to(4.2)f[2](y)=[y12-2y2y10],f[3](y)=[-2y3y12-2y130],g[1](y)=[y1y20],g[2](y)=[-3y3y1-3y3y20].
Then, it is straightforward to check that such a system satisfies condition (3.10), when the transformation(4.3)φ[2](z)=[z3z1-z12+z2z3z32],φ[3](z)=[-z2z12-z3z12-2z1z22-2z23-2z1z2z3],ψz[2](z)=0,ψu[1](z)=-2z3,ψz[3](z)=8z3z22+2z1z2z3,ψu[2](z)=4z32-2z1z2
is considered. Also the following relationships hold:(4.4)f̂[3](z)=[0-4z1z2z30],ĝ[2](z)=[0-z124z32].
Then, suppose that μ0=0, and let us assume that(4.5)ν1(μ)=ν1,ν2(μ)=ν2+μ,ν3(μ)=ν3
are such that the linear part of the system undergoes a Hopf bifurcation at μ=μ0. Hence, if we consider the class of controllers (3.23) featuring(4.6)r1(μ)=r1,r4(μ)=r4,ri(μ)=0,i=2,3,5,6,
we have that(4.7)M(μ0)=[-ν312(-r1+r4ν2)000-2r10-r1+2r4ν202ν3(-ν2)1/2000-r1+2r4ν2-2ν2012(-r1-r4ν2)3ν1-6(-ν2)3/20006(-ν2)3/23ν10].
Therefore,the supercritical bifurcation condition at μ=μ0 is given by(4.8)det⁡M(μ0)=3ν3(r1+r4ν2)(-r1+2r4ν2)(ν1ν2+2ν3(-ν2))-2ν2r1(r1-r4ν2)(9ν12+36(-ν2)3)<0,
defining the manifold of the pairs (r1,r4), which realize a proper control. The explicit form of the corresponding nonlinear part of u is obtained substituting the above functions and(4.9)v[2](z)=r1z12+r4z22,v[3](z)=0
in (3.19).


## 5. Conclusions

In this paper, we have introduced an approach to set up the Hopf bifurcation nature, without modifying the related equilibrium point features. This goal is achieved by introducing in the system specifically designed nonlinear terms, which only locally affect it in a close around of that fixed point. Then, the explicit form of classes of suitable control inputs has been obtained by exploiting local transformations, which lead the system to assume the ordinary differential equation form. In particular, we have considered third-order models, since they can be also used to represent certain classes of higher order systems. Indeed, they can be regarded as proper representations of more complex models characterized by an inner fast subsystem. Moreover, the control input has been designed via a frequency approach, which has not required the computation of the center manifold dynamics.



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