Chaos in the Piecewise Linear Saddle-Focus with Helicoidal Return

IFAC PapersOnLine 58-5 (2024) 60–65 ScienceDirect Available online at www.sciencedirect.com 2405-8963 Copyright © 2024 The Authors. This is an open access article under the CC BY-NC-ND license . Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2024.07.064 10.1016/j.ifacol.2024.07.064 2405-8963 Chaos in the Piecewise Linear Saddle-Focus with Helicoidal Return ⋆ Emilio Freire ∗, Enrique Ponce ∗, and Javier Ros ∗ ∗Departamento de Matem´atica Aplicada II, Universidad de Sevilla, SPAIN (e-mail: {efrem,eponcem,javieros}@us.es). Abstract: We introduce an interesting family of three-dimensional differential systems whose vector field is continuous and piecewise linear. A first analysis of its dynamics is presented, including a rigorous proof on the recurrence of orbits, and some necessary conditions for existence of homoclinic connections. Accordingly, parameter regions with Shilnikov chaos are detected. Keywords: Piecewise linear analysis, nonlinear systems, limit cycles, chaos, bifurcations. 1. INTRODUCTION The possibility of chaotic regimes in 3D autonomous piecewise linear differential systems is a well-known phenomenon, see Chua et al. (1986); Madan (1993). While several routes to chaos have been proposed, we are interested in the abrupt appearance of chaos through boundary equilibrium bifurcations, that is, when an equilibrium point crosses the boundary manifold that separates two different linearity zones of the vector field, see Glendinning (2018). Such a particular type of chaos generation is specific for piecewise linear systems and was pointed out in Example 5.8 from Di Bernardo et al. (2008). A particular case of boundary equilibrium bifurcation naturally appears associated to the zero-Hopf bifurcation of the origin in the family of symmetric, continuous piecewise linear systems written in the Lur’e form ˙x =AEx+bsat(x)=�t−10 m0−1 d00 �x+bsat(x), where x=(x,y,z)⊤,b∈R3,and the scalar saturation function ‘sat’ is given by sat(x) = sign(x)·min {|x|,1}. Here, the dot represents derivatives with respect to the time τ. As shown in Ponce et al. (2022), one can assume for the system such a generalized Li´enard form, equivalent to the observability form. If we define AC=�T−10 M0−1 D00 �,b=�T−t M−m D−d�, then the vector field for −1≤x≤1 becomes ˙x =ACx. Thus, the matrix AC(AE) rules the dynamics in the central (any external) zone, while T,M, and D(t,m, and d) are the coefficients of its characteristic polynomial. In Ponce et al. (2013), for such piecewise linear family, a specific one-parameter unfolding of the Hopf-zero bifurcation of the origin was analyzed, by assuming for the eigenvalues of ACthe configuration −ǫ, ǫρ ±jΩ, where ⋆Authors are partially supported by Spanish Ministerio de Ciencia e Innovaci´on, under grant of reference PID2021-123200NB-I00, and by Andalusian Consejer´ıa de Universidad, Investigaci´on e Innovaci´on under grant FEDER US-1380740. ρ∈Rand Ω >0 are fixed, and ǫbecomes the main bifurcation parameter. Thus, when ǫ= 0 the eigenvalues at the origin are 0 and ±jΩ, and there appears a bounded set completely filled by periodic orbis from which up to three limit cycles can bifurcate. Such eigenvalues imply T=T(ǫ) = (2ρ−1)ǫ,M=M(ǫ)=Ω 2+ρǫ2(ρ−2), D=D(ǫ)=−ǫ(ρ2ǫ2+Ω 2). Assuming d�= 0, it is easy to see that for ǫ= 0 one also has a degenerate pitchfork bifurcation of equilibria. In fact, by moving ǫ, we pass from an isolated equilibrium at the origin (dǫ < 0), to a continuum of equilibria with |x|≤1(ǫ= 0), and then to have three equilibria (the origin and a symmetric pair of points with |x|>1) when dǫ > 0. In fact, this phenomenon can be considered as a boundary equilibrium bifurcation. After the bifurcation, the additional appearance of one small limit cycle near each equilibrium point in the external zones was detected in some cases. To elucidate this phenomenon, we will assume in what follows d<0, so that the new equilibria appear for ǫ<0, and we will make the blow-up X=x−1 (−ǫ),Y=y−T(ǫ) (−ǫ),Z=z−M(ǫ) (−ǫ), where (X,Y,Z) are the new phase variables. As we are interested only in the dynamics near the new equilibria, we can discard for instance the original left zone x<−1, focussing our interest in a new system with only two zones, namely the ones with X≤0 and X>0. After some additional preparatory work, which involves some rescaling of variables, parameters, and time (not to be detailed here), and letting ǫ→0, we arrive to a continuous piecewise linear system with two zones, namely (X≤0)    ˙ X=−Y, ˙ Y=X−Z, ˙ Z=1, (X>0)    ˙ X=tX −Y, ˙ Y=mX −Z, ˙ Z=dX +1. (1) The goal of this paper is to present a first analysis of the dynamics of system (1), since it can explain the possible new phenomena associated to the aforementioned boundary equilibrium bifurcation. As it will be shown, this seemingly simple 3D continuous piecewise linear system with two zones can exhibit a very rich dynamical behaviour, including chaos. Chaos in the Piecewise Linear Saddle-Focus with Helicoidal Return ⋆ Emilio Freire ∗, Enrique Ponce ∗, and Javier Ros ∗ ∗Departamento de Matem´atica Aplicada II, Universidad de Sevilla, SPAIN (e-mail: {efrem,eponcem,javieros}@us.es). Abstract: We introduce an interesting family of three-dimensional differential systems whose vector field is continuous and piecewise linear. A first analysis of its dynamics is presented, including a rigorous proof on the recurrence of orbits, and some necessary conditions for existence of homoclinic connections. Accordingly, parameter regions with Shilnikov chaos are detected. Keywords: Piecewise linear analysis, nonlinear systems, limit cycles, chaos, bifurcations. 1. INTRODUCTION The possibility of chaotic regimes in 3D autonomous piecewise linear differential systems is a well-known phenomenon, see Chua et al. (1986); Madan (1993). While several routes to chaos have been proposed, we are interested in the abrupt appearance of chaos through boundary equilibrium bifurcations, that is, when an equilibrium point crosses the boundary manifold that separates two different linearity zones of the vector field, see Glendinning (2018). Such a particular type of chaos generation is specific for piecewise linear systems and was pointed out in Example 5.8 from Di Bernardo et al. (2008). A particular case of boundary equilibrium bifurcation naturally appears associated to the zero-Hopf bifurcation of the origin in the family of symmetric, continuous piecewise linear systems written in the Lur’e form ˙x =AEx+bsat(x)=�t−10 m0−1 d00 �x+bsat(x), where x=(x,y,z)⊤,b∈R3,and the scalar saturation function ‘sat’ is given by sat(x) = sign(x)·min {|x|,1}. Here, the dot represents derivatives with respect to the time τ. As shown in Ponce et al. (2022), one can assume for the system such a generalized Li´enard form, equivalent to the observability form. If we define AC=�T−10 M0−1 D00 �,b=�T−t M−m D−d�, then the vector field for −1≤x≤1 becomes ˙x =ACx. Thus, the matrix AC(AE) rules the dynamics in the central (any external) zone, while T,M, and D(t,m, and d) are the coefficients of its characteristic polynomial. In Ponce et al. (2013), for such piecewise linear family, a specific one-parameter unfolding of the Hopf-zero bifurcation of the origin was analyzed, by assuming for the eigenvalues of ACthe configuration −ǫ, ǫρ ±jΩ, where ⋆Authors are partially supported by Spanish Ministerio de Ciencia e Innovaci´on, under grant of reference PID2021-123200NB-I00, and by Andalusian Consejer´ıa de Universidad, Investigaci´on e Innovaci´on under grant FEDER US-1380740. ρ∈Rand Ω >0 are fixed, and ǫbecomes the main bifurcation parameter. Thus, when ǫ= 0 the eigenvalues at the origin are 0 and ±jΩ, and there appears a bounded set completely filled by periodic orbis from which up to three limit cycles can bifurcate. Such eigenvalues imply T=T(ǫ) = (2ρ−1)ǫ,M=M(ǫ)=Ω 2+ρǫ2(ρ−2), D=D(ǫ)=−ǫ(ρ2ǫ2+Ω 2). Assuming d�= 0, it is easy to see that for ǫ= 0 one also has a degenerate pitchfork bifurcation of equilibria. In fact, by moving ǫ, we pass from an isolated equilibrium at the origin (dǫ < 0), to a continuum of equilibria with |x|≤1(ǫ= 0), and then to have three equilibria (the origin and a symmetric pair of points with |x|>1) when dǫ > 0. In fact, this phenomenon can be considered as a boundary equilibrium bifurcation. After the bifurcation, the additional appearance of one small limit cycle near each equilibrium point in the external zones was detected in some cases. To elucidate this phenomenon, we will assume in what follows d<0, so that the new equilibria appear for ǫ<0, and we will make the blow-up X=x−1 (−ǫ),Y=y−T(ǫ) (−ǫ),Z=z−M(ǫ) (−ǫ), where (X,Y,Z) are the new phase variables. As we are interested only in the dynamics near the new equilibria, we can discard for instance the original left zone x<−1, focussing our interest in a new system with only two zones, namely the ones with X≤0 and X>0. After some additional preparatory work, which involves some rescaling of variables, parameters, and time (not to be detailed here), and letting ǫ→0, we arrive to a continuous piecewise linear system with two zones, namely (X≤0)    ˙ X=−Y, ˙ Y=X−Z, ˙ Z=1, (X>0)    ˙ X=tX −Y, ˙ Y=mX −Z, ˙ Z=dX +1. (1) The goal of this paper is to present a first analysis of the dynamics of system (1), since it can explain the possible new phenomena associated to the aforementioned boundary equilibrium bifurcation. As it will be shown, this seemingly simple 3D continuous piecewise linear system with two zones can exhibit a very rich dynamical behaviour, including chaos. Chaos in the Piecewise Linear Saddle-Focus with Helicoidal Return ⋆ Emilio Freire ∗, Enrique Ponce ∗, and Javier Ros ∗ ∗Departamento de Matem´atica Aplicada II, Universidad de Sevilla, SPAIN (e-mail: {efrem,eponcem,javieros}@us.es). Abstract: We introduce an interesting family of three-dimensional differential systems whose vector field is continuous and piecewise linear. A first analysis of its dynamics is presented, including a rigorous proof on the recurrence of orbits, and some necessary conditions for existence of homoclinic connections. Accordingly, parameter regions with Shilnikov chaos are detected. Keywords: Piecewise linear analysis, nonlinear systems, limit cycles, chaos, bifurcations. 1. INTRODUCTION The possibility of chaotic regimes in 3D autonomous piecewise linear differential systems is a well-known phenomenon, see Chua et al. (1986); Madan (1993). While several routes to chaos have been proposed, we are interested in the abrupt appearance of chaos through boundary equilibrium bifurcations, that is, when an equilibrium point crosses the boundary manifold that separates two different linearity zones of the vector field, see Glendinning (2018). Such a particular type of chaos generation is specific for piecewise linear systems and was pointed out in Example 5.8 from Di Bernardo et al. (2008). A particular case of boundary equilibrium bifurcation naturally appears associated to the zero-Hopf bifurcation of the origin in the family of symmetric, continuous piecewise linear systems written in the Lur’e form ˙x =AEx+bsat(x)=�t−10 m0−1 d00 �x+bsat(x), where x=(x,y,z)⊤,b∈R3,and the scalar saturation function ‘sat’ is given by sat(x) = sign(x)·min {|x|,1}. Here, the dot represents derivatives with respect to the time τ. As shown in Ponce et al. (2022), one can assume for the system such a generalized Li´enard form, equivalent to the observability form. If we define AC=�T−10 M0−1 D00 �,b=�T−t M−m D−d�, then the vector field for −1≤x≤1 becomes ˙x =ACx. Thus, the matrix AC(AE) rules the dynamics in the central (any external) zone, while T,M, and D(t,m, and d) are the coefficients of its characteristic polynomial. In Ponce et al. (2013), for such piecewise linear family, a specific one-parameter unfolding of the Hopf-zero bifurcation of the origin was analyzed, by assuming for the eigenvalues of ACthe configuration −ǫ, ǫρ ±jΩ, where ⋆Authors are partially supported by Spanish Ministerio de Ciencia e Innovaci´on, under grant of reference PID2021-123200NB-I00, and by Andalusian Consejer´ıa de Universidad, Investigaci´on e Innovaci´on under grant FEDER US-1380740. ρ∈Rand Ω >0 are fixed, and ǫbecomes the main bifurcation parameter. Thus, when ǫ= 0 the eigenvalues at the origin are 0 and ±jΩ, and there appears a bounded set completely filled by periodic orbis from which up to three limit cycles can bifurcate. Such eigenvalues imply T=T(ǫ) = (2ρ−1)ǫ,M=M(ǫ)=Ω 2+ρǫ2(ρ−2), D=D(ǫ)=−ǫ(ρ2ǫ2+Ω 2). Assuming d�= 0, it is easy to see that for ǫ= 0 one also has a degenerate pitchfork bifurcation of equilibria. In fact, by moving ǫ, we pass from an isolated equilibrium at the origin (dǫ < 0), to a continuum of equilibria with |x|≤1(ǫ= 0), and then to have three equilibria (the origin and a symmetric pair of points with |x|>1) when dǫ > 0. In fact, this phenomenon can be considered as a boundary equilibrium bifurcation. After the bifurcation, the additional appearance of one small limit cycle near each equilibrium point in the external zones was detected in some cases. To elucidate this phenomenon, we will assume in what follows d<0, so that the new equilibria appear for ǫ<0, and we will make the blow-up X=x−1 (−ǫ),Y=y−T(ǫ) (−ǫ),Z=z−M(ǫ) (−ǫ), where (X,Y,Z) are the new phase variables. As we are interested only in the dynamics near the new equilibria, we can discard for instance the original left zone x<−1, focussing our interest in a new system with only two zones, namely the ones with X≤0 and X>0. After some additional preparatory work, which involves some rescaling of variables, parameters, and time (not to be detailed here), and letting ǫ→0, we arrive to a continuous piecewise linear system with two zones, namely (X≤0)    ˙ X=−Y, ˙ Y=X−Z, ˙ Z=1, (X>0)    ˙ X=tX −Y, ˙ Y=mX −Z, ˙ Z=dX +1. (1) The goal of this paper is to present a first analysis of the dynamics of system (1), since it can explain the possible new phenomena associated to the aforementioned boundary equilibrium bifurcation. As it will be shown, this seemingly simple 3D continuous piecewise linear system with two zones can exhibit a very rich dynamical behaviour, including chaos. Chaos in the Piecewise Linear Saddle-Focus with Helicoidal Return ⋆ Emilio Freire ∗, Enrique Ponce ∗, and Javier Ros ∗ ∗Departamento de Matem´atica Aplicada II, Universidad de Sevilla, SPAIN (e-mail: {efrem,eponcem,javieros}@us.es). Abstract: We introduce an interesting family of three-dimensional differential systems whose vector field is continuous and piecewise linear. A first analysis of its dynamics is presented, including a rigorous proof on the recurrence of orbits, and some necessary conditions for existence of homoclinic connections. Accordingly, parameter regions with Shilnikov chaos are detected. Keywords: Piecewise linear analysis, nonlinear systems, limit cycles, chaos, bifurcations. 1. INTRODUCTION The possibility of chaotic regimes in 3D autonomous piecewise linear differential systems is a well-known phenomenon, see Chua et al. (1986); Madan (1993). While several routes to chaos have been proposed, we are interested in the abrupt appearance of chaos through boundary equilibrium bifurcations, that is, when an equilibrium point crosses the boundary manifold that separates two different linearity zones of the vector field, see Glendinning (2018). Such a particular type of chaos generation is specific for piecewise linear systems and was pointed out in Example 5.8 from Di Bernardo et al. (2008). A particular case of boundary equilibrium bifurcation naturally appears associated to the zero-Hopf bifurcation of the origin in the family of symmetric, continuous piecewise linear systems written in the Lur’e form ˙x =AEx+bsat(x)=�t−10 m0−1 d00 �x+bsat(x), where x=(x,y,z)⊤,b∈R3,and the scalar saturation function ‘sat’ is given by sat(x) = sign(x)·min {|x|,1}. Here, the dot represents derivatives with respect to the time τ. As shown in Ponce et al. (2022), one can assume for the system such a generalized Li´enard form, equivalent to the observability form. If we define AC=�T−10 M0−1 D00 �,b=�T−t M−m D−d�, then the vector field for −1≤x≤1 becomes ˙x =ACx. Thus, the matrix AC(AE) rules the dynamics in the central (any external) zone, while T,M, and D(t,m, and d) are the coefficients of its characteristic polynomial. In Ponce et al. (2013), for such piecewise linear family, a specific one-parameter unfolding of the Hopf-zero bifurcation of the origin was analyzed, by assuming for the eigenvalues of ACthe configuration −ǫ, ǫρ ±jΩ, where ⋆Authors are partially supported by Spanish Ministerio de Ciencia e Innovaci´on, under grant of reference PID2021-123200NB-I00, and by Andalusian Consejer´ıa de Universidad, Investigaci´on e Innovaci´on under grant FEDER US-1380740. ρ∈Rand Ω >0 are fixed, and ǫbecomes the main bifurcation parameter. Thus, when ǫ= 0 the eigenvalues at the origin are 0 and ±jΩ, and there appears a bounded set completely filled by periodic orbis from which up to three limit cycles can bifurcate. Such eigenvalues imply T=T(ǫ) = (2ρ−1)ǫ,M=M(ǫ)=Ω 2+ρǫ2(ρ−2), D=D(ǫ)=−ǫ(ρ2ǫ2+Ω 2). Assuming d�= 0, it is easy to see that for ǫ= 0 one also has a degenerate pitchfork bifurcation of equilibria. In fact, by moving ǫ, we pass from an isolated equilibrium at the origin (dǫ < 0), to a continuum of equilibria with |x|≤1(ǫ= 0), and then to have three equilibria (the origin and a symmetric pair of points with |x|>1) when dǫ > 0. In fact, this phenomenon can be considered as a boundary equilibrium bifurcation. After the bifurcation, the additional appearance of one small limit cycle near each equilibrium point in the external zones was detected in some cases. To elucidate this phenomenon, we will assume in what follows d<0, so that the new equilibria appear for ǫ<0, and we will make the blow-up X=x−1 (−ǫ),Y=y−T(ǫ) (−ǫ),Z=z−M(ǫ) (−ǫ), where (X,Y,Z) are the new phase variables. As we are interested only in the dynamics near the new equilibria, we can discard for instance the original left zone x<−1, focussing our interest in a new system with only two zones, namely the ones with X≤0 and X>0. After some additional preparatory work, which involves some rescaling of variables, parameters, and time (not to be detailed here), and letting ǫ→0, we arrive to a continuous piecewise linear system with two zones, namely (X≤0)    ˙ X=−Y, ˙ Y=X−Z, ˙ Z=1, (X>0)    ˙ X=tX −Y, ˙ Y=mX −Z, ˙ Z=dX +1. (1) The goal of this paper is to present a first analysis of the dynamics of system (1), since it can explain the possible new phenomena associated to the aforementioned boundary equilibrium bifurcation. As it will be shown, this seemingly simple 3D continuous piecewise linear system with two zones can exhibit a very rich dynamical behaviour, including chaos. Chaos in the Piecewise Linear Saddle-Focus with Helicoidal Return ⋆ Emilio Freire ∗, Enrique Ponce ∗, and Javier Ros ∗ ∗Departamento de Matem´atica Aplicada II, Universidad de Sevilla, SPAIN (e-mail: {efrem,eponcem,javieros}@us.es). Abstract: We introduce an interesting family of three-dimensional differential systems whose vector field is continuous and piecewise linear. A first analysis of its dynamics is presented, including a rigorous proof on the recurrence of orbits, and some necessary conditions for existence of homoclinic connections. Accordingly, parameter regions with Shilnikov chaos are detected. Keywords: Piecewise linear analysis, nonlinear systems, limit cycles, chaos, bifurcations. 1. INTRODUCTION The possibility of chaotic regimes in 3D autonomous piecewise linear differential systems is a well-known phenomenon, see Chua et al. (1986); Madan (1993). While several routes to chaos have been proposed, we are interested in the abrupt appearance of chaos through boundary equilibrium bifurcations, that is, when an equilibrium point crosses the boundary manifold that separates two different linearity zones of the vector field, see Glendinning (2018). Such a particular type of chaos generation is specific for piecewise linear systems and was pointed out in Example 5.8 from Di Bernardo et al. (2008). A particular case of boundary equilibrium bifurcation naturally appears associated to the zero-Hopf bifurcation of the origin in the family of symmetric, continuous piecewise linear systems written in the Lur’e form ˙x =AEx+bsat(x)=�t−10 m0−1 d00 �x+bsat(x), where x=(x,y,z)⊤,b∈R3,and the scalar saturation function ‘sat’ is given by sat(x) = sign(x)·min {|x|,1}. Here, the dot represents derivatives with respect to the time τ. As shown in Ponce et al. (2022), one can assume for the system such a generalized Li´enard form, equivalent to the observability form. If we define AC=�T−10 M0−1 D00 �,b=�T−t M−m D−d�, then the vector field for −1≤x≤1 becomes ˙x =ACx. Thus, the matrix AC(AE) rules the dynamics in the central (any external) zone, while T,M, and D(t,m, and d) are the coefficients of its characteristic polynomial. In Ponce et al. (2013), for such piecewise linear family, a specific one-parameter unfolding of the Hopf-zero bifurcation of the origin was analyzed, by assuming for the eigenvalues of ACthe configuration −ǫ, ǫρ ±jΩ, where ⋆Authors are partially supported by Spanish Ministerio de Ciencia e Innovaci´on, under grant of reference PID2021-123200NB-I00, and by Andalusian Consejer´ıa de Universidad, Investigaci´on e Innovaci´on under grant FEDER US-1380740. ρ∈Rand Ω >0 are fixed, and ǫbecomes the main bifurcation parameter. Thus, when ǫ= 0 the eigenvalues at the origin are 0 and ±jΩ, and there appears a bounded set completely filled by periodic orbis from which up to three limit cycles can bifurcate. Such eigenvalues imply T=T(ǫ) = (2ρ−1)ǫ,M=M(ǫ)=Ω 2+ρǫ2(ρ−2), D=D(ǫ)=−ǫ(ρ2ǫ2+Ω 2). Assuming d�= 0, it is easy to see that for ǫ= 0 one also has a degenerate pitchfork bifurcation of equilibria. In fact, by moving ǫ, we pass from an isolated equilibrium at the origin (dǫ < 0), to a continuum of equilibria with |x|≤1(ǫ= 0), and then to have three equilibria (the origin and a symmetric pair of points with |x|>1) when dǫ > 0. In fact, this phenomenon can be considered as a boundary equilibrium bifurcation. After the bifurcation, the additional appearance of one small limit cycle near each equilibrium point in the external zones was detected in some cases. To elucidate this phenomenon, we will assume in what follows d<0, so that the new equilibria appear for ǫ<0, and we will make the blow-up X=x−1 (−ǫ),Y=y−T(ǫ) (−ǫ),Z=z−M(ǫ) (−ǫ), where (X,Y,Z) are the new phase variables. As we are interested only in the dynamics near the new equilibria, we can discard for instance the original left zone x<−1, focussing our interest in a new system with only two zones, namely the ones with X≤0 and X>0. After some additional preparatory work, which involves some rescaling of variables, parameters, and time (not to be detailed here), and letting ǫ→0, we arrive to a continuous piecewise linear system with two zones, namely (X≤0)    ˙ X=−Y, ˙ Y=X−Z, ˙ Z=1, (X>0)    ˙ X=tX −Y, ˙ Y=mX −Z, ˙ Z=dX +1. (1) The goal of this paper is to present a first analysis of the dynamics of system (1), since it can explain the possible new phenomena associated to the aforementioned boundary equilibrium bifurcation. As it will be shown, this seemingly simple 3D continuous piecewise linear system with two zones can exhibit a very rich dynamical behaviour, including chaos. Chaos in the Piecewise Linear Saddle-Focus with Helicoidal Return ⋆ Emilio Freire ∗, Enrique Ponce ∗, and Javier Ros ∗ ∗Departamento de Matem´atica Aplicada II, Universidad de Sevilla, SPAIN (e-mail: {efrem,eponcem,javieros}@us.es). Abstract: We introduce an interesting family of three-dimensional differential systems whose vector field is continuous and piecewise linear. A first analysis of its dynamics is presented, including a rigorous proof on the recurrence of orbits, and some necessary conditions for existence of homoclinic connections. Accordingly, parameter regions with Shilnikov chaos are detected. Keywords: Piecewise linear analysis, nonlinear systems, limit cycles, chaos, bifurcations. 1. INTRODUCTION The possibility of chaotic regimes in 3D autonomous piecewise linear differential systems is a well-known phenomenon, see Chua et al. (1986); Madan (1993). While several routes to chaos have been proposed, we are interested in the abrupt appearance of chaos through boundary equilibrium bifurcations, that is, when an equilibrium point crosses the boundary manifold that separates two different linearity zones of the vector field, see Glendinning (2018). Such a particular type of chaos generation is specific for piecewise linear systems and was pointed out in Example 5.8 from Di Bernardo et al. (2008). A particular case of boundary equilibrium bifurcation naturally appears associated to the zero-Hopf bifurcation of the origin in the family of symmetric, continuous piecewise linear systems written in the Lur’e form ˙x =AEx+bsat(x)=�t−10 m0−1 d00 �x+bsat(x), where x=(x,y,z)⊤,b∈R3,and the scalar saturation function ‘sat’ is given by sat(x) = sign(x)·min {|x|,1}. Here, the dot represents derivatives with respect to the time τ. As shown in Ponce et al. (2022), one can assume for the system such a generalized Li´enard form, equivalent to the observability form. If we define AC=�T−10 M0−1 D00 �,b=�T−t M−m D−d�, then the vector field for −1≤x≤1 becomes ˙x =ACx. Thus, the matrix AC(AE) rules the dynamics in the central (any external) zone, while T,M, and D(t,m, and d) are the coefficients of its characteristic polynomial. In Ponce et al. (2013), for such piecewise linear family, a specific one-parameter unfolding of the Hopf-zero bifurcation of the origin was analyzed, by assuming for the eigenvalues of ACthe configuration −ǫ, ǫρ ±jΩ, where ⋆Authors are partially supported by Spanish Ministerio de Ciencia e Innovaci´on, under grant of reference PID2021-123200NB-I00, and by Andalusian Consejer´ıa de Universidad, Investigaci´on e Innovaci´on under grant FEDER US-1380740. ρ∈Rand Ω >0 are fixed, and ǫbecomes the main bifurcation parameter. Thus, when ǫ= 0 the eigenvalues at the origin are 0 and ±jΩ, and there appears a bounded set completely filled by periodic orbis from which up to three limit cycles can bifurcate. Such eigenvalues imply T=T(ǫ) = (2ρ−1)ǫ,M=M(ǫ)=Ω 2+ρǫ2(ρ−2), D=D(ǫ)=−ǫ(ρ2ǫ2+Ω 2). Assuming d�= 0, it is easy to see that for ǫ= 0 one also has a degenerate pitchfork bifurcation of equilibria. In fact, by moving ǫ, we pass from an isolated equilibrium at the origin (dǫ < 0), to a continuum of equilibria with |x|≤1(ǫ= 0), and then to have three equilibria (the origin and a symmetric pair of points with |x|>1) when dǫ > 0. In fact, this phenomenon can be considered as a boundary equilibrium bifurcation. After the bifurcation, the additional appearance of one small limit cycle near each equilibrium point in the external zones was detected in some cases. To elucidate this phenomenon, we will assume in what follows d<0, so that the new equilibria appear for ǫ<0, and we will make the blow-up X=x−1 (−ǫ),Y=y−T(ǫ) (−ǫ),Z=z−M(ǫ) (−ǫ), where (X,Y,Z) are the new phase variables. As we are interested only in the dynamics near the new equilibria, we can discard for instance the original left zone x<−1, focussing our interest in a new system with only two zones, namely the ones with X≤0 and X>0. After some additional preparatory work, which involves some rescaling of variables, parameters, and time (not to be detailed here), and letting ǫ→0, we arrive to a continuous piecewise linear system with two zones, namely (X≤0)    ˙ X=−Y, ˙ Y=X−Z, ˙ Z=1, (X>0)    ˙ X=tX −Y, ˙ Y=mX −Z, ˙ Z=dX +1. (1) The goal of this paper is to present a first analysis of the dynamics of system (1), since it can explain the possible new phenomena associated to the aforementioned boundary equilibrium bifurcation. As it will be shown, this seemingly simple 3D continuous piecewise linear system with two zones can exhibit a very rich dynamical behaviour, including chaos. Copyright © 2024 The Authors. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0/ ) Emilio Freire et al. / IFAC PapersOnLine 58-5 (2024) 60–65 61 Copyright © 2024 The Authors. This is an open access article under the CC BY-NC-ND license ( https://creativecommons.org/licenses/by-nc-nd/4.0/ ) 2. SOME PRELIMINARY RESULTS Regarding system (1), we will speak of ‘left’ subsystem when referring to the dynamics for X≤0 and ‘right’ one for the dynamics with X>0. A first result for the left dynamics is as follows. Lemma 1. Any orbit starting at (0,Y 0,Z 0) with Y0>0 enters the half-space X<0 and returns to the plane X= 0 at a point (0,Y 1,Z 1) with Y1<0, after a time 0<τ L<2π. Proof. By direct integration from the point (0,Y 0,Z 0) with Y0>0, where ˙ X<0, and while the orbit do not cross the plane X= 0, we can write X(τ)=−(1 + Y0) sin τ+Z0(1 −cos τ)+τ, Y(τ) = (1 + Y0) cos τ−Z0sin τ−1, Z(τ)=Z0+τ. (2) For 0 <τ<2πand putting τ=2θ, the condition X(τ) = 0 from (2) can be written as f(θ) := (1 + Y0) cos θ−Z0sin θ=θ sin θ. The above right-hand side is an increasing function that takes all the values from 1 to ∞when θ∈(0,π). We also have f(0) = 1 + Y0>1 and f(π)=−1−Y0<0, so that from the intermediate value theorem there exists a value 0<θ L<πsuch that X(τL)=X(2θL) = 0. After some manipulations, we get Y1=Y(2θL)=−Y0−2sin θL−θLcos θL sin θL <0, and the lemma follows.  In fact, from this result and the 2π-periodicity for Y(τ) in (2), while Z(τ) grows with the time, we see that orbits in the left zone have a kind of helicoidal behavior. This is the reason why we speak of helicoidad return in the title of the paper. This left system has the polynomial first integral H(X,Y,Z)=(X−Z)2+(Y+ 1)2, so that all the cylinders H(X,Y,Z)=k≥0 are invariant; each helicoidal orbit must belong to one of these cylinders. As mentioned, we assume in the following that d<0. This hypothesis is natural, since otherwise the derivative ˙ Zwould be always positive in the phase space and we could not have periodic orbits. Our system has so the equilibrium point (X,Y,Z)=(−1/d, −t/d, −m/d) at the right zone. Furthermore, under the assumption of complex eigenvalues for such equilibrium point, we can assure the existence of periodic orbits after some bifurcation of type focuscenter-limit cycle, see Carmona et al. (2005). In such a case, an equilibrium point of saddle- or node-focus type becomes a saddle- or node-center having a bounded period annulus in the focal plane, with the biggest periodic orbit tangent to the separation plane X= 0. This critical situation requires m>0 and appears for mt−d= 0, so that, if tis the bifurcation parameter, then its critical value is tc=d/m. By moving the bifurcation parameter taway from its critical value, the bounded linear center is distroyed but one limit cycle can bifurcate from the largest periodic orbit which is tangent to the separation plane. In fact, Theorem 1 in Carmona et al. (2005) implies the following result, where it is assumed m<1 for sake of definiteness. ˜ t ˜ d ˜ d=˜ t m˜ d=˜ t Fig. 1. In the plane (˜ t, ˜ d), for ˜ t<0, the cuspidal region corresponding to three real roots is far from the sectorial region of interest ˜ d<˜ t<m˜ d. Proposition 2. System (1) with 0 <m<1 and d<0 undergoes a focus-center-limit cycle bifurcation for t=tc; that is, from the linear center configuration that exists for X>0 when t=tc, one orbitally asymptotically stable limit cycle appears for t>t cand t−tcsufficiently small. The limit cycle predicted by Proposition 2 will persist for some range of values of twith t>t cuntil its possible disappearance in another bifurcation. The above family of cylinders is responsible for the existence of a center at infinity. From Theorem 2 in Freire et al. (2020), the folllowing result holds, where again the trace tis assumed to be the main bifurcation parameter. Proposition 3. System (1) with 0 <m<1 and d<0 undergoes a limit cycle bifurcation from infinity for t=d, so that one orbitally asymptotically stable limit cycle of great amplitude appears for d−t>0 and sufficiently small. Propositions 2 and 3 are two local results that would naively lead to claim that system (1) with 0 <m<1 and d<0 has one stable limit cycle for any value of twith tc<t<d, see Conjecture 3 in Freire et al. (2020). As shown later, the situation is not so simple, but at least the structure of eigenvalues does not change. Proposition 4. Assuming 0 <m<1 and d<0, the right subsystem has for tc<t<da real eigenvalue λ<0 and a complex eigenvalue pair σ±jω, with σ>0 and 0 <ω<1. Proof. Starting from the characteristic polynomial ξ3−tξ2+mξ −d=0, and rescaling the roots and coefficients so that ξ=m1/2µ, t=m1/2t,d=m3/2 d, we get the normalized polynomial µ3−tµ2+µ− d=0. The structure of eigenvalues changes trough the critical situation with one real root λ0and a double real root σ0. This boundary configuration implies t=λ0+2σ0, 1=2λ0σ0+σ2 0, and  d=λ0σ2 0. Solving for λ0in the second condition and substituting in the other two ones, we get the parameterization λ0=1−σ2 0 2σ0 ,t=1+3σ2 0 2σ0 , d=σ0(1 −σ2 0) 2, which defines the boundary of a cuspidal region in the halfplane ˜ t<0 where the three roots are real, see Fig. 1. Since the conditions tc<t<d<0 define in the normalized parameters the sectorial region  d<t<m  d<0,far from the mentioned cuspidal region, the system has always 62 Emilio Freire et al. / IFAC PapersOnLine 58-5 (2024) 60–65 1 1/λ −λ m=1 ω σ Fig. 2. Preliminary bifurcation set in the plane (σ, ω) for a fixed value of λ(the case λ=−1/2 is drawn). The arrows indicate the known bifurcations of periodic orbits, when entering the region with σ>0 and t<d. complex eigenvalues. Thus, we can write for the original parameters the equalities t=λ+2σ, m =2λσ +σ2+ω2,d=λ(σ2+ω2),(3) and so λ<0, while a simple computation gives mt−d=2σ[(λ+σ)2+ω2].(4) Therefore, sign(σ) = sign(mt−d) = sign(t−tc),and so σ>0, as stated. If we suppose ω≥1 then λω2≤λand from t<dwe get λ+2σ<λ(σ2+ω2)≤λσ2+λ, leading to the contradiction 2σ≤λσ2. The proposition follows.  Once we know that the structure of eigenvalues is fixed within our range of interest, we can take some of these eigenvalues as bifurcation parameters. We will assume that the real eigenvalue λis fixed, and draw in a parameter plane (σ, ω) the bifurcation points regarding Propositions 2 and 3. From (4), the bifurcation reported in Proposition 2 occurs at σ= 0. The condition t=dfor the bifurcation of Proposition 3 gives (σ−1/λ)2+ω2= 1+(1/λ)2, which represents a circle in the plane (σ, ω), with center at (1/λ, 0) and radius 1+(1/λ)2,seeFig.2.Therewehave also drawn the circle corresponding to m= 1, condition which is equivalent to (σ+λ)2+ω2=1+λ2. We show now a recurrence property for orbits which is much stronger than the given in Lemma 1. Proposition 5. If 0 <m<1, d<0 and tc<t<d, then the invariant focal plane of the right subsystem is Π(X,Y,Z) := λ2X−λY +Z+1/λ = 0, where λ<0 is its real eigenvalue. Any orbit of system (1) with initial point in the set {(X,Y,Z):X=0,Y > 0,Z ≤λY −1/λ}will return after a time 0 <τ<2πto a point of the planar set {(X,Y,Z):X=0,Y <0, Z < λY −1/λ}. Proof. Substituting (3) in matrix AE, we check that (λ2,−λ, 1) is a left eigenvector associated to the eigenvalue λfor AE. Thus, the expression for the focal plane comes from (λ2,−λ, 1)⊤(X−X,Y −Y,Z−Z) = 0. We see that d dτ Π(X,Y,Z)=λ2˙ X−λ˙ Y+˙ Z=λ3X−λ2Y+λZ +1, so that (d/dτ)Π(X,Y,Z)=λΠ(X,Y,Z), and the plane is invariant under the flow of the right subsystem. We consider now the points in the half straight line with Y>0 where this focal plane intersects the separation plane X= 0, see Fig. 3. Our claim is that their return points define a curve with Y<0 that is below the trace of such focal plane (the curve in red). Then the conclusion Y Z Fig. 3. On the separation plane X= 0, the curve in red corresponds to the return points through the flow of system (1) for the points of the trace of the focal plane with Y>0. The figure is for the case λ=−1/2. Each circle associates an initial point in black of an orbit with its return point to the separation plane in red. follows, since points below the trace will return below such a curve, and so all the more reason below the trace. We only sketch the proof of the claim, which is rather technical. Any orbit for a point (0,Y 0, λY0−1/λ) with Y0>0 in the trace, will return at a point (0,Y 1,Z 1) with Y1<0, according to Lemma 1. Imposing X(τ)=0and Z0=λY0−1/λ in (2), we get a monotone parameterization of Y0in terms of the return time τ, namely Y0(τ)=−1 λ·p(τ) q(τ)=−1 λ·1−cos τ−λ(τ−sin τ) sin τ−λ(1 −cos τ),(5) where p(τ) and q(τ)=p′(τ) are non-negative functions for 0<τ<τ max =2π+ 2 arctan(1/λ)∈(π,2π). Accordingly, using again (2), we write Y1(τ)=Y0(τ)q′(τ)+q(τ)/λ, Z1(τ)=λY0(τ)−1/λ +τ, (6) and we must study the curve (Y1(τ),Z 1(τ)) in the plane X= 0. By using extensively that p′(τ)=q(τ) and q′′(τ)=−q(λ)−λ, we compute the first derivatives of the parameterizations (6). Thus, we obtain that Y′ 1(τ)=p(τ) (1 −λq(τ)) λq(τ)2,Z ′ 1(τ)=p(τ)q′(τ) q(τ)2, so that dZ1 dY1 (τ)= λq′(τ) 1−λq(τ). Somehow longer computations give d2Z1 dY 2 1 (τ)=Z′′ 1Y′ 1−Z′ 1Y′′ 1 (Y′ 1)3(τ)=−λ2(1 + λ2)q(τ)3 p(τ) (1 −λq(τ))3. The conclusion is now a direct consequence of having for all τ∈(0,τ max) a negative second derivative, along with dZ1 dY1 (0) = λq′(0) 1−λq(0) =λ<0, so that the curve is tangent to the trace, see Fig. 3.  We end this section by emphasizing that in Fig. 3 one can put in correspondence each black point with its image in red by imposing that both points must belong to the same invariant cylinder under the flow of the left subsystem. Since each cylinder intersects the separation plane X=0 in a circle with center at (Y,Z)=(−1,0), it suffices to draw such circles to connect initial and return points. 3. CHAOS AND HOMOCLINIC CONNECTIONS Since the equilibrium point is a saddle-focus, the so called Shilnikov chaos could be expected, see Glendinning and Emilio Freire et al. / IFAC PapersOnLine 58-5 (2024) 60–65 63 Fig. 4. Chaos for λ=−1/2, ω=1/4, and σ≈0.028531. Fig. 5. A periodic orbit with 4 loops (2 loops) that appears for λ=−1/2, ω=1/4, and σ=0.12 (σ=0.13). Sparrow (1984); Gaspard et al. (1984); Silva (1993); Llibre et al. (2007). Effectively, we found by direct simulation the existence of chaotic regimes, see Fig. 4. In fact, by selecting λ=−1/2 and ω=1/4, and varying σ from zero to positive values, one see chaos for rather small values, as σ≈0.028531. Recall that Shilnikov condition is σ<|λ|. Such regimes arises from the stable limit cycle predicted by Proposition 2, after some cascade of doubling period bifurcations in a very small range; for bigger values of σwe have another inverse cascade of doubling period bifurcations, see Fig. 5. Since such chaotic regimes are associated to the existence of homoclinic orbits, we devote this section to look for such specific orbits. For a point in {(X,Y,Z):X=0,Y <0,Z ≤λY −1/λ}, its orbit will enter the region X>0, remaining below the focal plane until it generically hits a point of the planar set {(X,Y,Z):X=0,Y >0, Z < λY −1/λ}.We must except the point where the one-dimensional stable manifold of the equilibrium point intersects the plane X= 0. This point can be computed as follows. We find that (1,2σ, σ2+ω2)⊤ is one λ-eigenvector for matrix AE, so that �X(τ) Y(τ) Z(τ)�=  X Y Z +µ  1 2σ σ2+ω2 eλτ is a solution of the right subsystem for any µ∈Rsuch that X(τ)≥0. Thus, the one-dimensional stable invariant manifold of the equilibrium point is X−X 1=Y−Y 2σ=Z−Z σ2+ω2, intersecting the plane X= 0 at the point (0,Y H,Z H), with (YH,Z H)=�−1 σ2+ω2,−2σ σ2+ω2�. It is interesting to get the admissible locus of such point (YH,Z H); after some standard manipulations, we obtain that it should belong to the lower half of the ellipse �Y+1/(2ω2)�2 1/(4ω4)+Z2 1/ω2=1.(7) Clearly, if one orbit in the half-space X<0 hits the separation plane at the point (0,Y H,Z H), then it will Fig. 6. The maximal homoclinic orbit with the spiral on the focal plane tangent to the Z-axis at the point (0,0,−1/λ) for λ=−1/2, ω=1/4 and σ≈0.028531. follow the one-dimensional stable invariant manifold of the equilibrium point, which will be its asymptotic fate. Furthermore, if the mentioned orbit comes from a point at the trace of the focal plane, which is completely filled by orbits emanating from the focus, we should have a homoclinic orbit. Thus, one could think that the existence of homoclinic orbits is easily assured: if we superimpose the red curve of Fig. 3, which is exclusively determined by the value of λ, with the half-ellipse given in (7), which depends only on ω, then it is clear that for a sufficiently small value of ωthere will be an point (YH,Z H) where both curves intersect. The value of the third parameter σ will come directly from the condition ZH=2σYH. However, we should be conscious of an important fact: for any spiral on the focal plane arriving at a point belonging to its trace with the separation plane (0,Y 0,Z 0) with Y0>0, it really emanates from the focus only if, going backwards in time, such planar spiral is totally contained in the half-space X>0. Otherwise, the orbit will start at another point on the trace (0,Y,Z) with Y<0, and will not come from the focus. Therefore, we need to study which is the biggest planar spiral arriving at a point of the trace of the focal plane, such that going backwards in time it is completely contained in the half-space X>0. If we denote by (0,Ymax 0,Zmax 0) such an intersection point, then we must impose that for the point (0,Y 0,Z 0) at the trace whose orbit along the half space X<0 arrives to (0,Y H,Z H) the condition 0<Y 0≤Ymax 0(8) is fulfilled. Clearly, the required biggest planar spiral is the one that is tangent to the Z-axis at the point (0,0,−1/λ). To avoid computations in the three-dimensional phase space, we can take advantage that the target dynamics is confined to the focal plane Π, whose projection on the horizontal plane Z= 0 is ruled by the differential system ˙ X=(λ+2σ)X−Y, ˙ Y= [(λ+σ)2+ω2]X−λY +1/λ, (9) which is obtained by considering only the first two equations of the right subsystem and substituting in the second one the value of Zfor the focal plane. The solution starting at the origin at τ= 0 is given by �X(τ) Y(τ)�=�X Y�−eστ �Icos ωτ +sin ωτ ωB��X Y�, where Iis the identity matrix and Bis the matrix B=�λ+σ−1 (λ+σ)2+ω2−(λ+σ)�. The first positive value ˆτsuch that X(ˆτ) = 0 must satisfy 64 Emilio Freire et al. / IFAC PapersOnLine 58-5 (2024) 60–65 - (YH,Z H) Y Z 1 Fig. 7. The pre-image of the point (YH,Z H) should be on the arc of the circle centered at (−1,0) with Y>0 and Z<Z H(the thick arc). Such pre-image should also belong to the trace of the focal plane. We have drawn such traces for two different values of λ: only for the lower value homoclinic orbits are possible. X−eσˆτXcos ωˆτ+sin ωˆτ ω(λ+σ)X−Y=0, and using that Y=tX =(λ+2σ)X, we get a trascendental equation for ˆτ, which does not depend on λ, namely 1−eσˆτcos ωˆτ−σ ωsin ωˆτ=0.(10) Usually, this equation is studied by considering the equivalent condition cos θ−γsin θ= exp(−γθ),where θ=ωτ is a normalized time, and γ=σ/ω is the so called intrinsic focal parameter, see Chapter 1 in Ponce et al. (2022). It is not difficult to see that for γ>0 the first positive zero ˆ θ=ωˆτbelongs to the interval (π,2π). For Y(ˆτ), we have Y(ˆτ)=Y−eσˆτYcos ωˆτ+sin ωˆτ ω(0,1)⊤BX Y, which, after (10) and from Y=tX and X=−1/d, simplifies greatly to Y(ˆτ)=eσˆτsin ωˆτ ωX(−σ2−ω2)=eσˆτsin ωˆτ λω . In short, we get that Ymax 0(ˆτ)=eσˆτsin ωˆτ λω ,(11) where ˆτis the solution of (10) with π<ωˆτ<2π. 3.1 Some necessary conditions for homoclinic orbits In order to look for homoclinic orbits, it is useful to have some necessary conditions at our disposal. Regarding Fig. 7, if we assume that we have fixed σand ωin our parameter region of interest, then the point (YH,Z H) is completely determined. Clearly, to get a homoclinic connection, the orbit coming from an admissible point in the trace of the focal plane should hit the plane X= 0 at such point. This is only possible if the departure point of the trace belongs to the thick arc of points (Y,Z) which, being in the same circle with center at (−1,0) that passes through the point (YH,Z H), also satisfy Y>0, Z<Z H. Note that the arc for admissible pre-images of (YH,Z H) has its upper endpoint at (Y,ZH) with (Y+YH)/2=−1, that is Y=−2−YH. Thus, as a first condition that guarantees that the quoted arc is non-empty, we need for such a value to be positive, namely −2+ 1 σ2+ω2>0⇐⇒ σ2+ω2<1 2. Furthermore, there exists a critical value λcrit such that for λ≥λcrit the trace of the focal plane does not intersect the thick arc. This critical value can be computed by solving the equation −λ(−2−YH)+ZH+1/λ = 0, namely λ21 σ2+ω2−2+2σλ σ2+ω2−1=0.(12) If λis fixed, then the above equation represents in the plane (σ, ω) the circle σ−λ 1+2λ22 +ω2=2λ2(1 + λ2) (1 + 2λ2)2.(13) In short, for the existence of a homoclinic orbit, we get the two necessary conditions σ2+ω2<1 2,λ<λ crit =−σ+2σ2+ω2−2(σ2+ω2)2 1−2σ2−2ω2, to be taken into account in looking for or trying to avoid homoclinic orbits. Here, λcrit is the negative root of (12). 3.2 A computational algorithm We have build a continuation procedure to detect and follow the curve of parameter values for which a homoclinic orbit exists, as a next step to complete the bifurcation set of Fig. 2. Thus, we consider the four equations that come from (10) plus the backward integration conditions that guarantee that starting from the point (0,Y H,Z H) we arrive to a point (0,Y 0,Z 0) in the admissible segment of the trace of the focal plane. By changing τby −τin (2), we write 0=(1+YH) sin τ+ZH(1 −cos τ)−τ, Y0= (1 + YH) cos τ+ZHsin τ−1, λY0−1/λ =ZH−τ, (14) and by using τLand τRto distinguish different flight times, we get from (10), (14) and Y0=µY max 0, the four equations cos ωτR−(σ/ω) sin ωτR−exp(−στR)=0, 1−1 σ2+ω2sin τL−2σ(1 −cos τL) σ2+ω2−τL=0, 1−1 σ2+ω2cos τL−2σsin τL σ2+ω2−1−µY max 0(τR)=0, 1/λ −λµY max 0(τR)−2σ σ2+ω2−τL=0, (15) where we assume λfixed, and (τL,τ R,σ,ω,µ) are the five variables of the problem. From (8), we require 0 <µ≤1. After running a standard continuation algorithm for the above set of equations, we can improve the preliminary bifurcation set of Fig. 2 by drawing the curve of parameter values where a homoclinic orbit exists. For λ=−1/2, we show in Fig. 8, thanks to the proposed computational algorithm, the curve of points in the plane (σ, ω) where the system has a homoclinic connection. We emphasize the upper terminal point that corresponds to the maximal homoclinic orbit of Fig. 6, while for the lower endpoint we have stopped the continuation when σ≈0.156, ω≈0.018, and µ≈1.737 ∗10−12. 3.3 Final remarks Near the curve of parameter values for homoclinic orbits, there exists a lot of bifurcation curves, according with Emilio Freire et al. / IFAC PapersOnLine 58-5 (2024) 60–65 65 ω σ 1 Fig. 8. Magnification of region of interest of the bifurcation set of Fig. 2. We have added the homoclinic curve, the dashed arc of the circle that determines the necessary condition for the existence of homoclinic connections (13), and the curve of first period doubling bifurcation points, which surrounds the homoclinic curve. the theory of Shilnikov chaos, see again Glendinning and Sparrow (1984); Gaspard et al. (1984). Following another specific continuation procedure, in Fig. 8 we just have drawn the bifurcation curve that corresponds to the first period doubling bifurcation points; it emanates from the origin in a seemingly tangential way to the ω-axis, surrounding the homoclinic curve, to go back to the σ-axis at σ≈0.187. This last curve could be helpful as some safe boundary for not entering the chaotic parameter region and could also serve for stating a weakened version of Conjecture 3 in Freire et al. (2020). It is worthy to mention that in Fig. 8 the origin is a doublezero bifurcation point, from where the period doubling bifurcation curve seems to emanate, and deserves further study. The point at the σ-axis, where such a curve ends, is a natural candidate to be a Belyakov point, see Freire et al. (2009) and references therein. 4. CONCLUSION In the last decades, many authors have worked in looking for chaotic systems, see for instance Glendinning (2018); Barajas-Ram´ırez et al. (2021) and references therein. Contrarily to the quoted works on discontinuous models, we have found chaos in one of the simplest continuous piecewise linear models, when initially trying to check a natural conjecture on existence of stable limit cycles. For our model, we have shown in Proposition 5 an important recurrence property of orbits, by taking as reference the focal plane of the equilibrium, providing also a primary bifurcation set. Some necessary conditions for existence of homoclinic orbits have been derived and a specific continuation procedure was built in order to follow such a curve in an adequate parameter plane. The intrinsical interest of the model deserves a much deeper analysis regarding its usefulness in the characterization of boundary equilibrium bifurcations for 3D piecewise linear systems. In particular, this partial study paves the way for characterizing the direct transition to chaos in such bifurcations, as exemplified in the celebrated monography by Di Bernardo et al. (2008). REFERENCES J.G. Barajas-Ram´ırez, A. Franco-L´opez, H.G. Gonz´alez- Hern´andez. Generating Shilnikov Chaos in 3D Piecewise Linear Systems Applied Mathematics and Computation, 395, 125877, 2021. M. Di Bernardo, A. Champneys, C. J. Budd, and P. Kowalczyk. Piecewise-Smooth Dynamical Systems. Theory and Applications. Appl. Math. Sci. Series, 163, Springer London, 2008. V. Carmona, E. Freire, E. Ponce, J. Ros and F. Torres. Limit cycle bifurcation in 3D continuous piecewise linear systems whit two zones. Application to Chua’s circuit. International Journal of Bifurcation and Chaos, 15-10, 3153-3164, 2005. L.O. Chua, M. Komuro, and T. Matsumoto. The Double Scroll family. IEEE Transactions on Circuits and Systems, vol. 33, no. 11, 1072-1118, 1986. E. Freire, E. Ponce, and J. Ros. Following a saddle-node of periodic orbits bifurcation curve in Chua’s circuit. International Journal of Bifurcation and Chaos, 19, 487- 495, 2009. E. Freire, M. Ord´o˜nez, and E. Ponce. Bifurcations from a center at infinity in 3D piecewise linear systems with two zones. Physica D, 402, 132280, 2020. P. Gaspard, R. Kapral, and G. Nicolis. Bifurcation Phenomena near Homoclinic Systems: A Two-Parameter Analysis. Journal of Statistical Physics, 35, 697-727, 1984. P. Glendinning, and C. Sparrow. Local and Global Behavior near Homoclinic Orbits. Journal of Statistical Physics, 35, 645–696, 1984. P. Glendinning Shilnikov Chaos, Filippov Sliding and Boundary Equilibrium Bifurcations. European Journal of Applied Mathematics, 29, 757–777, 2018. J. Llibre, E. Ponce, and A.E. Teruel. Horseshoes near Homoclinic Orbits for Piecewise Linear Differential Systems in R3.International Journal of Bifurcation and Chaos, 17, 1171-1184, 2007. R.N. Madan. Chua’s circuit: A paradigm for Chaos. World Scientific Series on Nonlinear Science B, vol. 1, World Scientific, Singapore, 1993. E. Ponce, J. Ros, and E. Vela. Bifurcations in Continuous Piecewise Linear Systems. Applications to Lowdimensional Electronic Oscillators. RSME Springer Series, 7, Springer Nature Switzerland AG, 2022. E. Ponce, J. Ros, and E. Vela. Unfolding the fold-Hopf bifurcation in piecewise linear continuous differential systems with symmetry. Physica D, 250, 34–46, 2013. C.P. Silva. Shil’nikov’s Theorem–A Tutorial. IEEE Trans CAS-I, 40, 675–682, 1993.