In this work the homoclinic bifurcation of the family H={h (a,b)(x)=ax 2+b:a∈R/{0},b∈R} is studied. We proved that this family has a homoclinic tangency associated to x=0 of P1 for b=-2/a. Also we proved that W u( P 1) does not intersect the backward orbit of P1 for b>-2/a, but has intersection for b<-2/a with a>0. So H has this type of the bifurcation.

Local Unstable Set Unstable Set Homoclinic Point Homoclinic Orbit Non-Degenerate Homoclinic Tangency Homoclinic Bifurcation
1. Introduction

There are various definitions for the homoclinic bifurcation. In the sense of Devaney, the homoclinic bifurcation is a global type of bifurcations, that is, this type of bifurcation is a collection of local and simple types of bifurcations  (like, period-doubling and saddle-node of bifurcation  ).

According to    we have another definition of the homoclinic bifurcation via the notions of the unstable sets of a repelling periodic point (fixed point) and the intersection of this set with the backward orbits of this point.

The purpose of this work is to prove the family

H = { h a , b ( x ) = a x 2 + b : a ∈ ℝ / { 0 } , b ∈ ℝ } has homoclinic bifurcation at b = − 2 a following the second definition.

2. Definitions and Basic Concepts2.1. Definition 1: [<xref ref-type="bibr" rid="scirp.109549-ref6">6</xref>]

A fixed point P is said to be expanding for a map f, if there exists a neighborhood U ( P ) such that | f ′ ( x ) | > 1 for any x ∈ U ( P ) .

The neighborhood in the previous definition is exactly the local unstable set.

2.2. Definition 2: [<xref ref-type="bibr" rid="scirp.109549-ref7">7</xref>]

Let P be a repelling fixed point for a function f : I → I on a compact interval I ⊂ R , then there is an open interval about P on which f is one-to-one and satisfies the expansion property. | f ( x ) − f ( P ) | > | x − P | , ∀ x ∈ I where x ≠ P .

The interval in the previous definition is exactly the unstable set of P.

2.3. Definition 3: [<xref ref-type="bibr" rid="scirp.109549-ref8">8</xref>]

Let P is fixed point and f ′ ( P ) > 1 for a map f : ℝ → ℝ . A point q is called homoclinic point to P if q ∈ w l o c u ( P ) and there exists n > 0 such that f n ( q ) = P .

2.4. Definition 4: [<xref ref-type="bibr" rid="scirp.109549-ref9">9</xref>]

The union of the forward orbit of q with a suitable sequence of preimage of q is called the homoclinic orbit of P. That is O ( q ) = { P , ⋯ , q − 2 , q − 1 , q , q 1 , q 2 , ⋯ , q m = P } where q i + 1 = f ( q i ) for i ≤ m − 1 , q m = P and lim i → − ∞ q i = P .

2.5. Definition 5: [<xref ref-type="bibr" rid="scirp.109549-ref10">10</xref>]

The critical x point is non-degenerate if f ″ ( x ) ≠ 0 . The critical point x is degenerate if f ″ ( x ) = 0 .

2.6. Definition 6: [<xref ref-type="bibr" rid="scirp.109549-ref11">11</xref>]

Let f be a smooth map on I ⊂ R , and let p be a hyperbolic fixed point for the map f. If W u ( p ) intersects the backward orbit of p at a nondegenerate critical point x c r of f, then x c r is called a point of homoclinic tangency associated to p.

2.7. Definition 7: [<xref ref-type="bibr" rid="scirp.109549-ref3">3</xref>]

Let f φ be a smooth map on I ⊂ R , and let p be a hyperbolic fixed point for a map f φ . We say that f φ has homoclinic bifurcation associated to p at φ = φ ^ if:

1) For φ < φ ^ ( φ > φ ^ ), W u ( p ) and the backward orbit of p has no intersect.

2) For φ = φ ^ , f φ ^ has a point of homoclinic tangency x c r associated to p.

3) For φ > φ ^ ( φ < φ ^ ), the intersection of W u ( p ) with the backward orbit of p is nonempty.

3. Homoclinic Bifurcation of the Family H = { h a , b ( x ) = a x 2 + b : a ∈ ℝ / { 0 } , b ∈ ℝ }

In this section, we show that the family H has a point of homoclinic tangency associated to P1 at b = − 2 a , and H has a homoclinic bifurcation.

We need the following results proved in .

At the first, the fixed points of h a , b ( x ) are

P 1 = 1 + 1 − 4 a b 2 a , P 2 = 1 − 1 − 4 a b 2 a .

a) Proposition:

For h a , b ( x ) ∈ H with a > 0 the local unstable set of the fixed point P1 is w l o c u ( P 1 ) = ( 1 2 | a | , ∞ ) .

b) Lemma:

For h a , b ( x ) ∈ H , h a , b − 1 ( P 1 ) = ∓ P 1 − b a = ∓ P 1 where P 1 > b for a > 0.

c) Theorem:

For h a , b ( x ) ∈ H with a > 0 , the unstable set of the fixed point P1 is w u ( P 1 ) = ( 1 | a | − P 1 , ∞ ) .

d) Remark: 

The local unstable set of the fixed point P2 is w l o c u ( P 2 ) = ( − ∞ , − 1 2 | a | ) , and the unstable set of the fixed point P2 is w u ( P 2 ) = ( − ∞ , − 1 | a | − P 2 ) . In this work we will omit the result about P2 because ( h ′ a , b ( P 2 ) < − 1 , when b < − 3 4 a for a > 0 b > − 3 4 a for a < 0 ). Thus we will not care for the fixed point P2. (See definition (2.3)).

e) Remark:

For h a , b ( x ) ∈ H , h a , b − 2 ( P 1 ) = ∓ − P 1 − b a .

f) Proposition:

For h a , b ∈ H , if b < − ( 5 + 2 5 ) 4 a with a > 0 , then the second preimage of the fixed point P1 belongs to the local unstable set of P1.

g) Proposition:

For h a , b ∈ H , if − ( 5 + 2 5 ) 4 a ≤ b ≤ − 2 a with a > 0 , then the third preimage of the fixed point P1 belongs to the local unstable set of P1.

h) Theorem:

For the family H = { h a , b ( x ) = a x 2 + b : a > 0 } , there exist homoclinic points to the fixed point P1 whenever b ≤ − 2 a . Moreover h a , b − 2 ( P 1 ) = q 1 , 1 , h a , b − 3 ( P 1 ) = q 2 , 1 are the first homoclinic points for b < − ( 5 + 2 5 ) 4 a , − ( 5 + 2 5 ) 4 a ≤ b ≤ − 2 a respictivelity.

i) Example:

For h 1 , − 6 ( x ) = x 2 − 6 , a homoclinic orbit of a homoclinic point 3 is: O ( 3 ) = { 3 , − 3 , 3 , ⋯ , 3 } .

The main result:

3.1. Lemma 1

For h a , b ( x ) = a x 2 + b with a ∈ ℝ / { 0 } , the critical point of h a , b ( x ) is 0, and it is a non-degenerate critical point.

Proof:

Clearly that the critical point of h a , b ( x ) is zero.

Since a ≠ 0 , then

h ″ a , b ( x ) = 2 a ≠ 0 .

So h a , b ( x ) has a non-degenerate critical point at x = 0 . ∎

3.2. Lemma 2

If b = 0 of h a , b ( x ) ∈ H with a ∈ ℝ / { 0 } , then the backward orbit of the repelling fixed point P1 is undefined in ℝ .

Proof:

h a , 0 ( x ) = a x 2 , clearly P 1 = 1 a .

Now the first preimage of h a , 0 ( x ) is

h a , 0 − 1 ( x ) = ∓ x a , where x > 0 for a > 0 .

By Lemma (3-b), we have

h a , 0 − 1 ( 1 a ) = ∓ 1 a 2 = ∓ 1 a = ∓ P 1 .

But +P1 is a fixed point, and − P 1 = − 1 a ∉ w l o c u ( P 1 ) = ( 1 2 a , ∞ ) , see Proposition (3-a).

By Remark (3-e) we have

h a , 0 − 2 ( P 1 ) = ∓ − P 1 a = ∓ − 1 ą a = ∓ − 1 a 2 ∉ ℝ ,

since 1 a 2 > 0 , ∀ a ∈ ℝ / { 0 } .

Therefore h a , 0 − n ( P 1 ) are undefined in ℝ with n ≥ 2 .

Thus the backward orbit of the repelling fixed point P1 is undefined in

3.3. Theorem 1

For the family h a , b ( x ) = a x 2 + b , 0 belong to the backward orbit of P1 whenever b = − 2 a with a ∈ ℝ / { 0 } , and the backward orbit of P1 is:

h a , − 2 a − n ( P 1 = 2 a ) = { 2 a , − 2 a , 0 , 2 a , ⋯ } .

Proof:

We test the values of n which makes h a , b − n ( P 1 ) = 0 .

By Lemma (3-b), h a , b − 1 ( P 1 ) = ± P 1 .

So h a , b − 1 ( P 1 ) ≠ 0 .

Now suppose that h a , b − 2 ( P 1 ) = 0 , by Remark (3-e) then

h a , b − 2 ( P 1 ) = ∓ − P 1 − b a = 0 , thus

− P 1 − b a = 0

− P 1 − b = 0

P 1 = b .

Since the fixed point P 1 = 1 + 1 − 4 a b 2 a , therefore

1 + 1 − 4 a b 2 a = − b ,

then

1 + 1 − 4 a b = − 2 a b

1 − 4 a b = − 2 a b − 1

1 − 4 a b = 4 a 2 b 2 + 4 a b + 1

4 a 2 b 2 + 8 a b = 0 , which implies

4 a b ( a b + 2 ) = 0 , then either

b = 0 , but by the above Lemma (3.2) the backward orbit of P1 is undefined, so we omit this case.

Or a b + 2 = 0 , thus

b = − 2 a .

Now, P 1 = 2 a and to find the backward orbit of P1, we consider

h a , − 2 a − 1 ( x ) = ± a x + 2 a .

By Lemma (3-b) h a , b − 1 ( P 1 ) = ± P 1 , then

h a , − 2 a − 1 ( 2 a ) = ± 2 a . But + 2 a is a fixed point, therefor

h a , − 2 a − 1 ( 2 a ) = − 2 a .

So

h a , − 2 a − 2 ( 2 a ) = a ( − 2 a ) + 2 a = 0 .

h a , − 2 a − 3 ( 2 a ) = a ( 0 ) + 2 a = 2 a , and so on.

Therefore the backward orbit of P 1 = 2 a is:

h a , − 2 a − n ( P 1 = 2 a ) = { 2 a , − 2 a , 0 , 2 a , ⋯ } . ∎

3.4. Example

For h 1 , − 2 ( x ) = x 2 − 2 , 0 belongs to the backward orbit of P 1 = 2 (Figure 1), and the backward orbit of P1 is h 1 , − 2 − n ( 2 ) = { 2 , − 2 , 0 , 2 , ⋯ , 2 } .

3.5. Theorem 2

If b > − 2 a for h a , b ( x ) ∈ H with a > 0 , then there is no intersection of the backward orbit with the unstable set of P1.

Proof:

The backward orbit of P1

By Lemma (3-b) h a , b − 1 ( P 1 ) = ± P 1 , since +P1 is a fixed point, then we consider

h a , b − 1 ( P 1 ) = − P 1 .

By Remark (3-e), h a , b − 2 ( P 1 ) = ∓ − P 1 − b a .

If − P 1 > b , then by Theorem (3-h),

b ≤ − 2 a which is a contradiction with b > − 2 a . Therefore

− P 1 < b , which implies

h a , b − 2 ( P 1 ) ∉ ℝ .

So h a , b − n ( P 1 ) are undefined in ℝ with n ≥ 2 .

Thus the backward orbit of P1 is undefined .

So the intersection of W u ( P 1 ) with the backward orbit of P1 is also undefined. ∎

3.6. Theorem 3

If b = − 2 a for h a , b ( x ) ∈ H with a > 0 , then h a , − 2 a has a point of homoclinic tangency at 0 associated to P1.

Proof:

By Theorem (3.3), h a , − 2 a − n ( P 1 = 2 a ) = { 2 a , − 2 a , 0 , 2 a , ⋯ } .

By Theorem (3-c), W u ( P 1 ) = ( 1 a − P 1 , ∞ ) , then

W u ( P 1 = 2 a ) = ( 1 a − 2 a , ∞ ) , i.e.

W u ( P 1 = 2 a ) = ( − 1 a , ∞ ) . Now

h a , − 2 a − n ( 2 a ) intersects W u ( P 1 = 2 a ) at 0.

By Lemma (3.1) 0 is a non-degenerate critical point. So h a , − 2 a has a point of homoclinic tangency at 0 associated to P1. ∎

3.7. Theorem 4

If b < − 2 a for h a , b ( x ) ∈ H with a > 0 , then the backward orbit of P1 crosses the unstable set W u ( P 1 ) .

Proof:

First consider the backward orbit of P1.

By Lemma (3-b) h a , b − 1 ( P 1 ) = ± P 1 .

But + P1 is a fixed point, therefore we consider

h a , b − 1 ( P 1 ) = − P 1 .

By Remark (3-e), h a , b − 2 ( P 1 ) = ∓ − P 1 − b a .

Since b < − 2 a , then by Theorem (3-h)

h a , b − 2 ( P 1 ) ∈ ℝ .

Let h a , b − 2 ( P 1 ) = q 1 , 1 , h a , b − 3 ( P 1 ) = q 2 , 1 .

By Proposition (3-f), if b < − ( 5 + 2 5 ) 4 a , then q 1 , 1 ∈ W l o c u ( P 1 ) .

By Proposition (3-g), if − ( 5 + 2 5 ) 4 a ≤ b < − 2 a , then q 2 , 1 ∈ W l o c u ( P 1 ) .

Now since the local unstable set of the repelling fixed point contained in the unstable set of the repelling fixed point. Therefore

h a , b − n ( P 1 ) ∩ W u ( P 1 ) ≠ ∅ . ∎

Following examples explain the cases for b > − 2 a , b = − 2 a and b < − 2 a (with a > 0 ) respectively.

3.8. Example 1

For h 1 , − 1 ( x ) = x 2 − 1 , we have no intersection of the backward orbit of P1 with the unstable set of P1.

Solution:

Consider the fixed point of h 1 , − 1 ( x ) is P 1 = 1 + 5 2 , and

h 1 , − 1 − 1 ( x ) = ± x + 1 .

The backward orbit of P 1 = 1 + 5 2

h 1 , − 1 − 1 ( 1 + 5 2 ) = ± 1 + 5 2 , where + 1 + 5 2 is a fixed point, therefore we consider

h 1 , − 1 − 1 ( 1 + 5 2 ) = − 1 + 5 2 . Now

h 1 , − 1 − 2 ( 1 + 5 2 ) = ∓ − 1 + 5 2 + 1 ∉ ℝ .

So h 1 , − 1 − n ( 1 + 5 2 ) are undefined in ℝ with n ≥ 2 .

Thus the backward orbit of P1 is undefined.

So the intersection of W u ( 1 + 5 2 ) with the backward orbit of P1 is also undefined. ∎

3.9. Example 2

For h 1 , − 2 ( x ) = x 2 − 2 , then h 1 , − 2 has a point of tangency at 0 associated to P1.

Solution:

Consider the fixed point of h 1 , − 2 ( x ) is P 1 = 2 .

By Example (3.4), The backward orbit of P 1 = 2 is

h 1 , − 2 − n ( 2 ) = { 2 , − 2 , 0 , 2 , ⋯ , 2 } .

On the other hand, the unstable set of P 1 = 2 is

W u ( 2 ) = ( − 1 , ∞ ) , (see Theorem (3-c)). Now

h 1 , − 2 − n ( 2 ) intersects W u ( 2 ) at 0.

By Lemma (3.1), 0 is a non-degenerate critical point. So h 1 , − 2 has a point of tangency at 0 associated to P 1 = 2 . ∎

3.10. Example 3

For h 1 , − 6 ( x ) = x 2 − 6 , the backward orbit of P1 crosses the unstable set W u ( P 1 ) .

Solution:

First consider the fixed point P 1 = 3 .

The backward orbit of 3 is:

h 1 , − 6 − n ( 3 ) = { 3 , − 3 , 3 , ⋯ , 3 } (see Example (3-i)), with

h 1 , − 6 − 1 ( 3 ) = h 1 , − 2 2 ( 3 ) , and h 1 , − 6 − 2 ( 3 ) = 3 .

Since 3 is a homoclinic point of P 1 = 3 , then

3 ∈ W l o c u ( 3 ) .

Now since the local unstable set of the repelling fixed point P 1 = 3 contained in the unstable set of the repelling fixed point P 1 = 3 . Therefore

h 1 , − 6 − n ( 3 ) ∩ W u ( 3 ) ≠ ∅ . ∎

Note , the main theorem in the work :

3.11. Theorem 5

h a , b ( x ) = a x 2 + b , a > 0 has a homoclinic bifurcation associated to the repelling fixed point of h a , b , P 1 = 1 + 1 − 4 a b 2 a , at b = − 2 a .

Proof:

1) For b > − 2 a , by Theorem (3.5) the intersection of the backward orbit of P1 and the unstable set of P1 is undefined.

2) For b = − 2 a , by Theorem (3.6) h a , − 2 a has a point of homoclinic tangency associated to P1 at x = 0 .

3) For b < − 2 a , by Theorem (3.7) the backward orbit of P1 crosses the unstable set of P1, W u ( P 1 ) .

Therefore h a , b has a homoclinic bifurcation associated to P1 at b = − 2 a . ∎

3.12. Example

h 1 , − 2 ( x ) = x 2 − 2 has a homoclinic bifurcation associated to the repelling fixed point of h 1 , − 2 , P 1 = 2 , at b = − 2 .

h 1 , − 2 ( x ) has a homoclinic bifurcation associated to the repelling fixed point of h 1 , − 2 , P 1 = 2 , at b = − 2 . See examples (3.8), (3.9), (3.10).

3.13. Remark

For a < 0 , we have same results which proved above for a > 0 . In fact, we can prove in similar ways, that: h a , b ( x ) = a x 2 + b , a < 0 has a homoclinic bifurcation associated to the repelling fixed point of h a , b , P 1 = 1 + 1 − 4 a b 2 a , at b = − 2 a .

4. Conclusion

We conclude that the family H = { h a , b ( x ) = a x 2 + b : a ∈ ℝ / { 0 } , b ∈ ℝ } has homoclinic tangency associated to P1 at the critical point x = 0 . Also for b > − 2 a we have no intersection between the backward orbit of P1 and the unstable set of P1, and the backward orbit of P1 crosses the unstable set of P1 for b < − 2 a . So we have homoclinic bifurcation at b = − 2 a .

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Farris, S.M. and Abdul-Kareem, K.N. (2021) Homoclinic Bifurcation of a Quadratic Family of Real Functions with Two Parameters. Open Access Library Journal, 8: e7300. https://doi.org/10.4236/oalib.1107300

ReferencesGuo, S. and Wu, J. (2013) Bifurcation Theory of Functional Differential Equations, Vol. 10. Springer, New York. https://doi.org/10.1007/978-1-4614-6992-6Puu, T. and Sushko, I. (2006) Business Cycle Dynamics. Springer, New York. https://doi.org/10.1007/3-540-32168-3Block, L.S. and Coppel, W.A. (2006) Dynamics in One Dimension. Springer, New York.Devaney, R. (2018) An Introduction to Chaotic Dynamical Systems. CRC Press, Boca Raton, FL. https://doi.org/10.4324/9780429502309Gulick, D. (2012) Encounters with Chaos and Fractals. CRC Press, Boca Raton, FL. https://doi.org/10.1201/b11855Laura, G., Viktor, A., Iryna, S. and Fabio, T. (2019) Continuous and Discontinuous Piecewise-Smooth One-Dimensional Maps: Invariant Sets and Bifurcation Structures, Vol. 95. World Scientific, Singapore.Perko, L. (2013) Differential Equations and Dynamical Systems, Vol. 7. Springer Science & Business Media, Berlin Heidelberg.Bonatti, C., Díaz, L.J. and Viana, M. (2006) Dynamics beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective, Vol. 102. Springer Science & Business Media, Berlin Heidelberg.Abdul-Kareem, K.N. and Farris, S.M. (2020) Homoclinic Points and Homoclinic Orbits for the Quadratic Family of Real Functions with Two Parameters. Open Access Library Journal, 7, 1-18. https://doi.org/10.4236/oalib.1106170Abdul-Kareem, K.N. and Farris, S.M. (2020) Homoclinic Points and Bifurcation for a Quadratic Family with Two Parameters. MSc Thesis, Department of Mathematics, College of Computer and Mathematical Sciences, University of Mosul, Mosul, Iraq, 1-98.Onozaki, T. (2018) Nonlinearity, Bounded Rationality, and Heterogeneity. Springer, Berlin Heidelberg. https://doi.org/10.1007/978-4-431-54971-0Ghrist, R.W., Holmes, P.J. and Sullivan, M.C. (2006) Knots and Links in Three-Dimensional Flows. Springer, Berlin Heidelberg.Zhusubaliyev, Z.T. and Mosekilde, E. (2003) Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems. World Scientific, Singapore. https://doi.org/10.1142/5313