ABSTRACT
Consider the quadratic map () https://www.w3.org/1998/Math/MathML"> y n + 1 + B y n − 1 = 2 c y n + 2 y n 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750162/75f2bfa0-0ca3-4d38-8963-894d5bad8486/content/eq190.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> For |B| < 1 this is Henon’s dissipative map. For B = 0 the map is evidently just the logistic map, while for B = 1 the map is area conserving. This is seen by writing equation (6.1) in an equivalent form: () https://www.w3.org/1998/Math/MathML"> x n + 1 = y n + 1 − a x n 2 y n + 1 = b x n . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750162/75f2bfa0-0ca3-4d38-8963-894d5bad8486/content/eq191.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> The Jacobian matrix of this map is then given by () https://www.w3.org/1998/Math/MathML"> j = ( − 2 a x n 1 b 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750162/75f2bfa0-0ca3-4d38-8963-894d5bad8486/content/eq192.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and the Jacobian is just −b. Since this is a constant we may use equation (5.24) to find the following relation between the two les: () https://www.w3.org/1998/Math/MathML"> λ 1 + λ 2 = ln | b | . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750162/75f2bfa0-0ca3-4d38-8963-894d5bad8486/content/eq193.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Consequently it is not necessary to calculate more than the largest le in this case. The eigenvalue equation for the les is evidently a quadratic equation which may have either two real and different solutions, or two solutions that are complex conjugates of each other. In figure 6.1 the largest le is shown for fixed b. As may be seen, the dips in the exponent are no longer infinitely deep like in the logistic map. The characteristic flat regions are due to inequality (5.26) becoming an equality with the result λ1 = λ2 = ln(0.3)/2 = −0.602…. From figure 6.1 one may see that there are period doubling sequences very much as in the case of the logistic map. Since at least one le is negative the fixed points may be stable, i.e. have only a stable manifold (this is a stable node), or it may have a stable one-dimensional manifold plus an unstable one-dimensional manifold (this is a 59hyperbolic point or a saddle point). It is an easy matter to find the fixed points. They are given by () https://www.w3.org/1998/Math/MathML"> x 0 , 1 = ( − 1 + b ± ( 1 − b ) 2 + 4 a ) / 2 a . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750162/75f2bfa0-0ca3-4d38-8963-894d5bad8486/content/eq194.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> These are points in real space provided () https://www.w3.org/1998/Math/MathML"> a ≥ − ( 1 − b ) 2 / 4. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750162/75f2bfa0-0ca3-4d38-8963-894d5bad8486/content/eq195.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Calling the eigenvalues of the Jacobian matrix Λ, the eigenvalue equation is given by () https://www.w3.org/1998/Math/MathML"> Λ 2 + 2 a x Λ − b = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750162/75f2bfa0-0ca3-4d38-8963-894d5bad8486/content/eq196.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> The fixed point becomes unstable if |Λ| > 1. Taking Λ to be on the unit circle in the complex plane, i.e. Λ = eiφ , equation (6.7) splits into two equations () https://www.w3.org/1998/Math/MathML"> cos 2 φ + 2 a x cos φ − b = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750162/75f2bfa0-0ca3-4d38-8963-894d5bad8486/content/eq197.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and () https://www.w3.org/1998/Math/MathML"> sin 2 φ + 2 a x sin φ = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750162/75f2bfa0-0ca3-4d38-8963-894d5bad8486/content/eq198.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> The last of these equations has one solution sin φ = 0, cos φ = ±1. Inserting this into equation (6.8) shows that there is no solution for the positive sign except at () https://www.w3.org/1998/Math/MathML"> a = − ( 1 − b ) 2 / 4 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750162/75f2bfa0-0ca3-4d38-8963-894d5bad8486/content/eq199.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where the two fixed points coalesce. The negative sign gives Λ = −1 which corresponds to a period doubling bifurcation. From equation (6.7) one finds that (x 0, y 0) becomes unstable for () https://www.w3.org/1998/Math/MathML"> a > 3 ( 1 − b ) 2 / 4 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750162/75f2bfa0-0ca3-4d38-8963-894d5bad8486/content/eq200.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 60and (x 1, y 1) is always unstable. The largest Lyapunov exponent for the Hénon map for <italic>b</italic> = 0.3. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750162/75f2bfa0-0ca3-4d38-8963-894d5bad8486/content/fig6_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
