Dynamic Macroeconomic Theory Prof. Thomas Lux Bifurcation Theory
Bifurcation: qualitative change in the nature of the solution occurs if a parameter passes through a critical point (bifurcation or branch value).
Local Bifurcation Theory: Continuous Time Systems
Consider the system: y′ = ϕ(y,α) , (1) α where is a parameter. The equilibrium point ye of this system is given by solving ϕ α = (ye , ) 0 . (2) α Note: the equilibrium point ye , a stationary point, depends on the value of . At certain values of α the characteristics of the system changes, sometimes quite dramatically. By the implicit function theorem the equilibria are continuously differentiable functions of α : = α ye ye ( ) . (3) Some definitions: α (a) If, at an equilibrium point (ye,0 , 0 ) the Jacobian is zero and several branches of α equilibria come together, one says that (ye,0 , 0 ) is a point of bifurcation.
(b) Alternative definition: let Nα denote the number of equilibrium values of the system α α −ε α + ε when the parameter is equal to , then, if for any interval, ( 0 , 0 ) , Nα is not α constant, 0 is called a bifurcation value, and the system is said to undergo a bifurcation α α as passes through 0 . α (c) Another definition: a value 0 of equation (1) for which the solution of (1) is not structurally stable is a bifurcation value of α . (d) An equilibrium point at which no bifurcation occurs is called a hyperbolic fixed point. (e) The bifurcation diagram is a diagram in which the branches of equilibria are shown in α (ye , ) space.
1 Note: in the case of codimension one bifurcations1 the conditions on the Jacobian and its roots can be replaced with single partial derivatives.
1. Saddle-node (fold) bifurcation (prototype function: y′ = α − y 2 ) Consider the one parameter first-order differential equation y′ = f (y,α) (4) α = α and assume that when 0 there is an equilibrium ye,0 for which the following assumptions are satisfied: ∂f ( y ,α ) e,0 0 = 0, [ f (y,α) has a stationary point with respect to y at (y ,α ) ] (5.a) ∂y e,0 0
∂ 2 f (y ,α ) e,0 0 ≠ 0, [(y ,α ) is an extremum] (5.b) ∂y 2 e,0 0 ∂f (y ,α ) e,0 0 ≠ 0 . [ f (y,α) is not stationary with respect to α at (y ,α ) ] (5.c) ∂α e,0 0 Then, depending on the signs of the expression (5.b) and (5.c), there are α (i) no equilibria near (ye,0 , 0 ) when α < α α > α 0 ( 0 ); α (ii) two equilibria near (ye,0 , 0 ) for α > α each parameter value 0 α < α ( 0 ). These equilibria are hyperbolic; one of them is stable and the other unstable. Example: y′ = α − y 2
2. Transcritical bifurcation (prototype function: y′ = αy − y 2 ) Consider the one parameter first-order differential equation y′ = f (y,α) (6) α = α and assume that when 0 there is an equilibrium ye,0 for which the following assumptions
1 One-dimensional systems. 2 are satisfied: ∂f (y ,α ) e,0 0 = 0, [ f (y,α) has a stationary point with respect to y at (y ,α ) ] (7.a) ∂y e,0 0
∂ 2 f (y ,α ) e,0 0 ≠ 0, [(y ,α ) is an extremum] (7.b) ∂y 2 e,0 0
∂ 2 f (y ,α ) e,0 0 ≠ 0 . [Change in α shifts the phase curve] (7.c) ∂α∂y Then, depending on the signs of the expression (7.b) and (7.c) α < α (i) the equilibrium ye,0 is stable (unstable) when 0 α > α ( 0 );
(ii) the equilibrium ye,0 becomes unstable (stable) for each
parameter value α > α (α < α ), and a branch of 0 0 α additional stable (unstable) equilibria ye ( ) emerges.
Example: y′ = αy − y 2
3. Pitchfork bifurcation (prototype function: y′ = αy − y 3 ) Consider the one parameter first-order differential equation y′ = f (y,α) (8) α = α and assume that when 0 there is an equilibrium ye,0 for which the following assumptions are satisfied: ∂f (y ,α ) e,0 0 = 0, [ f (y,α) has a stationary point with respect to y at (y ,α ) ] (9.a) ∂y e,0 0
∂ 3 f (y ,α ) e,0 0 ≠ 0 , [Excluding the presence of a horizontal inflection at y ] (9.b) ∂y 3 e,0
∂ 2 f (y ,α ) e,0 0 ≠ 0 . [Shift of the phase curve] (9.c) ∂α∂y Then, depending on the signs of the expression (9.b) and (9.c) α < α α > α (iii) the equilibrium ye,0 is stable (unstable) when 0 ( 0 );
3 α > α (iv) the equilibrium ye,0 becomes unstable (stable) for each parameter value 0 α < α α 2 ( 0 ), and two branches of additional stable (unstable) equilibria ye ( ) emerge .
Example: y′ = αy − y 3
The Hopf bifurcations in continuous time
Note: the Hopf bifurcation requires at least a 2× 2 system to appear. Consider the 2× 2 system of first-order difference equations y′ = ϕ (y , y ,α), 1 1 1 2 (10) ′ = ϕ α y2 2 (y1 , y2 , ), and assume that for each α in the relevant range this system has an isolated equilibrium point = ye (y1e , y2e ) obtained by solving the system ϕ (y , y ,α) = 0, 1 1 2 (11) ϕ α = 2 (y1, y2 , ) 0.
The solution to (11) will give y1e , y2e as continuously differentiable functions of the parameter, namely (3), if the following Jacobian matrix of (11) is non singular at the equilibrium point ∂ϕ α ∂ϕ α 1 (y1 , y2 , ) 1 (y1 , y2 , ) ∂ ∂ α = y1 y2 J ( ) ∂ϕ α ∂ϕ α , (12) 2 (y1 , y2 , ) 2 (y1, y2 , ) ∂ ∂ y1 y2
2 The case in which the two additional equilibria were stable is called supercritical pitchfork. The case in which the two additional equilibria were unstable is called subcritical pitchfork. 4 Hopf bifurcation theorem
α α Assume that Jacobian matrix (12) evaluated at (ye,0 ( 0 ), 0 ) has the following properties: (i) it possesses a pair of simple complex conjugate eigenvalues θ (α) ± iω(α) that α θ α = become pure imaginary at the critical value 0 of the parameter – i.e., ( 0 ) 0, ω α ≠ while ( 0 ) 0; dθ (α) (ii) ≠ 0 ; dα α=α 0 THEN system (3) has a family of periodic solutions. α Note: the critical value 0 is called Hopf bifurcation point of system (10). When a stable cycle emerges, we have supercritical Hopf bifurcation (see fig. i), otherwise it is subcritical (see fig. ii). Note: (1) Conditions for determining the type of Hopf bifurcation (supercritical or subcritical) do exist, but involve the coefficients of third-order approximation to the nonlinear terms, which are typically undetermined in economic models (Perko L.: Differential Equation and Dynamical Systems. Springer-Verlag, Berlin, 1991. Chapter 4). (2) Conditions (i) and (ii) can also be applied to more general n× n systems of differential equations. In the case n ≥ 3 , the Hopf bifurcation theorem requires all remaining roots (except for the pair of complex conjugate roots under investigation) to have a negative real part as otherwise the system would be unstable anyway.
5 Local Bifurcation Theory: Discrete Time Systems
Note: in discrete time systems the root with unit modulus takes the place of the zero-real-part root of the Jacobian matrix. = α − 2 = +α − 2 1. Saddle-node (fold) bifurcation (prototype functions: yt+1 yt , yt+1 yt yt ). Consider the one parameter first-order difference equation = α yt +1 f (yt , ) (13) α = α and assume that when 0 there is an equilibrium ye,0 . We have fold bifurcation if the following hypotheses are satisfied: ∂f (y ,α ) e,0 0 = 1, (14.a) ∂y
∂2 f (y ,α ) e,0 0 ≠ 0 , (14.b) ∂y2 ∂f (y ,α ) e,0 0 ≠ 0 . (14.c) ∂α
= α − 2 = +α − 2 2. Transcritical bifurcation (prototype functions: yt+1 yt yt , yt+1 yt yt yt ). Consider the one parameter first-order difference equation = α yt +1 f ( yt , ) (15) α = α and assume that when 0 there is an equilibrium ye,0 . We have a transcritical bifurcation if the following hypotheses are satisfied: ∂f (y ,α ) e,0 0 = 1 ∂ y (16.a) ∂2 f (y ,α ) e,0 0 ≠ 0 , (16.b) ∂y2
∂2 f (y ,α ) e,0 0 ≠ 0 . (16.c) ∂α∂y
6 = α − 3 = +α − 3 3. Pitchfork bifurcation (prototype functions: yt+1 yt yt , yt+1 yt yt yt ). Consider the one parameter first-order difference equation = α yt +1 f ( yt , ) (17) α = α and assume that when 0 there is an equilibrium ye,0 . We have pitchfork bifurcation if the following hypotheses are satisfied: ∂f (y ,α ) e,0 0 = 1 ∂ y (18.a) ∂3 f (y ,α ) e,0 0 ≠ 0 , (18.b) ∂y3
∂2 f (y ,α ) e,0 0 ≠ 0 . (18.c) ∂αy
= α −α 2 4. Flip (period doubling) bifurcation (prototype functions: yt+1 yt yt , = − −α −α 3 yt+1 yt yt yt ).
Note: flip bifurcation can only arise in discrete dynamical systems. Consider the one parameter first-order difference equation = α yt +1 f ( yt , ) (19) α = α and assume that when 0 there is an equilibrium ye,0 for which the following hypotheses are satisfied: ∂f (y ,α ) e,0 0 = −1, (20.a) ∂y
∂f (y ,α ) ∂2 f (y ,α ) ∂2 f (y ,α ) e,0 0 e,0 0 + e,0 0 ≠ 2 2 0, (20.b) ∂α ∂y ∂αy
2 ∂2 f (y ,α ) ∂3 f (y ,α ) a ≡ 3 e,0 0 + 2 e,0 0 ≠ 0. (20.c) ∂ 2 ∂ 3 y y Then, depending of the signs of the expressions (20.b) and (20.c), α < α α > α (i) the equilibrium ye,0 is stable (unstable) when 0 ( 0 ); α > α (ii) the equilibrium ye,0 becomes unstable (stable) for each parameter value 0
7 α < α α ( 0 ), and a branch of additional stable (unstable) equilibria ye ( ) of order 2 emerges (two-cycle). Note: an equilibrium point (or fixed point) of order 2 is an equilibrium point of the following difference equitation3: = α = α α ≡ (2) α yt+2 f (yt+1, ) f ( f (yt , ), ) f (yt , ) . (21) When a in (20.c) is positive (negative), the emerging equilibrium points of order 2 are stable (unstable), and the flip bifurcation is said to be supercritical (subcritical) respectively.
Hopf bifurcation theorem for discrete-time systems
Note: unlike with the continuous time case the Hopf bifurcation theorem exists only for 2× 2 discrete time systems. Consider a 2× 2 non-linear difference system with one parameter = ϕ α yt+1 (yt , ) , (22) α = α and suppose that for each it has a smooth family of equilibrium points ye ye ( ) at which λ = θ α ± ω α α the eigenvalues are complex conjugate, 1,2 ( ) i ( ) . If there is a critical value 0 of the parameter such that
λ α = + θ 2 +ω 2 = λ j α ≠ = (i) 1,2 ( 0 ) 1, 1,2 ( 0 ) 1 for j 1,2,3,4 ;
d λ (α ) (ii) 1,2 0 ≠ 0 ; dα α =α 0 α THEN there is an invariant closed curve bifurcating from 0 .
3 An equilibrium point of order 2 is a point that repeats itself every two periods, i.e. a constant-amplitude alternation. 8