CENTER MANIFOLD AND NORMAL FORM THEORIES
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1 3 rd Sperlonga Summer School on Mechanics and Engineering Sciences 3-7 September 013 SPERLONGA CENTER MANIFOLD AND NORMAL FORM THEORIES ANGELO LUONGO 1
2 THE CENTER MANIFOLD METHOD Existence of an invariant manifold Linear systems The state-space N X of the linear system x () t Jx() t is direct sum of three invariant sub-spaces, i.e. X Xc Xs X, u where: Xc is the center subspace, of dimension N c, spanned by the (generalized) eigenvectors associated with non-hyperbolic eigenvalues of J; X s is the stable subspace, of dimension N s, spanned by the (generalized) eigenvectors associated with hyperbolic eigenvalues of J having negative real part; X u is the unstable subspace, of dimension N u, spanned by the (generalized) eigenvectors associated with non-hyperbolic eigenvalues of J having positive real part.
3 Nonlinear systems We consider the nonlinear system (in local form): x () t F( x(), t μ) admitting the critical equilibrium ( x 0, μc 0 ). We assume that J: F x ( 00, ) posses N c >0 critical eigenvalues, N s >0 stable eigenvalues and N u =0 unstable eigenvalues. The Center Manifold Theorem states that the asymptotic dynamics of the system around the equilibrium point x 0, at the critical value of the parameters μ μ c, takes place on a (critical) manifold M X c, which has the following properties: M c has dimension N c ; M c is tangent to the critical subspace Xc at x 0 ; M c is attractive, i.e. all the orbits tend to it when t. 3
4 The center manifold is therefore an N c -dimensional surface in the N=N c +N s -dimensional state-space. Example: To analyze the asymptotic dynamics, it needs: (a) to find the center manifold M c ; (b) to obtain the reduced N c dimensional equations governing the motion on M (bifurcation equations). c 4
5 Dependence of the CM on parameters Since we are interested not only in the dynamics at μ μ c, but also at the dynamics at μ close to μ c, we can use the trick to consider μ as additional critical variables, by considering the extended dynamical system: x () t F( x(), t μ()) t μ () t 0 Therefore, the critical subspace becomes X c : X c P with P : { μ} the parameter space. Hence: M c has dimension N c +M; M c is tangent to the critical subspace X c ; M c is attractive, i.e. all the orbits tend to it when t. 5
6 Reduction process Equations of motion, expanded: By expanding x () t F( x() t, μ) (and ignoring the dummy equations μ 0 ) for small x() t and μ close to μ c, we have: x () t Jx() t f( x(), t μ), f O( x() t, μ ) Linear transformation of the variables: After letting x(): t ( x (), t x ()) t, with x () t X critical variables and x s c s c () t X stable variables, a proper linear transformation uncouples the linear s part of the equations (t omitted): x c Jc 0 xc fc( xc, xs, μ) s s s( c, s, ) x 0 Js x f x x μ c 6
7 Center manifold: active and passive coordinates Cartesian equations for the CM: Tangency requirements: x h( x, μ ) s c h( 0, 0) 0, h ( 0, 0) 0, h ( 0, 0) 0. x The equation expresses the stable variables x s as (unknown) functions of the critical variables x c ; therefore x c are also said active coordinates, and x s passive coordinates. Time derivative of x s : The passive character of x s also holds for time-derivatives. By using the chain rule and the upper partition of the equations of motion: d xs h( xc, μ) hx ( xc, μ) xc d t h ( x, μ)( Jx f( x, hx ( ), μ)) x c c c c c c μ 7
8 Equation for the center manifold: The lower-partition of the equation of motion supplies the equation for determining M : c h ( x, μ)( Jx f( x, hx ( ), μ)) Jhx ( ) f( x, hx ( ), μ ) x c c c c c c s c s c c This equation can be solved, e.g., by using power series expansions, (starting from degree- polynomial, due to the required tangency): hx (, μ) α z α z, z: { x, μ} c 3 3 c Bifurcation equations: The upper-partition governs the dynamics on M c : x J x f ( x, h( x ), μ) c c c c c c 8
9 Example 1: a static bifurcation Nonlinear, -dimensional system: x x xy cx y 0 1y bx 3 0 Critical point, critical and stable coordinates: Equation for the Center Manifold: 0, x { x}, x { y} c c s y hx (, ) h( x, )[ xxhx (, ) cx] hx (, ) bx x 3 Power series expansion: y h( x, ) { x x } { x }
10 Zeroing separately the different powers: x : b0 0 x: 0 11 : 0 By solving for α s coefficients, at the lowest order, we have: 0 3 ybx x O( (, ) ) Bifurcation equation: x x( bc) x 3 If b+c<0, a supercritical (stable) pitchfork occurs; if b+c>0, a sub-critical (unstable) pitchfork occurs; if b+c=0, higher-order terms must be evaluated. 10
11 Example : a dynamical bifurcation 3-dimensional nonlinear system: x 1 0 x xz y 1 0 y yz z 0 0 1z x kxy y Equation for CM: z x y xy x y O(3) z-equation (passive coordinate), transformed: ( ) xy ( x y ) x y ( x y xy x y) x kxy y O(3)
12 Zeroing separately the different powers in the z-equation: x 31 1 : k /5 y : k /5 xy : ( 1) 3k 0 3 k /5 x : y : Bifurcation equations: x x yx k x k xy k y y y x y k x k xy k y [(1 /5) /5 (1 /5) ] [(1 / 5) / 5 (1 / 5) ] 1
13 THE NORMAL FORM THEORY Scope: To use a smooth nonlinear coordinate transformation, in order to put the bifurcation equation in the simplest form. Algorithm: Equations of motion: x Jx f( x) Transformed Equations: y Jy g( y) Near-identity transformation: x y h( y ) where x (possibly) includes the dummy variables µ. 13
14 o Transformed equation in the h(y) unknown: By differentiating the nearly-identity transformation: x [ I h y ( y)] y the equation of motion is transformed into: [ I h ( y)][ Jyg( y)] J[ yh( y)] f( yh( y)) y or: h ( y) Jy Jh( y) f ( y h( y)) [ Ih ( y)] g( y ) y y which is a differential equation for h(y) for any given g(y). A series solution is often necessary. 14
15 o By letting: fy ( ) f( y) f3( y) gy ( ) g( y) g3( y) hy ( ) h ( y) h ( y) 3 with the homogeneous polynomial of k-degree: Mk Mk Mk f ( y) p ( y), g ( y) p ( y), h ( y) p ( y) k km km k km km k km km m1 m1 m1 15
16 o Zeroing the independent monomials: where (if J is diagonal): L o Resonance: L γ α β k k k k diag[ ], : ( m ), m k k i i j j i j j1 j1 N j 0,0,, i, i,, 0 Since 1 i for some i (i.e. L k is singular); hence, β k 0 must be taken for compatibility, and resonant terms survive in the normal form!!! N 16
17 Example 1: System independent of parameters o System at a Hopf bifurcation: 3 3 x i 0 x 31x 3x x 33xx 34x, x, 3 3 x 0 i x 31x 3x x33xx 34x o Normal form: o Near-identity transformation: y iy y y y yy y x y y y y yy y
18 o Equation for the coefficients: i i i o Solution: 0, o Normal form: y iy3 y y The term y y is resonant, since 1 1, i.e. ( i) ( i) i 18
19 o Amplitude equation: Changing the variables according to: it yt () At ()e, At () the normal form is transformed into: [ At () iat ()]e iat ()e A() t At ()e it it (1) it 3 or: A t () 3 A () t A() t This is called Amplitude Modulation Equation (AME). Since A 3 O( A ), the AME describes a slow modulation. Therefore, the change of variable filters the fast dynamics. 19
20 Example : Non-diagonalizable system o Double-zero bifurcation equations: x 1 0 1x1 1x1 xx 1 3x x 0 0 x 4x1 5xx 1 6x o Normal Form: y 1 0 1y1 1y1 y1 y 3 y y 0 0 y 4 y1 5 y1 y 6 y o Near-Identity transformation: x1 y1 1y1 y1 y 3 y x y 4 y1 5 y1 y 6 y 0
21 o By zeroing the coefficients of the three monomials in the two transformed equations: Since Rank[ L ] 4 : K ( L ) span{ u, u }, u : (0,1,0,0,0,1), u (0,0,1,0,0,0) T 1 1 T K ( L ) span{ v, v }, v (,0,0,0,1,0), v (0,0,0,1,0,0) T T T 1 1 1
22 o Compatibility condition: ( ) ( ) 0, It is possible to take and 1 0 or 5 0. o By taking 5 0, / the NF reads (Takens normal form): 1 y 1 0 1y1 ( 1 5) y1 y 0 0 y 4 y 1 o By taking 1 0, the NF reads (Bogdanov normal form): y 1 0 1y 0 1 y 0 0 y 4 y1 ( 5 1) y1 y The Normal Form is not unique!!
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