Stability results for non-autonomous dynamical systems Cecilia Gonz´ alez Tokman ( Collaborators: G. Froyland, R. Murray & A. Quas ) New Developments in Open Dynamical Systems and Their Applications Banff International Research Station, 19 March 2018
Intro Non-autonomous systems and MET Stability Motivation � To develop mathematical tools –analytical and numerical– to analyse and understand transport and mixing phenomena in (non-autonomous) dynamical systems. 13/09/15 20/09/15 http://earth.nullschool.net Cecilia Gonz´ alez Tokman (UQ) Stability results for non-autonomous dynamical systems
Intro Non-autonomous systems and MET Stability Transfer Operators � Powerful analytical tool to investigate global properties of dynamical systems, by considering densities , or ensembles of trajectories. �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� �������������� f ������������� ������������� �������������� �������������� Lf ������������� ������������� �������������� �������������� ������������� ������������� �������������� �������������� ������������� ������������� �������������� �������������� ������������� ������������� �������������� �������������� ������������� ������������� �������������� �������������� ������������� ������������� �������������� �������������� ������������� ������������� x Tx � Linear operators encoding the global dynamics, acting on a linear (Banach, Hilbert) space X , � � L : X → X, f · g ◦ T dm = L f · g dm. Cecilia Gonz´ alez Tokman (UQ) Stability results for non-autonomous dynamical systems
Intro Non-autonomous systems and MET Stability Transfer Operators � Very useful for numerical analysis of dynamical systems, e.g. via Markovian models. Numerical approximations to invariant measure of a dynamical system via transfer operators (blue) and long trajectories (red). � Ulam discretisation scheme: P = { B 1 , . . . , B k } partition of the state space into bins , k 1 � � � � E P ( f ) = 1 B j f dm 1 B j . m ( B j ) j =1 Cecilia Gonz´ alez Tokman (UQ) Stability results for non-autonomous dynamical systems
Intro Non-autonomous systems and MET Stability Transfer Operators, Quasi-compactness � Also useful for the analytical study of transport phenomena in dynamical systems. � L is quasi-compact if there exists 0 ≤ k < 1 , called essential spectral radius of L , such that, outside the disc of radius k : ◦ The spectrum of L consists of isolated eigenvalues : 1 = γ 1 , . . . , γ m , m ≤ ∞ , ���������������� ���������������� � � ��� ��� �� �� 1 ���������������� ���������������� � � ��� ��� �� �� ���������������� ���������������� ��� ��� ���������������� ���������������� ��� ��� ���������������� ���������������� ��� ��� ���������������� ���������������� ��� ��� ���������������� ���������������� ��� ��� ���������������� ���������������� ��� ��� ���������������� ���������������� ��� ��� such that | γ 1 | ≥ | γ 2 | ≥ · · · ≥ | γ m | > k , and k ���������������� ���������������� ��� ��� ���������������� ���������������� � � ��� ��� ���������������� ���������������� ��� ��� � � ���������������� ���������������� � � ��� ��� ���������������� ���������������� ��� ��� ���������������� ���������������� ��� ��� ���������������� ���������������� � � ��� ��� �� �� �� �� � � ���������������� ���������������� � � �� �� �� �� � � � � � � � � ◦ Finite-dimensional corresponding � � � � � � � � generalised eigenspaces : E 1 , . . . , E m . Cecilia Gonz´ alez Tokman (UQ) Stability results for non-autonomous dynamical systems
Intro Non-autonomous systems and MET Stability Transfer Operators, Spectral Properties � It is now known that for a rich class of transformations T (including piecewise smooth expanding/hyperbolic maps) and appropriate X , L is quasi-compact. Furthermore, 1 = γ 1 simple � Ergodic system ; f 1 ∈ E 1 � Density of physical invariant measure . n − 1 1 1 � � g ( T j x ) lim =: lim nS n g ( x ) = gf 1 dm, m a.e. x ∈ I. n n →∞ n →∞ j =0 Cecilia Gonz´ alez Tokman (UQ) Stability results for non-autonomous dynamical systems
Intro Non-autonomous systems and MET Stability Transfer Operators, Spectral Properties � It is now known that for a rich class of transformations T (including piecewise smooth expanding/hyperbolic maps) and appropriate X , L is quasi-compact. Furthermore, 1 = γ 1 simple � Ergodic system ; | γ 2 | < 1 � Mixing system ; | γ 2 | � Rate of mixing ; f 1 ∈ E 1 � Density of physical invariant measure . Cecilia Gonz´ alez Tokman (UQ) Stability results for non-autonomous dynamical systems
Intro Non-autonomous systems and MET Stability Transfer Operators, Spectral Properties � It is now known that for a rich class of transformations T (including piecewise smooth expanding/hyperbolic maps) and appropriate X , L is quasi-compact. Furthermore, 1 = γ 1 simple � Ergodic system ; | γ 2 | < 1 � Mixing system ; | γ 2 | � Rate of mixing ; f 1 ∈ E 1 � Density of physical invariant measure . Cecilia Gonz´ alez Tokman (UQ) Stability results for non-autonomous dynamical systems
Intro Non-autonomous systems and MET Stability Transfer Operators, Spectral Properties � It is now known that for a rich class of transformations T (including piecewise smooth expanding/hyperbolic maps) and appropriate X , L is quasi-compact. Furthermore, 1 = γ 1 simple � Ergodic system ; | γ 2 | < 1 � Mixing system ; | γ 2 | � Rate of mixing ; f 1 ∈ E 1 � Density of physical invariant measure . Cecilia Gonz´ alez Tokman (UQ) Stability results for non-autonomous dynamical systems
Intro Non-autonomous systems and MET Stability Transfer Operators, Spectral Properties � It is now known that for a rich class of transformations T (including piecewise smooth expanding/hyperbolic maps) and appropriate X , L is quasi-compact. Furthermore, 1 = γ 1 simple � Ergodic system ; | γ 2 | < 1 � Mixing system ; | γ 2 | � Rate of mixing ; f 1 ∈ E 1 � Density of physical invariant measure . Cecilia Gonz´ alez Tokman (UQ) Stability results for non-autonomous dynamical systems
Intro Non-autonomous systems and MET Stability Transfer Operators, Spectral Properties � It is now known that for a rich class of transformations T (including piecewise smooth expanding/hyperbolic maps) and appropriate X , L is quasi-compact. Furthermore, 1 = γ 1 simple � Ergodic system ; | γ 2 | < 1 � Mixing system ; | γ 2 | � Rate of mixing ; f 1 ∈ E 1 � Density of physical invariant measure . Cecilia Gonz´ alez Tokman (UQ) Stability results for non-autonomous dynamical systems
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