Michael Margaliot School of Electrical Engineering, Tel Aviv University, Israel

1. A Generalization of Positive Linear Systems with Applications to Nonlinear Systems: Invariant Sets and the Poincare-BendixonProperty Michael Margaliot School of Electrical Engineering,Tel Aviv University, Israel Joint work with Eyal Weiss

2. Outline • Introduction to the theory of cooperative systems and totally positive matrices • A new class of dynamical systems: k-positive linear systems • Applications to the analysis of nonlinear dynamical systems: invariant sets and asymptotic properties Questions are welcome at any time

3. Positive Linear Systems • Consider • The state-variables represent nonnegative quantities, e.g., probabilities, densities, concertation of molecules,.... • Let • The linear system is called positive if

4. Linear Positive Systems The linear system is called positive if i.e. i.e. i.e. every off-diagonal entry of is nonnegative. Such a matrix is called Metzler. Note: a sign-pattern condition.

5. Linear Positive Systems The linear system is called positive if Positive systems have special and useful properties e.g. asymptotic stability iff there exists a diagonal matrix such that See A. Rantzer. Distributed control of positive systems, 2012.

6. Linear Positive Systems and Nonlinear Systems: the Variational Equation • Consider Let • Let denote the solution at time with • Let Then

7. Linear Positive Systems and Nonlinear Systems • Consider Let • For • This is the variational equation. If is Metzler for all then the nonlinear system is called cooperative.

8. Linear Positive Systems and Nonlinear Systems • Consider Let • For • If is Metzler for all then the nonlinear system is called cooperative. Then

9. Cooperative Nonlinear Systems • Consider Let • If is Metzler for all then the nonlinear system is called cooperative. • TheoremIf the system is strongly cooperative then for almost any initial condition the trajectory either leaves every compact set or converges to the set of equilibria. See M. W. Hirsch and H. Smith. Monotone dynamical systems, 2004.

10. The Main Idea – A Generalization of Positive Systems The linear system is called positive if i.e. its flow maps vectors with zero sign variations to vectors with zero sign variations. Definition: The linear system is k-positive if its flow maps vectors with up to (k-1)-sign variations to vectors with up to (k-1)-sign variations.

11. The Main Idea Definition: The linear system is k-positive if its flow maps vectors with up to (k-1)-sign variations to vectors with up to (k-1)-sign variations. • Example 1-positive is just positive. • Example 2-positive: maps vectors with up to 1-sign variations to vectors with up to 1-sign variations

12. Number of Sign Variations in a Vector • For a vector with no zero entries, the number of sign variations is clear: • For a general vector let where is the vector obtained from after deleting all zero entries. For example, • Question: what linear maps preserve the number of sign variations?

13. Sign Variation Diminishing Property (VDP) of Totally Nonnegative (TN) Matrices • Definitionminor = determinant of a square submatrix. • A matrix is called TN if all its minors are nonnegative. • Theorem (VDP). If is TN then See: Krein and Gantmacher. Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems (English Translation, 2002).

14. Definition of k-Positive Linear Systems • For a positive integer let This is the set of vectors with up to k-1 sign variations. Example: Definition is k-positive if is an invariant set of the dynamics, i.e., Example: a 1-positive system is just a positive system.

15. k-Positive Linear Systems • For let Definition We say that is k-positive if • Question 1: When is the system k-positive? • Question 2: What are the properties of such systems? • Question 3: What are the implications for nonlinear dynamical systems?

16. When is a System k-positive? Consider with continuous. Four different cases: • 1-positive; • k-positive with k odd and 1<k<n-1; • k-positive with k even and 1<k<n-1; • (n-1)-positive. In each case we have a necessary and sufficient sign-pattern condition on

17. When is a System k-positive? • Consider with continuous. • Theorem The system is (n-1)-positive if has thesign-pattern: where here n=4 (similar for every n).

18. When is a System k-positive? • Theorem is k-positive, with 1<k<n-1 and k oddif here n=5 (similar for every n). Note: A(t) is Metzler. Intuitive interpretation: a circular tridiagonal positive structure.

19. When is a System k-positive? • Theorem is k-positive, with 1<k<n-1 and k even if here n=5 (similar for every n). Note: A(t) is not Metzler. • Intuitive interpretation: a tridiagonal positive structure with negative feedback from 1 to n and n to 1.

20. Example: (n-1)-Positive System • Consider with continuous. • Theorem The system is (n-1)-positive iff where here n=4 (similar for every n).

21. Example: an (n-1)-positive Linear System • Consider: with: Its sign pattern is • This is an (n-1)-positive pattern i.e. a 3-positive system.

22. Example Cont’d • This is a 3-positive system. • Here Note:

23. A Note on the Proof of These Sign-Pattern Results Theorem Let Fix . The following are equivalent: • All minors of of order have the same sign; • For any with we have See Ben-Avraham, Sharon, Zarai & Margaliot, Dynamical systems with a cyclic sign variation diminishing property, IEEE TAC, 2019.

24. A Note on the Proof of theSign-Pattern Results Theorem Let Fix The following are equivalent: • All minors of of order have the same sign; • For any with we have We require that if then Thus, all minors of order of must have the same sign. This yields a Metzler condition on some “lifted” matrix that gives the sign-pattern conditions.

25. Applications to Nonlinear Systems: the Variational Equation • Consider the nonlinear dynamical system whose trajectories evolve on a convex invariant set and assume it admits a unique solution • Assume that is and define the Jacobian • Fix and let • For denote

26. Applications to Nonlinear Systems: the Variational Equation • Using these definitions yields: • This “LTV” is the associated variationalsystem • If the variational system is a positive system then the nonlinear system is called a cooperative system • Definition The nonlinear system is k-cooperative if the variational system (*) is k-positive. (*)

27. Implications of k-cooperativity • Explicit invariant sets • 2-cooperative systems satisfy the Poincare-Bendixon property: a bounded trajectory that does not converge to an equilibrium point converges to a limit cycle. Poincare-BendixonThm. for 2D flow

28. Implications of k-cooperativity: Invariant Sets • Theorem if a system is k-cooperative then for any with we have • Corollary If 0 is an equilibrium of then for any with we have

29. Implications of 2-cooperativity: Poincare-BendixonProperty This is based on combining the work of Sanchez (2009) with a geometric analysis of The set is a cone of rank k. In particular, is a cone of rank 2. See L. A. Sanchez. Cones of rank 2 and the Poincare-Bendixson property for a new class of monotone systems, JDE, 2009.

30. Cones of Rank k Definition: A set is called a cone of rank k if (1) • is closed • includes a subspace of dimension k and no subspace with a larger dimension. Example: is not a cone of rank k is a cone of rank 1

31. Cones of Rank k Definition: A set is called a cone of rank k if (1) • is closed • includes a subspace of dimension k and no subspace with a larger dimension. In dynamical systems with invariant cones of rank k the trajectories can be projected to the k-dimensional subspace. If this projection is one-to-one then the trajectories behave like in a k-dimensional system.

32. Implications of 2-cooperativity: Poincare-Bendixon Property Fact is a cone of rank k: (a) • is a closed set • contains a subspace of dimension k Example Suppose that n=3 and k=2. Then

33. Implications of 2-cooperativity: Poincare-Bendixon Property Consider If and is irreducible then every bounded solution of the system converges to an equilibrium point or a limit cycle. This generalizes a result of Mallet-Paret & Smith (JDE, 1990).

34. Further Research • When is the system k-positive? • What are the properties of such systems? • What are the implications for nonlinear systems? • What are k-positive control systems? • Other types of dynamical systems? • More applications?

35. Conclusion • We introduced a generalization of linear positive systems and nonlinear cooperative systems • Based on easy to verify sign-pattern conditions Useful in fields such as systems biology. • Many open questions…

36. More Details • Weiss and Margaliot. A generalization of linear positive systems with applications to nonlinear systems: invariant sets and the Poincare-Bendixonproperty, submitted. Available: https://arxiv.org/abs/1902.01630 • Margaliot and Sontag. Revisiting totally positive differential systems: a tutorial and new results, Automatica, 2019. Thank you!